{"id":2282,"date":"2023-05-03T04:38:16","date_gmt":"2023-05-03T04:38:16","guid":{"rendered":"https:\/\/vartmaaninstitutesirsa.com\/?page_id=2282"},"modified":"2024-03-09T13:02:49","modified_gmt":"2024-03-09T13:02:49","slug":"chapter-11-thermal-properties-of-matter","status":"publish","type":"page","link":"https:\/\/vartmaaninstitutesirsa.com\/index.php\/chapter-11-thermal-properties-of-matter\/","title":{"rendered":"Chapter 11 Thermal Properties of Matter"},"content":{"rendered":"

Chapter 11 Thermal Properties of Matter<\/h1>\n

Chapter 11 Thermal Properties of Matter: Heat, temperature, thermal expansion; thermal expansion of solids, liquids and gases,
\nanomalous expansion of water; specific heat capacity; Cp, Cv \u2013 calorimetry; change of state \u2013 latent heat capacity.<\/p>\n

Heat transfer-conduction, convection and radiation, thermal conductivity, qualitative ideas of
\nBlackbody radiation, Wein\u2019s displacement Law, Stefan\u2019s law, Green house effect.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/em><\/strong><\/p>\n

\n

FLUID DYNAMICS<\/span><\/u><\/strong><\/p>\n

In the order to describe the motion of a fluid, in principle, one might apply Newton\u2019s laws to a particle (a small volume element of fluid) and follow its progress in time. This is a difficult approach. Instead, we consider the properties of the fluid, such as velocity, pressure, at fixed points in space. In order to simplify the discussion we take several assumptions:<\/span><\/p>\n

(i) The fluid is non viscous<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>(ii) the flow is steady<\/span><\/p>\n

(iii) The flow is non rotational<\/span>\u00a0\u00a0\u00a0<\/span>(IV) The fluid is incompressible.<\/span><\/p>\n

42.\u00a0<\/span><\/strong>\u00a0<\/span><\/u><\/strong>Viscosity:<\/span><\/u><\/strong><\/p>\n

It is an opposing force which comes into play when a layer of liquid slide over another layer of liquid.<\/span><\/em><\/p>\n

Some phenomenon associated with viscosity.<\/span><\/p>\n

\uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>During swimming we experience some resistance that is due to viscosity.<\/span><\/p>\n

\uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Water moves faster as compared to honey through a hole because viscosity of water is smaller than viscosity of honey.<\/span><\/p>\n

\uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>We can walk faster in air as compared to water due to viscosity.<\/span><\/p>\n

Coefficient of Viscosity<\/span><\/u><\/strong>:<\/span><\/u><\/em><\/strong><\/p>\n

Suppose two layers of a liquid\u00a0<\/span>\"\"\u00a0and\u00a0<\/span>\"\"\u00a0moving with velocities\u00a0<\/span>\"\"\u00a0and\u00a0<\/span>\"\"\u00a0and\u00a0<\/span>\"\"\u00a0are the heights of the layer from the ground.<\/span><\/p>\n

\"\"Then according to Newton the viscous force between two layers depends upon area of layers and velocity gradient as<\/span>\u00a0\u00a0<\/span><\/p>\n

\u00a0<\/span>\"\"<\/p>\n

\"\"<\/p>\n

Combining eq. (i) and (ii)<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

\u00a0<\/span>\"\"<\/p>\n

Or<\/span>\u00a0\u00a0<\/span>\"\"<\/p>\n

Where\u00a0<\/span>\"\"\u00a0is called coefficient of viscosity. Here\u00a0<\/span>\"\"\u00a0sign indicate that viscous opposes the change in motion.<\/span><\/p>\n

If<\/span>\u00a0<\/span>\"\"\u00a0\u00a0<\/span>and<\/span>\u00a0\u00a0<\/span>\"\"\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

Then\u00a0<\/span>\u00a0<\/span>\"\"<\/p>\n

Hence coefficient of viscosity may be defined as the viscous force acting on unit area of the layer having unit velocity gradient perpendicular to direction of flow of the liquid<\/span><\/em>.<\/span><\/p>\n

\uf076<\/span>\u00a0\u00a0\u00a0<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/em><\/strong>\u00a0\u00a0\u00a0<\/span>In\u00a0<\/span>\"\"\u00a0\u00a0<\/span>\"\"<\/p>\n

\uf076<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>In SI unit of\u00a0<\/span>\"\"Nsm<\/span>-2<\/sup><\/span>\u00a0Or<\/span>\u00a0\u00a0\u00a0<\/span>decapoise<\/span><\/p>\n

\uf076<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>1decapoise =<\/span>\"\"\u00a0<\/span><\/p>\n

\uf076<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>(1decapoise =1 Pascal second (<\/span>\"\"))<\/span><\/p>\n

Question22:<\/span><\/em><\/strong>\u00a0<\/span><\/strong>A metal plate 1 m<\/span><\/strong>2<\/sup><\/span><\/strong>\u00a0in area rests on a layer of castor oil (<\/span><\/strong>\uf068<\/span>\u00a0= 1 decapoise) 0.1 cm thick. Calculate the horizontal force required to move the plate with a speed of<\/span><\/strong>\u00a0\u00a0\u00a0<\/span><\/strong>3 cm\/s.\u00a0<\/span><\/strong><\/p>\n

Solution:<\/span><\/em><\/strong>\u00a0<\/span><\/em><\/strong>\"\"\u00a0<\/span><\/em><\/strong>where\u00a0<\/span>\uf068<\/span>\u00a0= 15.5 poise\u00a0<\/span>A\u00a0<\/span><\/em>= 100 cm<\/span>2<\/sup><\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/em><\/strong><\/p>\n

\u00a0\u00a0<\/span><\/em><\/strong>\"\"30s<\/span>\u20131<\/sup><\/span>\u00a0<\/span><\/p>\n

\u00a0\u00a0<\/span><\/strong>F<\/span><\/em>\u00a0=\u00a0<\/span>\uf02d<\/span>1\u00a0<\/span>\uf0b4<\/span>\u00a01\u00a0<\/span>\uf0b4<\/span>\u00a030 =\u00a0<\/span>\uf02d<\/span>30<\/span>\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

so force required =\u00a0<\/span>30 N<\/span><\/strong>\u00a0<\/span><\/p>\n

Question 23: A plate of area 2 m<\/span><\/strong>2<\/sup><\/span><\/strong>\u00a0moves horizontally with a speed of 2 m\/s by applying a horizontal tangential force over the free surface of a liquid, the depth of the liquid is 1 m and the liquid in contact with the bed is stationary. Coefficient of viscosity of liquid is 0.01 poise. Find the tangential force needed to move the plate.\u00a0<\/span><\/strong><\/p>\n

\"\"Sol: Apply the Newton\u2019s formula for the frictional force between two layers of a liquid.<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

Velocity gradient<\/span>\u00a0<\/span>\"\"<\/p>\n

AS, |<\/span>\"\"<\/p>\n

\"\"<\/p>\n

So, to keep the plate moving, a force of\u00a0<\/span>\"\"\u00a0must be applied.\u00a0<\/span><\/p>\n

43.<\/span><\/strong>\u00a0\u00a0<\/span><\/u><\/strong>Effect of Temperature and Pressure on the Viscosity:<\/span><\/u><\/strong><\/p>\n

    \n
  • The viscosity of a liquid decrease with increase in temperature and increase with decrease in temperature i.e.\u00a0<\/span>\"\"\u00a0\u00a0<\/span>\"\"<\/li>\n
  • The Viscosity of gases increase with increase in temperature and decrease with decrease in temperature i.e.<\/span>\u00a0\u00a0<\/span>\"\"\u00a0\u00a0<\/span>\"\"<\/li>\n
  • Except water, viscosity of liquid increase with increase in pressure and decrease with decrease in pressure.<\/span><\/li>\n
  • Viscosity of gases is independent of pressure.<\/span><\/li>\n<\/ul>\n

    44.<\/span><\/strong>\u00a0Some Applications of Viscosity:<\/span><\/u><\/strong><\/p>\n

      \n
    • Since viscosity of a liquid depends on temperature, so we can use proper lubricant in different reason (hot or cold).<\/span><\/li>\n
    • Liquid of high viscosity are used as shock absorber and buffers of train.<\/span><\/li>\n
    • Viscosity is also helpful in determining the molecular weight and shapes of organic molecules.<\/span><\/li>\n
    • It finds an important role in blood circulation in arteries and veins of human body.<\/span><\/li>\n<\/ul>\n

      45.<\/span><\/strong>\u00a0Similarities and Differences between Viscosity and Solid Friction:<\/span><\/u><\/strong><\/p>\n

      Similarities:<\/span><\/u><\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/strong>(1)<\/span>\u00a0\u00a0<\/span>\u00a0<\/span><\/strong>Both oppose the relative motions.<\/span><\/p>\n

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>(2)<\/span>\u00a0\u00a0\u00a0<\/span>Both come into play when one layer move over the second layer.<\/span><\/p>\n

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>(3)<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Both are due to intermolecular interactions.<\/span><\/p>\n

      Dissimilarities:<\/span><\/u><\/strong><\/p>\n

        \n
      • Viscosity depends upon the area of the layers while friction does not depend upon the area contact.<\/span><\/li>\n
      • Viscosity is independent on normal reaction between two layers but friction is directly proportional to normal reaction between two layers.<\/span><\/li>\n
      • Viscosity depends between relative velocities of two layers while friction does not depend upon the relative velocities of layers.<\/span><\/li>\n<\/ul>\n

        46.<\/span><\/strong>\u00a0Poiseuille\u2019s Formula:<\/span><\/u><\/strong><\/p>\n

        According to Poiseuille\u2019s Formula, the volume of a liquid flowing per second through a horizontal capillary tube of length<\/span>\"\", radius\u00a0<\/span>\"\"\u00a0under a pressure\u00a0<\/span>\"\"\u00a0applied across its end is given by<\/span><\/p>\n

        As<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

        This relation is called Poiseuille\u2019s Formula.<\/span><\/p>\n

        Derivation of Poiseuille\u2019s Formula on the Basis of Dimensional Analysis:<\/span><\/u><\/strong><\/p>\n

        The volume of liquid flowing through capillary tube per second depends upon\u00a0<\/span><\/p>\n

        (i) Coefficient of Viscosity of the Liquid<\/span>\u00a0\u00a0<\/span><\/p>\n

        \u00a0<\/span>(ii) Radius of the Tube<\/span>\u00a0\u00a0\u00a0<\/span><\/p>\n

        (iii) Pressure Gradient set up along the capillary tube.<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

        \u00a0<\/span>i.e.<\/span>\u00a0<\/span>\"\"<\/p>\n

        Where dimension formula of various quantities are<\/span>\u00a0\u00a0\u00a0<\/span><\/p>\n

        \u00a0<\/span>\"\"<\/p>\n

        Comparing the power of M, L, T on both sides we get<\/span>\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

        \u00a0\u00a0<\/span>\"\"\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

        \u00a0<\/span>\"\"\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

        \u00a0<\/span>\"\"<\/p>\n

        On solving we get\u00a0<\/span>\"\"\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

        \u27f9<\/span>\u00a0\u00a0<\/span>\u00a0<\/span>\"\"<\/p>\n

        \"\"<\/p>\n

        Where K is constant of proportionality and its value is found\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

        so<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

        Which is the required expression for Poiseuille\u2019s Formula?<\/span><\/p>\n

        47.<\/span><\/strong>\u00a0Stokes Law:<\/span><\/u><\/strong><\/p>\n

        According to stokes law, the force acting on a small sphere of radius\u00a0<\/span>\"\"\u00a0moving with uniform velocity\u00a0<\/span>\"\"\u00a0through fluid of viscosity\u00a0<\/span>\"\"\u00a0is given by<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

        Derivation of Stokes Law on the basis of dimensional Analysis:<\/span><\/u><\/strong><\/p>\n

        It is found that viscous force on a body moving in liquid depends upon\u00a0<\/span><\/p>\n

          \n
        1. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Coefficient of viscosity \"\"\u00a0of the liquid.<\/li>\n
        2. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Radius \"\"\u00a0of the spherical body.<\/li>\n
        3. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Velocity \"\"\u00a0of the body<\/li>\n<\/ol>\n

          So let<\/span>\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

          The above eq. may be written dimension as<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

          \u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

          Now comparing the coefficient of M,L,T on both sides, we get<\/span><\/p>\n

          \"\"<\/p>\n

          \"\"<\/p>\n

          \"\"\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

          \u00a0<\/span>On solving\u00a0<\/span>\"\"\u00a0\u00a0<\/span><\/p>\n

          \u00a0<\/span>\u27f9<\/span>\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

          Where K is constant of proportionality and its value is found\u00a0<\/span>\"\"\u00a0\u00a0<\/span><\/p>\n

          So<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

          Which is the required stokes law.<\/span><\/p>\n

          Validity of Stokes Law:<\/span><\/u><\/strong><\/p>\n

          \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>The fluid should have infinite extension and stream line.<\/span><\/p>\n

          \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>The body should not slip on liquid and perfectly rigid smooth.<\/span><\/p>\n

          \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>The size of the body should be small.<\/span><\/p>\n

          Question 24:<\/span><\/strong>\u00a0\u00a0<\/span><\/strong>An air bubble of diameter 2 mm rises steadily through a solution of density\u00a0<\/span><\/strong>\"\"\u00a0at the rate of<\/span><\/strong>\"\".Calculate the coefficient of viscosity of the solution. The density of air is negligible.<\/span><\/strong>\u00a0\u00a0<\/span><\/strong><\/p>\n

          Sol: As the air bubble rises with constant velocity, the net force on it is zero. The force of buoyancy B is equal to the weight of the displaced liquid.<\/p>\n

          Thus\"\"<\/p>\n

          This force is upward. The viscous force acting downward is\"\"<\/p>\n

          \"\".<\/p>\n

          \u00a0For uniform velocity\"\"<\/p>\n

          \"\"<\/p>\n

          \"\"<\/p>\n

          Question25:<\/span><\/em><\/strong>\u00a0An air bubble of diameter 2 cm rises through a long cylindrical column of a viscous liquid, and<\/span><\/strong><\/p>\n

          Travels at 0.21 cms<\/span><\/strong>\uf02d<\/sup><\/span><\/strong>1<\/sup><\/span><\/strong>. If the density of the liquid is 189 kg m<\/span><\/strong>\u20133<\/sup><\/span><\/strong>, find its coefficient of viscosity.\u00a0<\/span><\/strong><\/p>\n

          Solution:<\/span><\/em><\/strong>\u00a0\u00a0<\/span>Weight of the bubble is equal to the viscous force.<\/span><\/p>\n

          \u00a0<\/span>\"\"\u00a0\u00a0\u00a0<\/span><\/em><\/strong><\/p>\n

          Or<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"\u00a0<\/span>\u2026(i)<\/span><\/p>\n

          \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

          Given:<\/span>\u00a0<\/span>r<\/span><\/em>\u00a0= 10<\/span>\uf02d<\/sup><\/span>2<\/sup><\/span>\u00a0m<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>v<\/span><\/em>\u00a0= 0.21\u00a0<\/span>\uf0b4<\/span>\u00a010<\/span>\uf02d<\/sup><\/span>2<\/sup><\/span>\u00a0m\/s\u00a0<\/span>g<\/span><\/em>\u00a0= 10 m\/s<\/span>2<\/sup><\/span>\u00a0<\/span><\/p>\n

          \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/em><\/p>\n

          Substituting these values in (i) we have,<\/span>\u00a0\u00a0<\/span><\/p>\n

          \"\"\u00a0<\/span>\uf0bb<\/span>\u00a0<\/span>20 kgm<\/span><\/strong>\uf02d<\/sup><\/span><\/strong>1<\/sup><\/span><\/strong>\u00a0<\/sub><\/span><\/strong>s<\/span><\/strong>\uf02d<\/sup><\/span><\/strong>1<\/sup><\/span><\/strong><\/p>\n

          48.<\/span><\/strong>\u00a0Terminal Velocity:<\/span><\/u><\/strong><\/p>\n

          When a body falls in liquid then against the motion of body a viscous force and a upward thrust acts on the body. At this situation the maximum constant velocity acquired by the body while falling into the viscous medium is called terminal velocity.\u00a0<\/span><\/p>\n

          Suppose a spherical body of mass m, density\u00a0<\/span>\"\"\u00a0and radius\u00a0<\/span>\"\"\u00a0falls into a viscous fluid of density\u00a0<\/span>\"\"\u00a0and coefficient of viscosity<\/span>\"\". Then various forces acting on the body are:<\/span><\/p>\n

          \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/p>\n

           <\/p>\n

          \"Thermal<\/p>\n

           <\/p>\n

           <\/p>\n

           <\/p>\n

           <\/p>\n

           <\/p>\n

           <\/p>\n

           <\/p>\n

          Upward force T and Viscous force F. Downward forces W. Now net down ward force acting on the body is\u00a0<\/span><\/p>\n

          \u00a0\u00a0<\/span>\"\"<\/p>\n

          Or<\/span>\u00a0<\/span>\"\"<\/p>\n

          Or<\/span>\u00a0<\/span>\"\"<\/p>\n

          Or<\/span>\u00a0<\/span>\"\"<\/p>\n

          Or<\/span>\u00a0<\/span>\"\"<\/p>\n

          This is the required expression for terminal velocity.<\/span><\/p>\n

          Clearly if body is denser then liquid then body will move into the liquid and if density of body is smaller than density of liquid then body will float.\u00a0<\/span><\/em><\/p>\n

          Question 26:<\/strong>\u00a0\u00a0<\/strong>Two spherical raindrops of equal size are falling vertically through air with a terminal velocity of<\/strong>\u00a0\u00a0\u00a0<\/strong>1 m\/s. What would be the terminal speed if these two drops were to coalesce to form a large spherical drop?\u00a0<\/strong><\/p>\n

          Sol: Use the formula for terminal velocity for spherical body.<\/p>\n

          \u00a0\"\".<\/p>\n

          \u00a0Let r be the radius of small rain drops and R the radius of large drop.<\/p>\n

          Equating the volume, we have<\/p>\n

          \u00a0\"\"<\/p>\n

          \"\"<\/p>\n

          Or\u00a0 \"\"<\/p>\n

          \"\"<\/p>\n

          \"\"<\/p>\n

          49.<\/span><\/strong>Streamline, Laminar and Turbulent Flow\u00a0<\/span><\/u><\/strong><\/p>\n

          \"\"(1)\u00a0<\/span>Stream line flow<\/span><\/u><\/strong>:<\/span>\u00a0\u00a0<\/span><\/p>\n

          \u00a0<\/span>Stream line flow of a liquid is that flow in which each element of the liquid passing through a point travels along the same path and with the same velocity as the preceding element passes through that point.\u00a0<\/span><\/p>\n

          \u00a0<\/span>Path ABC is streamline as shown in the figure and\u00a0<\/span>\"\",\u00a0<\/span>\"\"\u00a0and\u00a0<\/span>\"\"\u00a0are the velocity of the liquid particle at A, B and C point respectively.\u00a0<\/span><\/p>\n

          (2<\/span>)<\/span><\/strong>\u00a0\u00a0<\/span><\/strong>Laminar flow:<\/span><\/u><\/strong>\u00a0<\/span><\/u><\/p>\n

          \u00a0<\/span>If a liquid is flowing over a horizontal surface with a steady flow and moves in the form of layers of different velocities which do not mix with each other, then the flow of liquid is called laminar flow. In this flow, the velocity of liquid flow is always less than the critical velocity of the liquid. The laminar flow is generally used synonymously with streamlined flow.<\/span>\u00a0\u00a0\u00a0<\/span><\/p>\n

          \u00a0\u00a0<\/span>\"\"<\/u><\/p>\n

           <\/p>\n

           <\/p>\n

           <\/p>\n

          Turbulent flow<\/u><\/strong>:<\/p>\n

          When a liquid moves with a velocity greater than its critical velocity, the motion of the particles of liquid becomes disordered or irregular. Such a flow is called a turbulent flow. In a turbulent flow, the path and the velocity of the particles of the liquid change continuously.\u00a0<\/span><\/p>\n

          50 .\u00a0<\/span><\/strong>Critical Velocity and Reynolds\u2019s Number\u00a0<\/span><\/u><\/strong><\/p>\n

          The critical velocity is that velocity of liquid flow up to which its flow is streamlined and above which its flow becomes turbulent.\u00a0<\/span><\/p>\n

          Reynolds\u2019s number is a pure number which determines the nature of flow of liquid through a pipe.<\/span><\/em>\u00a0\u00a0<\/span><\/em><\/p>\n

          It is defined as the ratio of the inertial force per unit area to the viscous force per unit area for a flowing fluid.<\/span>\u00a0\u00a0<\/span>As<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

          \u00a0<\/span>\"\"<\/p>\n

          If the value of Reynolds number\u00a0<\/span><\/u><\/p>\n

          \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Lies between 0 to 2000, the flow of liquid is streamline or laminar.\u00a0<\/span><\/p>\n

          \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Lies between 2000 to 3000 the flow of liquid is unstable and changing from streamline to turbulent flow.<\/span>\u00a0\u00a0<\/span><\/p>\n

          \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Above 3000, the flow of liquid is definitely turbulent<\/span><\/p>\n

          51 .<\/span><\/strong>Equation of Continuity<\/span><\/u><\/strong><\/p>\n

          It states that, the volume of liquid flowing per unit time is constant through different cross-sections of the container of the liquid. Thus, if\u00a0<\/span>\"\"\u00a0and\u00a0<\/span>\"\"\u00a0are velocities of fluid at respective points A and B of areas of cross-sections\u00a0<\/span>\"\"\u00a0and\u00a0<\/span>\"\"\u00a0and\u00a0<\/span>\"\"and\u00a0<\/span>\"\"\u00a0be the radius respectively.\u00a0<\/span><\/p>\n

          Then the equation of continuity is given by<\/span><\/p>\n

          \u00a0<\/span>\"\"\u00a0\u00a0<\/span>… (i)\u00a0<\/span><\/p>\n

          If the same liquid is flowing, then<\/span>\u00a0\u00a0<\/span>\"\"<\/p>\n

          then the equation (i) can be written<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

          \u00a0<\/span>\"\"\u00a0\u00a0\u00a0<\/span>.. (ii)<\/span><\/p>\n

          \u00a0\u00a0<\/span>\u21d2<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

          \u21d2<\/span>\"\"<\/p>\n

            \n
          • Equation of continuity represents the law of conservation of mass of moving fluids.<\/span><\/li>\n<\/ul>\n

            \"\"\u00a0\u00a0<\/span>(General equation of continuity)<\/span><\/p>\n

              \n
            • This equation is applicable to actual liquids or to other fluids which are not incompressible.\u00a0<\/span><\/li>\n<\/ul>\n

              Question 27: Water is flowing through a horizontal tube of non-uniform cross-section. At a place, the radius of the tube is 1.0 cm and the velocity of water is 2 m\/s. What will be the velocity of water, where the radius of the pipe is 2.0 cm?<\/span><\/strong>\u00a0\u00a0<\/span><\/strong><\/p>\n

              Sol: Using equation of continuity,\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

              \u21d2<\/span>\u00a0<\/span>\"\"<\/p>\n

              Question 28:<\/span><\/strong>\u00a0\u00a0<\/span><\/strong>Figure shows a liquid being pushed out of a tube by pressing a piston. The area of cross-section of the piston is 1.0 cm<\/span><\/strong>2<\/sup><\/span><\/strong>\u00a0and that of the tube at the outlet is 20 mm<\/span><\/strong>2<\/sup><\/span><\/strong>. If the piston is pushed at a speed of 2 cm-s<\/span><\/strong>-1<\/sup><\/span><\/strong>, what is the speed of the outgoing liquid?\u00a0<\/span><\/strong><\/p>\n

              \u00ad<\/span>Sol: From the equation of continuity\u00a0<\/span>\"\"=10cm\/s\u00a0<\/span><\/p>\n

              52.<\/span><\/strong>\u00a0DIFFERENT FORM OF ENERGY IN FLUID FLOW<\/span><\/u><\/strong><\/p>\n

                \n
              1. Pressure Energy<\/li>\n<\/ol>\n

                \u00a0<\/span><\/strong>If P is the pressure on the area A of a fluid, and the liquid moves through a distance due to this pressure,\u00a0<\/span><\/p>\n

                Then<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                \"\"<\/p>\n

                The volume of the liquid is Al. Pressure energy per unit volume of liquid<\/span><\/p>\n

                \u00a0<\/span>\"\"<\/p>\n

                  \n
                1. \u00a0<\/span>Kinetic Energy<\/li>\n<\/ol>\n

                  If a liquid of mass m and volume V is flowing with velocity v,<\/span><\/p>\n

                  \u00a0<\/span>then the kinetic energy is\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                  Kinetic energy per unit volume of liquid<\/span><\/p>\n

                  \u00a0<\/span>=\u00a0<\/span>\"\"=<\/span>\"\"\u00a0\u00a0<\/span><\/p>\n

                  Here, \u03c1 is the density of liquid.<\/span><\/p>\n

                    \n
                  1. \u00a0<\/span>Potential energy<\/li>\n<\/ol>\n

                    \u00a0<\/span><\/strong>If a liquid of mass m is at a height h from the reference line (h = 0), then its potential energy is mgh.\u00a0<\/span><\/p>\n

                    \u2234<\/span>\u00a0Potential energy per unit volume of the liquid<\/span><\/p>\n

                    \u00a0<\/span>=\u00a0<\/span>\"\"<\/p>\n

                    53 .<\/span><\/strong>\u00a0BERNOULLI\u2019S THEOREM<\/span><\/u><\/strong><\/p>\n

                    When a non-viscous and an incompressible fluid flows in a streamlined motion from one place to another in a container, then the total energy (pressure + kinetic + potential) of the fluid per unit volume is constant at every point of its path.\u00a0<\/span><\/p>\n

                    Total energy = pressure energy + Kinetic energy + Potential energy\u00a0<\/span><\/p>\n

                    =\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                    Where P is pressure, V is volume, M is mass and h is height from a reference level.<\/span><\/p>\n

                    The total energy per unit volume\u00a0<\/span><\/p>\n

                    =\u00a0<\/span>\"\"<\/p>\n

                    Where \u03c1 is density, Thus if a liquid of density \u03c1, pressure\u00a0<\/span>\"\"\u00a0at a height\u00a0<\/span>\"\"\u00a0which flows with velocity\u00a0<\/span>\"\"\u00a0to another point in streamline motion where the liquid has pressure\u00a0<\/span>\"\", at height\u00a0<\/span>\"\"\u00a0which flows with velocity\u00a0<\/span>\"\",<\/span><\/p>\n

                    \u00a0<\/span>Then<\/span>\u00a0 \u00a0<\/span>\"\"<\/p>\n

                    Derivations:-\u00a0<\/span><\/strong><\/p>\n

                    To prove it, consider a liquid flowing steadily through a tube of non-uniform area of cross-section as shown in fig. If\u00a0<\/span>\"\"and<\/span>\"\"are the pressures at the two ends of the tube respectively, work done in pushing<\/span>\u00a0\u00a0<\/span>the volume V of incompressible fluid from point A to B through the tube will be<\/span><\/p>\n

                    \"C:\\Users\\vartmaan\\AppData\\Local\\Microsoft\\Windows\\Temporary\"\"<\/p>\n

                    \"\"<\/p>\n

                    This work is used by the fluid in two ways.\u00a0<\/span><\/p>\n

                    (a) In changing the potential energy of mass m (in the volume V) from\u00a0<\/span>\"\"\u00a0to\u00a0<\/span>\"\"\u00a0\u00a0<\/span><\/p>\n

                    i.e.<\/span>\u00a0\u00a0<\/span>\"\"\u00a0\u00a0<\/span>\u2026… (ii)<\/span><\/p>\n

                    (b) In changing the kinetic energy from<\/span><\/p>\n

                    \u00a0<\/span>\"\"\u00a0to<\/span>\u00a0\u00a0<\/span>\"\",<\/span>\u00a0\u00a0<\/span><\/p>\n

                    i.e.,<\/span>\u00a0\u00a0<\/span>\"\"K =<\/span>\"\"<\/p>\n

                    \"\"\u00a0\u00a0\u00a0\u00a0<\/span>(iii)\u00a0<\/span><\/p>\n

                    Now as the fluid is non-viscous, by conservation of mechanical energy<\/span><\/p>\n

                    \u00a0\u00a0<\/span>\"\"=<\/span>\"\"\u00a0<\/span><\/p>\n

                    i.e.,<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"+<\/span>\"\"<\/p>\n

                    Or<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"+<\/span>\"\"<\/p>\n

                    Or<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"+<\/span>\"\"<\/p>\n

                    Or<\/span>\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                    Or\u00a0<\/span>\"\"<\/p>\n

                    This equation is the so called Bernoulli’s equation and represents conservation of mechanical energy in case of moving fluids.\u00a0<\/span><\/p>\n

                    54 .<\/span><\/strong>Applications of Bernoulli’s Theorem:-\u00a0<\/span><\/u><\/strong><\/p>\n

                      \n
                    1. Attraction between two closely parallel moving boats (or buses)<\/u><\/strong>:<\/li>\n<\/ol>\n

                      \"\"When two boats or buses move in the same direction, the water (or air) in the region between them moves faster than that on the remote sides. The pressure between them is reduced and hence due to pressure difference they are pulled towards each other creating the so called attraction.\u00a0<\/span><\/p>\n

                        \n
                      1. Working of an aeroplane:-<\/u><\/strong><\/li>\n<\/ol>\n

                        \"\"The wings of the aeroplane are of specific shape when the aeroplane runs, air passes at higher speed over it as compared to its lower surface. This difference of air speeds above and below the wings creates a pressure difference, due to which an upward force called ‘dynamic lift\u2019 acts on the plane. If this force becomes greater than the weight of the plane, the plane will rise up.<\/span><\/p>\n

                          \n
                        1. Action of atomizer :-<\/u><\/strong>\u00a0<\/span><\/li>\n<\/ol>\n

                          \"\"The action of carburetor, paint-gun, scent-spray or insect-sprayer is based on Bernoulli’s principle. In all these, by means of motion of a piston P in a cylinder C, high speed air is passed over a tube T dipped in liquid L to be sprayed. High speed air creates low pressure over the tube due to which liquid (paint, scent, insecticide or petrol) rises in it and is then blown off in very small droplets with expelled air.<\/span><\/p>\n

                            \n
                          1. Blowing off roofs by wind storms:-<\/u>\u00a0\u00a0\u00a0<\/u><\/li>\n<\/ol>\n

                            During a tornado or hurricane, when a high speed wind blows over a straw or tin roof, it creates a low pressure (P) in accordance with Bernoulli’s principle. However, the pressure below the roof (i.e., inside the room) is still atmospheric. So due to this difference of pressure, the roof is lifted up and is then blown off by the wind.<\/span><\/p>\n

                              \n
                            1. Heart attack\u00a0<\/u><\/li>\n<\/ol>\n

                              due to access colestrol veins becomes narrows . presser outside the veins become more and blockage of veins occurs due to which heart stop working.<\/span>\u00a0\u00a0<\/span><\/p>\n

                                \n
                              1. Venturimeter<\/u><\/li>\n<\/ol>\n

                                \"\"A<\/span>\u00a0\u00a0<\/span>Venturimeter used to measure flow speed in a pipe of non-uniform cross-section. We apply Bernoulli\u2019s equation to the wide (point 1) and narrow (point 2) parts of the pipe, with<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                From Bernoulli theorem\u00a0<\/span><\/p>\n

                                \u00a0<\/span>\"\"<\/p>\n

                                From the continuity equation<\/span>\"\"<\/p>\n

                                Substituting and rearranging, we get<\/span><\/p>\n

                                \u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                                The pressure difference is also equal to \u03c1gh, where h is the difference in liquid level in the two tubes.\u00a0<\/span><\/p>\n

                                Substituting in equation (i), we get<\/span><\/p>\n

                                \u00a0<\/span>\"\"<\/p>\n

                                Question 29: Calculate the rate of flow of glycerin of density\u00a0<\/span><\/strong>\"\"\u00a0through the conical section of a pipe, if the radii of its ends are 0.1 m and 0.04 m and the pressure drop across its length is\u00a0<\/span><\/strong>\"\"<\/p>\n

                                Sol: Apply the equation of continuity. Where area of cross-section is larger, the velocity of fluid is lesser and vice-versa. From continuity equation,<\/span><\/p>\n

                                \u00a0\u00a0<\/span>\"\"<\/p>\n

                                \"\"<\/p>\n

                                \"\"<\/p>\n

                                From Bernoulli\u2019s equation,\u00a0<\/span><\/p>\n

                                \u00a0<\/span>\"\"<\/p>\n

                                \"\"<\/p>\n

                                \"\".<\/span>\u00a0\u00a0<\/span>.. (ii)\u00a0<\/span><\/p>\n

                                Solving equations (i) and (ii), we get\u00a0<\/span><\/p>\n

                                \u00a0\u00a0<\/span>\"\"<\/p>\n

                                \u00a0<\/span>Rate of volume flow through the tube<\/span><\/p>\n

                                \u00a0\u00a0<\/span>\"\"<\/p>\n

                                \"\"<\/p>\n

                                Question 30: Figure shows a liquid of density 1200 kg m<\/span><\/strong>\u20133<\/sup><\/span><\/strong>\u00a0flowing steadily in a tube of varying cross section. The cross section at a point A is 1.0 cm<\/span><\/strong>2<\/sup><\/span><\/strong>\u00a0and that at B is 20 mm<\/span><\/strong>2<\/sup><\/span><\/strong>, the points A and B are in the same horizontal plane. The speed of the liquid at A is 10 cm s<\/span><\/strong>-1<\/sup><\/span><\/strong>. Calculate the difference in pressure at A and<\/span><\/strong>\u00a0B.<\/span>\u00a0\u00a0<\/span><\/p>\n

                                Sol: From equation of continuity. The speed\u00a0<\/span>\"\"\u00a0at B is given by,<\/span><\/p>\n

                                \u00a0<\/span>\"\"<\/p>\n

                                \u00a0\u00a0<\/span>Or<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                Or<\/span>\"\"<\/p>\n

                                By Bernoulli equation,\u00a0<\/span>\"\"<\/p>\n

                                \"\"Here<\/span>\"\".<\/span>\u00a0\u00a0\u00a0<\/span><\/p>\n

                                Thus<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                \"\"<\/p>\n

                                55 .<\/span><\/strong>TORRICELLI\u2019S THEOREM or Velocity of Efflux\u00a0<\/span><\/u><\/strong><\/p>\n

                                \"\"\u00a0\u00a0<\/span>If a liquid is filled in a vessel up to height H and a hole is made at a depth h below the free surface of the liquid as shown in fig. then taking the level of hole as reference level\u00a0<\/span><\/p>\n

                                (i.e., zero point of potential energy) and applying Bernoulli’s principle to the liquid just inside and outside the hole (assuming the liquid to be at rest inside)<\/span><\/p>\n

                                \u00a0<\/span>We get<\/span>\u00a0\u00a0<\/span>\"\"<\/p>\n

                                \u21d2<\/span>\"\"<\/p>\n

                                This is same speed as that an object would acquire in falling from rest through a distance h and called\u00a0<\/span>velocity of efflux or velocity of flow<\/span><\/strong><\/p>\n

                                This result was first given by Torricelli, so this is known as Torricelli’s theorem.\u00a0<\/span><\/p>\n

                                Range up to which liquid will fall =\u00a0<\/span>\"\"<\/p>\n

                                Or<\/span>\u00a0\u00a0<\/span>\"\"<\/p>\n

                                Now,<\/span>\u00a0\u00a0<\/span>\"\"<\/p>\n

                                \u21d2<\/span>\"\"\u00a0<\/span><\/p>\n

                                Or<\/span>\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                \u21d2<\/span>\"\"<\/p>\n

                                  \n
                                • Range is max. If<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/li>\n<\/ul>\n

                                  \"\"<\/p>\n

                                  So<\/span>\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                  \u21d2<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                  I.e. Range would be the same when the hole is at a height h or at a height H \u2013 h from the ground or from the top of the beaker.<\/span><\/p>\n

                                  R is maximum at\u00a0<\/span>\"\"\u00a0\u00a0<\/span>and\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                                  \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>The velocity of efflux is independent of the nature of liquid, quantity of liquid in the vessel and the area of orifice.\u00a0<\/span><\/p>\n

                                  \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Greater is the distance of the hole from the free surface of liquid, greater will be the velocity of efflux<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>[i.e.\u00a0<\/span>\"\"<\/p>\n

                                  15. CALORIMETRY AND THERMAL EXPANSION<\/span><\/u><\/strong><\/p>\n

                                  57 .\u00a0<\/span><\/strong>\u00a0<\/span><\/u><\/strong>DEFINITION OF HEAT<\/span><\/u><\/strong><\/p>\n

                                  Heat is energy in transient. Heat energy flows from one body to another body due to their temperature difference. It is measured in units of calories.\u00a0<\/span><\/p>\n

                                  The SI unit is Joule.<\/span>\u00a0 \u00a0<\/span>1 calorie = 4.2J<\/span><\/strong><\/p>\n

                                  Illustration 26: What is the difference between heat and temperature?<\/span><\/strong>\u00a0\u00a0\u00a0<\/span><\/strong><\/p>\n

                                  Sol:<\/span><\/p>\n

                                    \n
                                  • Temperature is associated with kinetic energy of atoms\/molecule while heat is energy in transit.<\/span>\u00a0\u00a0<\/span><\/li>\n
                                  • \u00a0<\/span>Temperature is the measure of the average kinetic energy of the molecules or atoms in a substance. Heat is the flow of energy from one body to another as a result of a temperature difference.\u00a0<\/span><\/li>\n
                                  • \u00a0<\/span>It is important to point out that matter does not contain heat; it contains molecular kinetic energy and not heat. Heat flows and it is the energy that is being transferred. Once heat has been transferred to an object, it ceases to be heat. It becomes internal energy.\u00a0<\/span><\/li>\n<\/ul>\n

                                    58 .<\/span><\/strong>PRINCIPLE OF CALORIMETRY\u00a0<\/span><\/u><\/strong><\/p>\n

                                    When two bodies at different temperatures are mixed, heat will pass from the body at a higher temperature to the body at a lower temperature until the temperature of the mixture becomes constant.<\/span><\/p>\n

                                    \u00a0<\/span><\/em>The principle of calorimetry implies that heat lost by the body at a higher temperature is equal to the heat gained by the other body at a lower temperature assuming that there is no loss of heat in the surroundings.<\/span><\/em><\/p>\n

                                    \u00a0<\/span><\/strong><\/p>\n

                                    59.<\/span><\/strong>\u00a0HEAT CAPACITY\u00a0<\/span><\/u><\/strong><\/p>\n

                                    The heat capacity of a body is defined as the amount of heat required to raise its temperature by 1\u00b0C. It is also known as the thermal capacity of the body.\u00a0<\/span><\/p>\n

                                    Suppose a body has mass m and specific heat c. Heat capacity = Heat required to raise the temperature of the body by 1\u00b0C = mc \u00d7 1 =mc<\/span><\/p>\n

                                    \u2234<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Heat capacity =mc<\/span><\/p>\n

                                    Hence heat capacity of a body (solid or liquid) is equal to the product of its mass and specific heat. Clearly, the SI unit of heat capacity is J\/\u00b0C or J\/K.\u00a0<\/span><\/p>\n

                                    \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>The greater the mass of a body, the greater is its heat capacity.\u00a0<\/span><\/p>\n

                                    60.<\/span><\/strong>\u00a0SPECIFIC HEAT CAPACITY\u00a0<\/span><\/u><\/strong><\/p>\n

                                    When we supply heat to a solid substance (or liquid) its temperature increases.<\/span>\u00a0\u00a0<\/span>It is found that the amount of heat Q absorbed by the solid substance (or liquid), is<\/span><\/p>\n

                                    (i) Directly proportional to the mass (m) of the substance<\/span><\/p>\n

                                    \u00a0<\/span>i.e.,<\/span>\u00a0\u00a0<\/span>Q\u00a0<\/span>\u221d<\/span>\u00a0m\u00a0<\/span><\/p>\n

                                    (ii) Directly proportional to the rise in temperature (\u2206T)\u00a0<\/span><\/p>\n

                                    i.e.<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Q<\/span>\u221d<\/span>\u00a0\u2206T<\/span>\u00a0\u00a0<\/span><\/p>\n

                                    Combining the two factors,\u00a0<\/span><\/p>\n

                                    we have,\u00a0<\/span>\"\"\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\u2026\u2026. (i)\u00a0<\/span><\/p>\n

                                    Or<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Q = cm \u2206T\u00a0<\/span><\/p>\n

                                    Where c is constant of proportionality and is called specific heat capacity or simply specific heat of the substance. From eq. (i),\u00a0<\/span><\/p>\n

                                    we have\u00a0<\/span>\"\"<\/p>\n

                                    If m = 1 kg and \u2206T = 1\u00b0C, then c = Q.<\/span><\/p>\n

                                    Hence the specific heat of a solid (or liquid) may be defined as the amount of heat required to raise the temperature of 1kg of solid (or liquid) through 1\u00b0C (or 1K). SI unit of specific heat is J kg\u00a0<\/span>-1<\/sup><\/span>\u00a0\u00b0C<\/span>-1<\/sup><\/span>\u00a0or J kg\u00a0<\/span>-1<\/sup><\/span>\u00a0K<\/span>-1<\/sup><\/span><\/p>\n

                                      \n
                                    • Specific heat capacity is the property of material and heat capacity is property of a given body.\u00a0<\/span><\/li>\n<\/ul>\n

                                      61 .<\/span><\/strong>DEFINITION OF CALORIE<\/span><\/u><\/strong><\/p>\n

                                      The amount of heat needed to increase the temperature of 1 g of water from 14.5\u00b0C to 15.5\u00b0C at a pressure of<\/span><\/em>\u00a0\u00a0\u00a0<\/span><\/em>1 atm is called 1 calorie.<\/span><\/em>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/em><\/p>\n

                                      1 kilo calorie =103 calories; 1 calorie = 4.186 Joule<\/span><\/p>\n

                                      \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Thermal capacity of a body is the quantity of heat required to raise its temperature through 1\u00b0C and is equal to the product of mass and specific heat of the body.<\/span><\/p>\n

                                      \"\"\u00a0<\/span><\/p>\n

                                      \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>(Be careful about unit of temperature, use units according to the given units of s)<\/span><\/p>\n

                                      Question 31: A geyser heats water flowing at the rate of 3.0 liters per minute from 27\u00b0C to 77\u00b0C. If the geyser operates on a gas burner, what is the rate of combustion of the fuel if its heat of combustion is\u00a0<\/span><\/strong>\"\"\u00a0J g<\/span><\/strong>-1<\/sup><\/span><\/strong>?\u00a0<\/span><\/strong><\/p>\n

                                      Sol: The total heat required to increase the temperature of the water is equal to the heat supplied by the combustion of gas per minute. Mass of 3 liters of water =3kg\u00a0<\/span><\/p>\n

                                      \u2234<\/span>Mass of water flowing per minute,\u00a0<\/span>\"\"\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

                                      Rise of temperature,<\/span>\"\"\u00a0<\/span><\/p>\n

                                      Heat absorbed by water per minute= mc \u2206 \u03b8 =3000\u00d71\u00d7 50cal\u00a0<\/span><\/p>\n

                                      = 3000\u00d7 1\u00d7 50\u00d7 4.2\u00a0<\/span>\"\"\u00a0= 630000\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                                      \u2234<\/span>\u00a0Heat supply by gas burner= 630000\u00a0<\/span>\"\"and heat of combustion of fuel = 4.0\u00d7 104\u00a0<\/span>\"\"<\/p>\n

                                      \u00a0<\/span>\u2234<\/span>\u00a0Rate of combustion of fuel =\u00a0<\/span>\"\"<\/p>\n

                                      62. MOLAR SPECIFIC HEAT CAPACITY (C) FOR SOLIDS OR LIQUIDS<\/span><\/strong><\/p>\n

                                      The molar specific heat of a solid (or liquid) is defined as the amount of heat required to raise the temperature of 1 mole of the solid (or liquid) through 1\u00b0C (or 1K). It is denoted by the symbol C.<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

                                      Therefore, the amount of heat Q required to raise the temperature of n moles of a solid (or liquid) through a temperature change \u2206T<\/span>\u00a0\u00a0\u00a0<\/span>is given by;<\/span><\/p>\n

                                      Q =nC \u2206T<\/span><\/p>\n

                                      \u00a0<\/span>It is clear that SI unit of C is J mol<\/span>-1<\/sup><\/span>\u00a0K<\/span>-1<\/sup><\/span>.\u00a0<\/span><\/p>\n

                                      For any material of mass m and molecular mass M,<\/span><\/p>\n

                                      \u00a0<\/span>the number of moles<\/span>\u00a0\u00a0<\/span>\"\"<\/p>\n

                                      \"\"<\/p>\n

                                      \"\"<\/p>\n

                                      Eq. (i) gives the relation between molar specific heat C and the ordinary specific heat.<\/span><\/p>\n

                                      63. MOLAR SPECIFIC HEAT CAPACITY FOR THE GASES\u00a0<\/span><\/strong><\/p>\n

                                      The amount of heat required to increase the temperature of 1 mole of a gas through 1\u00b0C is called molar heat capacity.\u00a0<\/span><\/em><\/p>\n

                                      The number of moles, n, in mass m of the gas is given by<\/span><\/p>\n

                                      \"\"<\/p>\n

                                      1 Molar Specific Heat at Constant Volume,\u00a0<\/span><\/u><\/strong>\"\"<\/p>\n

                                      :If\u00a0<\/span>\"\"\u00a0is the heat required to raise the temperature of mass m gm or n moles of gas of molecular weight M at constant volume through temperature\u00a0<\/span><\/p>\n

                                      \u00a0\u00a0<\/span>\u2206 T,<\/span>\"\"\u00a0<\/span><\/p>\n

                                      Where<\/span>\u00a0\u00a0<\/span>\"\"\u00a0molar specific is heat at constant volume and is equal to<\/span>\u00a0\u00a0<\/span>\"\"\u00a0.<\/span><\/p>\n

                                      2 Molar specific heat at Constant Pressure, Cp:<\/span><\/strong><\/p>\n

                                      If\u00a0<\/span>\"\"\u00a0is the heat required to raise the temperature of mass m gm or n moles of gas of molecular weight M at constant pressure through temperature<\/span><\/p>\n

                                      \u2206 T,\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                                      Where\u00a0<\/span>\"\"\u00a0\u00a0<\/span>molar specific is heat at constant volume and is equal to\u00a0<\/span>\"\"\u00a0.<\/span><\/p>\n

                                      For monatomic gases,<\/span><\/p>\n

                                      \u00a0\u00a0<\/span>\"\"and<\/span>\"\"\u00a0<\/span><\/p>\n

                                      For diatomic gas<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

                                      Mayer\u2019s relation<\/span><\/strong>\u00a0gives,<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                      Illustration 32 : How much heat is required to raise the temperature of an ideal mono atomic gas by 10 K if the gas is maintained at constant pressure?<\/span><\/strong>\u00a0\u00a0<\/span><\/strong><\/p>\n

                                      \u00a0<\/span>Sol: The process is at constant pressure here. Formula for heat capacity of gas at constant pressure is used.\u00a0<\/span><\/p>\n

                                      The heat required is given by<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

                                      \u00a0<\/span>\"\"\u00a0\u00a0\u00a0<\/span><\/p>\n

                                      Here<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>n=1 & \u2206T<\/span>\u00a0\u00a0<\/span>=10 K;<\/span>\"\"\u00a0\u00a0\u00a0<\/span><\/p>\n

                                      \u00a0<\/span>So\u00a0 <\/span>\"\".\u00a0<\/span>\u00a0<\/span><\/p>\n

                                      64. LATENT HEAT<\/span><\/strong><\/p>\n

                                      The amount of heat required to change a unit mass of a substance completely from one state to another at constant temperature is called the latent heat of the substance.<\/span><\/em>\u00a0<\/span><\/p>\n

                                      If a substance of mass m required heat Q to change completely from one state to another at constant temperature,\u00a0<\/span><\/p>\n

                                      Then the<\/span>\u00a0\u00a0\u00a0<\/span>latent heat<\/span>\"\"<\/p>\n

                                      The SI unit of latent heat of a substance is J kg<\/span>-1<\/sup><\/span><\/p>\n

                                      .\u00a0<\/span>There are two types of latent heats:\u00a0<\/span><\/strong><\/p>\n

                                        \n
                                      1. \u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Latent heat of fusion<\/u><\/strong>:.<\/li>\n<\/ol>\n

                                        The amount of heat required to change the unit mass of solid mass into its liquid state at constant temperature is called the latent heat of fusion of the solid.<\/span><\/em><\/p>\n

                                          \n
                                        • For example, the latent heat of fusion of ice is 334 J\/kg.<\/span><\/li>\n<\/ul>\n

                                          \u00a0<\/span>It means to change 1 kg of ice at 0\u00b0C into liquid water at 0\u00b0C, we must supply 334 KJ of heat.\u00a0<\/span><\/p>\n

                                            \n
                                          1. \u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Latent heat of vaporization:.<\/u><\/li>\n<\/ol>\n

                                            The amount of heat required to change the unit mass of a liquid into its gaseous state at constant temperature is called latent heat of vaporization of the liquid.<\/span><\/p>\n

                                            Question 33: A piece of ice of mass 100 g and at temperature 0\u00b0C is put in 200 g of water at 25\u00b0C. How much ice will melt as the temperature of the water reaches 0\u00b0C? The specific heat capacity of water =4200 JK\u00a0<\/span><\/strong>-1<\/sup><\/span><\/strong>\u00a0and the specific latent heat of fusion of ice= 3.4\u00d7 10<\/span><\/strong>5<\/sup><\/span><\/strong>\u00a0JK\u00a0<\/span><\/strong>-1<\/sup><\/span><\/strong><\/p>\n

                                            Sol:<\/span>\u00a0\u00a0<\/span>Total heat lost by the water equal to the total heat gained by the ice.<\/span><\/p>\n

                                            The heat released as the water cools down from 25\u00b0C to 0\u00b0C is<\/span><\/p>\n

                                            \u00a0<\/span>\"\"<\/p>\n

                                            The amount of ice melted by this much heat is given by<\/span><\/p>\n

                                            \u00a0<\/span>\"\"<\/p>\n

                                            65. WATER EQUIVALENT<\/span><\/u><\/strong>:-<\/span><\/strong>\u00a0<\/span><\/p>\n

                                            The water equivalent of a body is defined as the mass of water that will absorb or lose the same amount of heat as the body for the same rise or fall in temperature.\u00a0<\/span><\/em><\/p>\n

                                            The water equivalent of a body is measured in kg in SI unit and in g in C.G.S. units.\u00a0<\/span><\/p>\n

                                            Consider a body of mass m and specific heat required to raise the temperature of the body through \u2206T is<\/span>\u00a0\u00a0<\/span><\/p>\n

                                            \u00a0<\/span>Q= cm \u2206T<\/span>\u00a0\u00a0\u00a0<\/span>\u2026\u2026\u2026\u2026\u2026… (i)\u00a0<\/span><\/p>\n

                                            Suppose w is water equivalent of this body. Then, by definition,\u00a0<\/span><\/p>\n

                                            \"\"\u00a0\u00a0<\/span><\/p>\n

                                            From eqn. (i) And (ii),\u00a0<\/span><\/p>\n

                                            we have,<\/span>\u00a0\u00a0<\/span>\"\"<\/p>\n

                                            \u21d2<\/span>\u00a0\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                                            Thus the water equivalent of a body is numerically equal to the product of the mass of the body and its specific heat<\/span><\/em><\/p>\n

                                            \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Water equivalent and heat capacity of a body are numerically equal.<\/span><\/p>\n

                                            Question 34: The water equivalent of a body is 10 kg. What does it mean?<\/span><\/strong>\u00a0\u00a0\u00a0<\/span><\/strong><\/p>\n

                                            Sol: It means that if a body is heated through say 5\u00b0C, it will absorb the same amount of heat as absorbed by 10 kg of water when heated through 5\u00b0C.<\/span><\/p>\n

                                            66.<\/span><\/strong>\u00a0MECHANICAL EQUIVALENT OF HEAT<\/span><\/u><\/strong>:-<\/span><\/u><\/p>\n

                                            If mechanical work W produces the same temperature change as heat H,\u00a0<\/span><\/p>\n

                                            We write, W=JH.\u00a0<\/span><\/p>\n

                                            Here J is called mechanical equivalent of heat. It is clear that if W and H are both measured in the same unit then J=1.<\/span><\/p>\n

                                            \u00a0<\/span>If W is measured in joule (work done by a force of 1 N in displacing an object by 1 m in its direction) and H in calorie (heat required to raise the temperature of 1 g of water by 1\u00b0C) then J is expressed in joule per calorie.\u00a0<\/span><\/p>\n

                                            The value of J gives how many joules of mechanical work is needed to raise the temperature of 1 g of water by 1\u00b0C.<\/span><\/p>\n

                                            67. TEMPERATURE SCALES\u00a0<\/span><\/u><\/strong><\/p>\n

                                            1 Kelvin Temperature Scale<\/span><\/u><\/strong>:-\u00a0<\/span><\/p>\n

                                            Kelvin is a temperature scale designed such that zero K is defined as absolute zero and the size of one unit is the same as the size of one degree Celsius.\u00a0<\/span><\/p>\n

                                            Water freezes at 273.15K; water boils at 373. 15K. for calculation purposes,\u00a0<\/span><\/p>\n

                                            we take\u00a0<\/span>\"\"<\/p>\n

                                            \"\"<\/p>\n

                                            \u00a0<\/span><\/p>\n

                                            \u00a0<\/span><\/p>\n

                                            \u00a0<\/span><\/p>\n

                                            \u00a0<\/span><\/p>\n

                                            \u00a0<\/span><\/p>\n

                                            \u00a0<\/span><\/p>\n

                                            \u00a0<\/span><\/p>\n

                                            \u00a0<\/span><\/p>\n

                                            \u00a0<\/span><\/p>\n

                                            \u00a0<\/span><\/p>\n

                                            \u00a0<\/span><\/p>\n

                                              \n
                                            • Absolute<\/span><\/u><\/strong>\u00a0\u00a0<\/span><\/u><\/strong>zero :<\/span><\/u><\/strong>\u00a0<\/span><\/li>\n<\/ul>\n

                                              At absolute zero, a hypothetical temperature, all molecular movement stops<\/span><\/p>\n

                                              \"\"<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>All actual temperatures are above absolute zero.<\/span><\/p>\n

                                              2 Celsius Temperature Scale:-<\/span><\/u><\/strong><\/p>\n

                                              Temperature Scale according to which the temperature difference between the reference temperature of the freezing and boiling of water is divided into 100 degrees.\u00a0<\/span><\/p>\n

                                              The freezing point is taken as zero degrees Celsius and the boiling point as 100 degrees Celsius<\/span><\/em>.\u00a0<\/span><\/p>\n

                                                \n
                                              • The Celsius scale is widely known as the centigrade scale because it is divided into 100 degrees.<\/span><\/li>\n<\/ul>\n

                                                3 Fahrenheit Scale:-<\/span><\/u><\/strong><\/p>\n

                                                Fahrenheit temperature scale is a scale based on 32 for the freezing point of water and 212 for the boiling point of water, the interval between the two being divided into 180 parts.\u00a0<\/span><\/p>\n

                                                \"\"<\/p>\n

                                                \u00a0<\/span><\/p>\n

                                                \u00a0<\/span><\/p>\n

                                                \u00a0<\/span><\/p>\n

                                                \u00a0<\/span><\/p>\n

                                                For easy conversion of temperature units, remember the following equation<\/span><\/p>\n

                                                \"\"<\/p>\n

                                                Here C, F and K are respectively temperature in Celsius, Fahrenheit and Kelvin scale. .<\/span><\/p>\n

                                                Question 35:<\/span><\/strong>\u00a0\u00a0<\/span><\/strong>Express a temperature of 60\u00b0 F in degree Celsius and in Kelvin.<\/span><\/strong>\u00a0\u00a0\u00a0<\/span><\/strong><\/p>\n

                                                  \n
                                                1. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/li>\n
                                                2. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/li>\n<\/ol>\n

                                                  Question 36: Calculate the temperature which has the same value on (i) the Celsius and Fahrenheit (ii) Fahrenheit and Kelvin scales<\/span><\/strong><\/p>\n

                                                  Sol: (i)<\/span>\u00a0\u00a0\u00a0<\/span>Let the required temperature be x\u00b0,\u00a0<\/span><\/p>\n

                                                  Now\u00a0<\/span>\"\"<\/p>\n

                                                  \"\"\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

                                                  Or<\/span>\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                  \u2234<\/span>\"\"<\/p>\n

                                                  \u21d2<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                  \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Let the required temperature be x\u00b0 from relation<\/p>\n

                                                  \u00a0<\/span>\"\"<\/p>\n

                                                  On solving by taking<\/span>\"\"\u00a0<\/span><\/p>\n

                                                  We get,<\/span>\u00a0\u00a0<\/span>X = 574.6<\/span><\/p>\n

                                                  68.<\/span><\/strong>Triple Point of Water<\/span><\/u><\/strong><\/p>\n

                                                  The triple point of water is that unique temperature and pressure at which water can coexist in equilibrium between the solid, liquid and gaseous states.\u00a0<\/span><\/p>\n

                                                  The pressure at the triple point of water is\u00a0<\/span>4.58 mm of Hg<\/span><\/strong>\u00a0and the temperature is\u00a0<\/span>273.16K<\/span><\/strong>\u00a0(or 0.01\u00b0C).\u00a0<\/span><\/p>\n

                                                  THERMAL EXPANSION<\/span><\/u><\/strong><\/p>\n

                                                  69. DEFINITION OF THERMAL EXPANSION<\/span><\/strong><\/p>\n

                                                  \"\"It is the expansion due to increase in temperature. Most substances expand when they are heated. Thermal expansion is a result of the change in average separation between the atoms of an object.<\/span><\/p>\n

                                                  \u00a0<\/span>Atoms of an object can be imagined to be connected to one another by springs as shown in figure… As the temperature of the solid increases, the atoms oscillate with greater amplitudes, as a result the average separation between them increases, and the object expands,\u00a0<\/span><\/p>\n

                                                  Thermal expansion arises from the nature of the potential energy curve.\u00a0<\/span><\/p>\n

                                                  70.<\/span><\/strong>\u00a0THERMAL EXPANSION OF SOLIDS<\/span><\/u><\/strong><\/p>\n

                                                  1 Linear Expansion<\/span><\/u><\/strong><\/p>\n

                                                  When a solid substance is heated, most of them generally expand. If a solid has a length\u00a0<\/span>\"\"and has a very small area of cross-section, at a temperature<\/span>\"\", its length increases to\u00a0<\/span>\"\"\u00a0when its temperature is increased by \u2206T.<\/span><\/p>\n

                                                  \u00a0<\/span>The increase in length,<\/span>\u00a0\u00a0<\/span>\"\"\u00a0,\u00a0<\/span><\/p>\n

                                                  Also<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                  So<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                                                  Where \u03b1 is the coefficient of linear expansion which is given by<\/span><\/p>\n

                                                  \"\"\u00a0<\/span><\/p>\n

                                                  The\u00a0<\/span>coefficient of linear expansion is equal to the increase in length per unit length per degree rise of temperature<\/span><\/em>.<\/span><\/p>\n

                                                  \u00a0<\/span>The SI unit of \u03b1 is \/\u00b0C or \/K.\u00a0<\/span><\/p>\n

                                                  \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Its value is different for different solid materials.<\/span><\/p>\n

                                                  \u00a0<\/span>2 Superficial Expansion or Surface Expansion:-<\/span><\/u><\/strong><\/p>\n

                                                  If a solid plate of area\u00a0<\/span>\"\"\u00a0and of very small thickness is heated through a temperature \u2206T so that its area increases to<\/span>\"\", then the increase in area \u2206A is given by<\/span><\/p>\n

                                                  \"\"<\/p>\n

                                                  Or<\/span>\"\"\u00a0\u00a0<\/span><\/p>\n

                                                  Here \u03b2 is called the coefficient of superficial expansion.<\/span>\u00a0\u00a0<\/span>2 \u03b2= \u03b1<\/span><\/p>\n

                                                  Hence\u00a0<\/span>the coefficient of superficial expansion of a solid may be defined as the fractional change in surface area<\/span><\/em>\u00a0\u00a0<\/span><\/em>\"\"\u00a0\u00a0\u00a0<\/span><\/em>per degree change in temperature<\/span><\/em>.\u00a0<\/span><\/p>\n

                                                  Its SI unit is also \/\u00a0<\/span>0<\/sup><\/span>C or \/K.<\/span><\/p>\n

                                                  3 Volume Expansion:-<\/span><\/u><\/strong><\/p>\n

                                                  If a solid of initial volume\u00a0<\/span>\"\"\u00a0at any temperature is heated so that its volume is increased to\u00a0<\/span>\"\"with increase of temperature \u2206T,\u00a0<\/span><\/p>\n

                                                  The increase in volume, \u2206V, is given by<\/span>\u00a0\u00a0<\/span><\/p>\n

                                                  \u00a0<\/span>\"\"<\/p>\n

                                                  Or<\/span>\"\"<\/p>\n

                                                  Here \u03b3 is called the coefficient of volume or cubical expansion. \u03b3 = 3 \u03b1.\u00a0<\/span><\/p>\n

                                                  As the temperature of solid increases, the oscillation of atoms increases which results in an increase of average distance between atoms due to which the volume increases.<\/span><\/p>\n

                                                    \n
                                                  • If\u00a0<\/span>\"\"\u00a0is the density of a solid at 0\u00b0C and\u00a0<\/span>\"\"\u00a0is its density<\/span>\"\", then for a constant mass m of the solid,<\/span>\u00a0\u00a0<\/span><\/li>\n<\/ul>\n

                                                    \u00a0<\/span>\"\"\u00a0\u00a0\u00a0\u00a0<\/span>and<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                                                    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Where\u00a0<\/span>\"\"\u00a0and\u00a0<\/span>\"\"\u00a0are its respective volume at 0\u00b0C & T\u00b0<\/span>\"\".<\/span><\/p>\n

                                                    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>So<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\u27f9<\/span>\"\"<\/p>\n

                                                    Hence\u00a0<\/span>coefficient of cubical expansion of a solid may be defined as the fractional change in volume<\/span><\/em>\u00a0\u00a0<\/span><\/em>\"\"\u00a0per degree change in temperature<\/span><\/em>.\u00a0<\/span><\/p>\n

                                                    Its SI unit is \/ \u00b0C or \/K.\u00a0<\/span><\/p>\n

                                                    71 .<\/span><\/strong>RELATION BETWEEN COEFFICIENTS OF EXPANSION<\/span><\/u><\/strong><\/p>\n

                                                    We shall now show that for solid, the approximate relations between \u03b1, \u03b2 and \u03b3 are:\u00a0<\/span>\u03b2= 2\u03b1, \u03b3= 3\u03b1;<\/span><\/strong>\u00a0<\/span><\/p>\n

                                                      \n
                                                    1. \u00a0 \u00a0 \u00a0 \u00a0<\/span>Relation between \u03b2 and \u03b1.\u00a0<\/span><\/span><\/li>\n<\/ol>\n

                                                      Consider a square plate of side\u00a0<\/span>\"\"\u00a0at 0\u00b0C<\/span>\u00a0\u00a0\u00a0<\/span>and\u00a0<\/span>\"\"\u00a0at<\/span>\u00a0\u00a0\u00a0<\/span>t\u00b0C<\/span>\u00a0\u00a0<\/span><\/p>\n

                                                      As length<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                                                      Area of plate at t\u00b0C,\u00a0<\/span>\"\"<\/p>\n

                                                      Also Area of plate at t\u00b0C,\u00a0<\/span>\"\"<\/p>\n

                                                      \u21d2<\/span>\"\"<\/p>\n

                                                      Or<\/span>\u00a0\u00a0\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                                                      Since the value of \u03b1 is small, the term\u00a0<\/span>\"\"\u00a0may be neglected<\/span><\/p>\n

                                                      So<\/span>\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                      Or<\/span>\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                      .\u00a0<\/span>\u2234<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\u03b2= 2 \u03b1<\/span><\/strong>\u00a0\u00a0<\/span><\/p>\n

                                                      \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>The result is altogether general because any flat surface can be regarded as a collection of small squares.\u00a0<\/span><\/p>\n

                                                      \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Relation between \u03b3 and \u03b1.<\/strong><\/p>\n

                                                      \u00a0<\/span>Consider a cube of side<\/span>\u00a0\u00a0\u00a0<\/span>\"\"\u00a0At 0\u00b0C and\u00a0<\/span>\"\"\u00a0at t\u00b0C\u00a0<\/span><\/p>\n

                                                      As length<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                                                      Volume of cube at 0<\/span>0<\/sup><\/span>C,<\/span>\u00a0\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                                                      Volume of cube at t \u00b0C,<\/span>\u00a0\u00a0<\/span>\"\"\u00a0\u00a0<\/span><\/p>\n

                                                      \u00a0\u00a0<\/span>Also Volume of cube at t \u00b0C,\u00a0<\/span>\"\"<\/p>\n

                                                      \u21d2<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"\u00a0\u00a0<\/span><\/p>\n

                                                      Or<\/span>\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                      Since the value of \u03b1 is small, we can neglect the higher power of \u03b1.\u00a0<\/span><\/p>\n

                                                      As<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                      \u2234<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\u00a0<\/span>3 \u03b1\u00a0<\/span>\"\"t = \u03b3<\/span>\"\"\u00a0t\u00a0<\/span><\/p>\n

                                                      Or<\/span>\u00a0\u00a0\u00a0<\/span>\u03b3 =3\u03b1<\/span><\/strong><\/p>\n

                                                      Hence<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                      \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Again, result is general because any solid can be regarded as a collection of small cubes.<\/span><\/p>\n

                                                      72 . VARIATION OF DENSITY WITH TEMPERATURE\u00a0<\/span><\/u><\/strong><\/p>\n

                                                      As we know<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"(for a given mass)<\/span><\/p>\n

                                                      So we may write<\/span>\u00a0\u00a0<\/span><\/p>\n

                                                      \u00a0\u00a0<\/span>\"\"<\/p>\n

                                                      \u2234<\/span>\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                      Or<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                      By Binomial theorem as \u03b3 is small.\u00a0<\/span><\/p>\n

                                                      \u00a0<\/span>So\u00a0<\/span>\"\"<\/p>\n

                                                        \n
                                                      • Most substances expand when they are heated,<\/span><\/li>\n<\/ul>\n

                                                        \u00a0<\/span>i.e.. Volume of a given mass of a substance increases on heating, so the density should decrease<\/span><\/p>\n

                                                        Question 37:<\/span><\/strong>\u00a0\u00a0<\/span><\/strong>A glass flask of volume 200 cm<\/span><\/strong>3<\/sup><\/span><\/strong>\u00a0is just filled with mercury at 20\u00b0 C. How much mercury will over flow when the temperature of the system is raised to 100\u00b0C? The coefficient of volume expansion of glass is 1.2\u00d710<\/span><\/strong>-5<\/sup><\/span><\/strong>\/\u00b0C and that of mercury is 18\u00d7 10<\/span><\/strong>-5<\/sup><\/span><\/strong>\/\u00b0C.<\/span><\/strong>\u00a0\u00a0<\/span><\/strong><\/p>\n

                                                        Sol:\u00a0 The increase in the volume of the flask is\u00a0 \"\"<\/p>\n

                                                        \"\"<\/p>\n

                                                        The increase in the volume of the mercury is<\/p>\n

                                                        \u00a0\"\"<\/p>\n

                                                        \u2234<\/span> The volume of the mercury that will overflow<\/p>\n

                                                        \u00a0\"\"<\/p>\n

                                                        Question 38:\u00a0<\/span><\/strong>A sheet of brass is 40 cm long and 8 cm broad at 0 \u00b0C. If the surface area at 100\u00b0C is 320.1 cm2, find the coefficient of linear expansion of brass.<\/strong>\u00a0\u00a0<\/strong><\/p>\n

                                                        Sol: Surface area of sheet at 0\u00b0C, \"\"<\/p>\n

                                                        Surface area of sheet at 100\u00b0C, \"\"<\/p>\n

                                                        Rise in temperature\"\"<\/p>\n

                                                        Increase in surface is \"\"<\/p>\n

                                                        \"\"<\/p>\n

                                                        Coefficient of surface expansion \u03b2 is given by\"\"<\/p>\n

                                                        \"\"<\/p>\n

                                                        Coefficient of linear expansion,<\/p>\n

                                                        \u00a0\"\"<\/p>\n

                                                        73,<\/span><\/strong>\u00a0THERMAL STRESS<\/span><\/u><\/strong><\/p>\n

                                                        If the ends of rods of length\u00a0<\/span>\"\"\u00a0are rigidly fixed and it is heated, its length\u00a0<\/span>\"\"\u00a0tends to increase due to increase in temperature \u2206T , but it is prevented from expansion. It results in setting up compressive or tensile stress in the rod which is called the thermal stress.<\/span><\/p>\n

                                                        \u00a0<\/span>As<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                        \"\"<\/p>\n

                                                        \"\"<\/p>\n

                                                        74. PRACTICAL APPLICATION OF THERMAL EXPANSION OF SOLIDS<\/span><\/u><\/strong><\/p>\n

                                                        There are a large number of important practical applications of thermal expansion of solids. However, we shall brief only a few of them by way of illustration.<\/span><\/p>\n

                                                          \n
                                                        • While laying the railway tracks, a small gap is left between the successive lengths of the rails.<\/span>\u00a0\u00a0<\/span>This gap is provided to allow for the expansion of the rails during summer. If no gap is left, these expansions cause the rails to buckle (tight).<\/span><\/li>\n
                                                        • When the iron tyre of a wheel to be put on the wheel, the tyre is made slightly smaller in diameter than that of wheel. The iron tyre is first heated uniformly till its diameter becomes more than that of the wheel and is then slipped over the wheel. On cooling the tyre contracts and makes a tight fit on the wheel.<\/span><\/li>\n
                                                        • \u00a0<\/span>In bridges, one end is rigidly fastened to its abutment while the other rests on rollers. This provision allows the expansion and contraction to take place during changes in temperature.<\/span><\/li>\n
                                                        • The concrete roads and floor are always made in sections and enough space is provided between the sections. This provision allows expansion and contraction to take place due to change in temperature.<\/span><\/li>\n
                                                        • Effect of temperature on the time period of a pendulum:\u00a0<\/span><\/u>\u00a0<\/span>As the temperature is increased length of the pendulum and hence, time period gets increased or a pendulum clock becomes slow and it\u2019s loses the time. Similarly, if the temperature is decreased the length and hence, the time period gets decreased. A pendulum clock on this case runs fast and it gains the time.\u00a0<\/span><\/li>\n<\/ul>\n

                                                          \"\"\u00a0\u00a0<\/span>\u2022\u00a0<\/span>If a solid object has a hole in it, what happens to size of the hole, when the temperature of the object increases?\u00a0<\/span><\/strong><\/p>\n

                                                          A common misconception is that if the object expands, the hole will shrink because material expands into the hole. But the, truth is that if the Object expands the hole will expand too, because every linear dimension of an object change in the same way when the temperature changes.\u00a0<\/span><\/p>\n

                                                          \u00a0<\/span><\/p>\n

                                                          7<\/span>5. THERMAL EXPANSION OF LIQUIDS\u00a0<\/span><\/u><\/strong><\/p>\n

                                                          As a liquid in a vessel acquires the shape of the vessel, its heating increases the volume of the vessel initially due to expansion of the vessel which decreases the level of the liquid initially. When the temperature of the liquid is increased further, it increased the volume of the liquid.\u00a0<\/span><\/p>\n

                                                          \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Thus the apparent expansion of the liquid is lesser than the real expansion of the liquid which gives a value of coefficient of real expansion more than that for the coefficient of apparent expansion.\u00a0<\/span><\/p>\n

                                                          The coefficient of real expansion, of a liquid is defined as the real increase in volume per degree rise of temperature per unit original volume of the liquid.\u00a0<\/span>\"\",\u00a0<\/span><\/p>\n

                                                          \u00a0<\/span>\"\"\u00a0Of a liquid is defined as the ratio of the observed increase in volume of the liquid with respect to the original level before heating per degree rise of temperature to the original volume of the liquid.\u00a0<\/span><\/p>\n

                                                          So<\/span>\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                          It is clear that\u00a0<\/span>\"\"\u00a0\u00a0<\/span>and both are measured unit \u00b0C<\/span>-1<\/sup><\/span><\/p>\n

                                                          . It can be shown that:\u00a0<\/span>\"\"\u00a0<\/span><\/p>\n

                                                          Where\u00a0<\/span>\"\"is the coefficient of cubical expansion of glass (or material of the container).\u00a0<\/span><\/p>\n

                                                          \u00a0<\/span><\/strong><\/p>\n

                                                          HEAT TRANSFER<\/span><\/u><\/strong><\/p>\n

                                                          76. INTRODUCTION<\/span><\/u><\/strong><\/p>\n

                                                          Heat can be transformed from one place to another place by the three processes – conduction, convection and radiation.<\/span><\/p>\n

                                                            \n
                                                          • In\u00a0<\/span>conduction,<\/span><\/strong>\u00a0the heat flows from a place of higher temperature to a place of lower temperature through a stationary medium. The molecules of the medium oscillate about their equilibrium positions more violently at a place of higher temperature and collide with the molecules of adjacent position,. Thus heat can be transmitted by collision of molecules.\u00a0<\/span><\/li>\n
                                                          • In the cases of liquids and gases, the heat is transferred not only by collision but also by motion of heated molecules which carry the heat in such media. This process is called\u00a0<\/span>convection<\/span><\/strong>.\u00a0<\/span><\/li>\n
                                                          • Radiation is mode of transfer of heat in which the heat travels directly from one place to another without the role of any intervening medium. The heat from the sun propagates mostly through vacuum to reach the earth by the process of\u00a0<\/span>radiation<\/span><\/strong>.<\/span><\/li>\n<\/ul>\n

                                                            77. CONDUCTION<\/span><\/u><\/strong><\/p>\n

                                                            \"\"In conduction, the heat flows from a place of higher temperature to a place of lower temperature through a stationary medium.\u00a0<\/span><\/p>\n

                                                            \"\"The quantity of heat conducted Q in time t across a slab of length L, area of cross-section A and steady state temperature\u00a0<\/span>\"\"\u00a0and\u00a0<\/span>\"\"\u00a0at respective hot and cold ends is given by<\/span>\"\"<\/p>\n

                                                            \"\",<\/span>\u00a0\u00a0<\/span>\u2026\u2026.(1)<\/span><\/p>\n

                                                            Here k is the coefficient of thermal conductivity\u00a0<\/span><\/p>\n

                                                            Coefficient of thermal conductivity is equal to the quantity of heat flowing per unit time through unit area of cross-section of a material per unit length along the direction of flow of heat.<\/span><\/p>\n

                                                            Units of k are kilocalorie\/meter second degree centigrade or J.m<\/span>-1<\/sup><\/span>sec<\/span>-1<\/sup><\/span>\u00a0K<\/span>-1<\/sup><\/span>.\u00a0<\/span><\/p>\n

                                                            In C.G.S. units of k is expressed in calcm<\/span>-1<\/sup><\/span>\u00a0(<\/span>\u2103<\/span>)<\/span>-1<\/sup><\/span>\u00a0sec<\/span>–<\/sup><\/span>\"\"\u00a0<\/sup><\/span>\"\"<\/p>\n

                                                            \u2234 From(1)<\/span>\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                            Or<\/span>\"\"<\/p>\n

                                                            The quantity\u00a0<\/span>\"\"\u00a0is called the temperature gradient.\u00a0<\/span><\/p>\n

                                                            The minus sign indicates that\u00a0<\/span>\"\"\u00a0is negative along the direction of the heat flow, i.e., heat flows from a higher temperature to a lower one.<\/span><\/p>\n

                                                            \"\"<\/p>\n

                                                            Here \u2206T = temperature difference (TD) and\u00a0<\/span>\"\"\u00a0Thermal resistance of the rod.\u00a0<\/span><\/p>\n

                                                            This relation is mathematically equivalent to Ohm\u2019s Law and can be used very effectively in solving problems effectively by considering temperature analogous to potential and heat transferred per unit time as current. Heat flow through a conducting rod Current flow through a resistance\u00a0<\/span><\/p>\n

                                                            \u00a0<\/span><\/p>\n

                                                            78. NEWTON\u2019S LAW OF COOLING<\/span><\/u><\/strong><\/p>\n

                                                            The rate of cooling of a body is directly proportional to the difference of temperature of the body over its surroundings. If a body at temperature\u00a0<\/span>\"\"\u00a0is placed in surroundings at lower temperature<\/span>\"\",<\/span><\/p>\n

                                                            The rate of cooling is given by<\/span><\/p>\n

                                                            \"\"\u00a0\u00a0<\/span><\/p>\n

                                                            Where\u00a0<\/span>\"\"\u00a0is the quantity of heat lost in time dt.\u00a0<\/span><\/p>\n

                                                            Newton\u2019s law of cooling gives<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

                                                            \"\"<\/p>\n

                                                            Where k is constant<\/span><\/p>\n

                                                            If a body of mass m and specific heat s loses a temperature\u00a0<\/span>\"\"\u00a0in time<\/span>\"\",\u00a0<\/span><\/p>\n

                                                            Then<\/span>\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                            \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Newton\u2019s law of cooling can also be thought in the context of Stefan-Boltzmann law by considering the temperature difference between the body and the surroundings very close to zero, i.e. it can be considered as a special case of the latter.\u00a0<\/span><\/p>\n

                                                            79. GROWTH OF ICE ON PONDS<\/span><\/u><\/strong><\/p>\n

                                                            When temperature of the atmosphere falls below 0\u00b0C, the water in the pond starts freezing. Let at time t thickness of ice in the pond is y and atmospheric temperature is -T\u00b0C. The temperature of water in contact with the lower surface of ice will be 0\u00b0C.\u00a0<\/span><\/p>\n

                                                            \u00a0<\/span>Using\u00a0<\/span>\"\"<\/p>\n

                                                            \"\"<\/p>\n

                                                            Or<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                            \"\"\u00a0<\/span><\/p>\n

                                                            Where L\u00a0<\/span>=<\/span>\u00a0Latent heat of fusion<\/span><\/p>\n

                                                            And hence time taken by ice to grow a thickness<\/span><\/p>\n

                                                            \u00a0\u00a0<\/span>y\u00a0<\/span>\"\"<\/p>\n

                                                            \"\"\u00a0\u00a0<\/span><\/p>\n

                                                              \n
                                                            • Time does not depend on the area of pond. Time taken by ice to grow on ponds is independent of area of the pond and it is only dependent only the thickness of ice sheet.\u00a0<\/span><\/li>\n<\/ul>\n

                                                              \u00a0<\/span><\/p>\n

                                                              80. CONVECTION<\/span><\/u><\/strong><\/p>\n

                                                              In this process, actual motion of heated material results in transfer of heat from one place to another<\/span><\/em>.\u00a0<\/span><\/p>\n

                                                              \u00a0<\/span>For example,\u00a0<\/span><\/p>\n

                                                                \n
                                                              • In a hot air blower, air is heated by a heating element and is blown by a fan. The air carries the heat wherever it goes.\u00a0<\/span><\/li>\n
                                                              • When water is kept in a vessel and heated on a stove, the water at the bottom gets heated due to conduction through the vessel\u2019s bottom. Its density decreases and consequently it rises. Thus, the heat is carried from bottom to the top by the actual movement of the parts of the water.\u00a0<\/span><\/li>\n<\/ul>\n

                                                                \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>If the heated material is forced to move, say by a blower or by a pump, the process of heat transfer is called\u00a0<\/span>forced convection<\/span><\/u>.<\/span><\/p>\n

                                                                \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>If the material moves due to difference in density, it is called\u00a0<\/span>natural or free convection<\/span><\/u>.\u00a0<\/span><\/p>\n

                                                                81. RADIATION<\/span><\/u><\/strong><\/p>\n

                                                                All objects radiate energy continuously in the form of electromagnetic waves, the transfer of energy in form of radiation which does not require a medium.\u00a0<\/span><\/p>\n

                                                                \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>The best known example of this process is the radiation from Sun.\u00a0<\/span><\/p>\n

                                                                Stefan\u2019s law<\/span><\/u><\/strong><\/p>\n

                                                                The rate at which an object radiates energy is proportional to the fourth power of its absolute temperature<\/span><\/em>. This is known as the\u00a0<\/span>Stefan\u2019s law<\/span><\/u>\u00a0.<\/span><\/p>\n

                                                                As<\/span>\u00a0\u00a0\u00a0<\/span>\u00a0<\/span><\/strong>\"\"<\/p>\n

                                                                e lies between 0 and 1 and is called emissivity of the object and \u03c3 is universal constant called Stefan\u2019s constant, which has the value,\u00a0<\/span><\/p>\n

                                                                \"\"\u00a0<\/span><\/p>\n

                                                                82. PERFECTLY BLACK BODY<\/span><\/u><\/strong><\/p>\n

                                                                \"\"A body that absorbs the entire radiation incident upon it and has as emissivity equal to 1 is called a perfectly black body<\/span><\/em>. A black body is also an ideal radiator. It implies that if a black body and an identical another body is kept at the same temperature, then the black body will radiate maximum power as is obvious from equation<\/span><\/p>\n

                                                                \u00a0\u00a0<\/span>\"\"<\/p>\n

                                                                This is also because e=1 for a perfectly black body while for any other body, e<1.\u00a0<\/span><\/p>\n

                                                                  \n
                                                                • Always remember that black body is a perfect absorber and emitter of light. At temperatures higher than the surrounding, it is the most shining thing and at lower temperatures it is the darkest thing.\u00a0<\/span><\/li>\n
                                                                • There is no perfect black body. Materials like black velvet or lamp black come close to being ideal black bodies, but the best practical realization of an ideal black body is a small hole leading into a cavity, as this absorbs 98% of the radiation incident on them.\u00a0<\/span><\/li>\n<\/ul>\n

                                                                  83. ABSORPTIVE POWER \u2018a\u2019<\/span><\/u><\/strong><\/p>\n

                                                                  \u00a0<\/span><\/em>\u201cIt is defined as the ratio of the radiant energy absorbed by a body in a given time to the total radiant energy incident on it in the same interval of time.\u201d<\/span><\/em>\u00a0\u00a0<\/span><\/em><\/p>\n

                                                                  \u00a0\u00a0<\/span>\"\"<\/p>\n

                                                                  As a perfectly black body absorbs all radiations incident on it, the absorptive power of perfectly black body is maximum and unity.\u00a0<\/span><\/p>\n

                                                                  84. SPECTRAL ABSORPTIVE \u2018<\/span><\/u><\/strong>\"\"\u2019<\/span><\/p>\n

                                                                  This absorptive power \u2018a\u2019 refers to radiations of all wavelengths (or the total energy) while the spectral absorptive power is the ratio of radiant energy absorbed by a surface to the radiant energy incident on it for a particular wavelength \u03bb. It may have different values for different wavelengths for a given surface.\u00a0<\/span><\/p>\n

                                                                  85. EMISSIVE POWER \u2018e\u2019<\/span><\/u><\/strong><\/p>\n

                                                                  \u00a0<\/span>\u201cFor a given surface it is defined as the radiant energy emitted per second per unit area of the surface.\u201d<\/span><\/p>\n

                                                                  \u00a0<\/span>It has the units of<\/span>\u00a0\u00a0<\/span>\"\"\u00a0,<\/span><\/p>\n

                                                                  \u00a0<\/span>for a black body\u00a0<\/span>\"\"<\/p>\n

                                                                  Note: Absorptive power is dimensionless quantity where emissive power is not.<\/span><\/p>\n

                                                                  \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Don\u2019t confuse it with the emissivity e which is different from it, although both have the same symbols e.<\/span><\/p>\n

                                                                  86.<\/span><\/u><\/strong>\u00a0\u00a0<\/span><\/u><\/strong>SPECTRAL EMISSIVE POWER\u00a0<\/span><\/u><\/strong><\/p>\n

                                                                  \u00a0<\/span>Similar to the definition of the spectral absorptive power, it is emissive power for a particular wavelength \u03bb.\u00a0<\/span><\/p>\n

                                                                  Thus,<\/span>\u00a0\u00a0\u00a0<\/span>\"\"<\/p>\n

                                                                  87 . KIRCHHOFF\u2019S LAW\u00a0<\/span><\/u><\/strong><\/p>\n

                                                                  The ratio of emissive power to absorptive power is the same for all bodies at a given temperature and is equal to the emissive power E of a blackbody at that temperature.<\/span><\/em><\/p>\n

                                                                  Thus<\/span>\"\"<\/p>\n

                                                                  \uf046<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>Kirchhoff\u2019s law tells that if a body has high emissive power, it should also have high absorptive power to have the ratio e\/a same. Similarly, a body having low emissive power should have low absorptive power.\u00a0<\/span><\/p>\n

                                                                  88. STEFANS-BOLTZMANN LAW<\/span><\/u><\/strong><\/p>\n

                                                                  The energy of thermal radiation emitted per unit time by a blackbody of surface area A is given by<\/span><\/p>\n

                                                                  \"\"<\/p>\n

                                                                  Here<\/span>\"\"\u00a0is a universal constant known as Stefan Boltzmann constant.\u00a0<\/span><\/p>\n

                                                                  The measured value of \u03c3 is 5.67\u00d710<\/span>-8<\/sup><\/span>\u00a0Wm<\/span>-2<\/sup><\/span>\u00a0K<\/span>-4<\/sup><\/span>.\u00a0<\/span><\/p>\n

                                                                  89. WIEN\u2019S DISPLACEMENT LAW<\/span><\/u><\/strong><\/p>\n

                                                                  At ordinary temperatures (below about 600<\/span>\u2103<\/span>), the thermal radiation emitted by bodies is invisible, most of them lie in wavelengths longer than visible light.. As the temperature of the black body increases, two different behaviors are observed. The first effect is that the peak of the distribution shifts to shorter wavelengths. This shift is found to satisfy the following relationship called Wien\u2019s displacement law.\u00a0<\/span><\/p>\n

                                                                  \"\"\u00a0.<\/span><\/p>\n

                                                                  \u00a0<\/span>Here b is a constant called Wien\u2019s constant.\u00a0<\/span><\/p>\n

                                                                  The value of this constant in SI unit is\u00a0 <\/span>\"\".\u00a0<\/span><\/p>\n

                                                                  Question39:<\/span><\/strong>\u00a0\u00a0<\/span><\/strong>The light from the sun is found to have a maximum intensity near the wavelength of 470 nm. Assuming that the surface of the sun emits as a blackbody, calculate the temperature of the surface of the sun.\u00a0<\/span><\/strong><\/p>\n

                                                                  Sol: For a blackbody<\/span>\"\"\u00a0\u00a0\u00a0<\/span><\/p>\n

                                                                  Thus,\u00a0<\/span>\"\"<\/p>\n

                                                                  90. SOLAR CONSTANT AND TEMPERATURE OF SUN<\/span><\/u><\/strong><\/p>\n

                                                                  Solar constant is defined as the amount of radiation received from the sun at the earth per minute per\u00a0<\/span><\/em>\"\"\u00a0of a surface placed at right angle to the solar radiation at a mean distance of the earth from the sun<\/span><\/em>.<\/span><\/p>\n

                                                                  \u00a0<\/span>Assuming that the absorption of solar radiation by the atmosphere near the earth is negligible,<\/span><\/p>\n

                                                                  \u00a0<\/span>The value of solar constant, S, is equal to\u00a0<\/span>\"\"<\/p>\n

                                                                  \u00a0<\/span><\/p>\n

                                                                  \n

                                                                  1<\/strong><\/p>\n

                                                                  \u00a0<\/em><\/strong><\/p>\n<\/div>\n<\/div>\n

                                                                  \n","protected":false},"excerpt":{"rendered":"

                                                                  Chapter 11 Thermal Properties of Matter Chapter 11 Thermal Properties of Matter: Heat, temperature, thermal expansion; thermal expansion of solids, liquids and gases, anomalous expansion of water; specific heat capacity; Cp, Cv \u2013 calorimetry; change of state \u2013 latent heat capacity. Heat transfer-conduction, convection and radiation, thermal conductivity, qualitative ideas of Blackbody radiation, Wein\u2019s displacement […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"default","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"class_list":["post-2282","page","type-page","status-publish","hentry"],"aioseo_notices":[],"_hostinger_reach_plugin_has_subscription_block":false,"_hostinger_reach_plugin_is_elementor":false,"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false},"uagb_author_info":{"display_name":"sandeep.soni484@gmail.com","author_link":"https:\/\/vartmaaninstitutesirsa.com\/index.php\/author\/sandeep-soni484gmail-com\/"},"uagb_comment_info":0,"uagb_excerpt":"Chapter 11 Thermal Properties of Matter Chapter 11 Thermal Properties of Matter: Heat, temperature, thermal expansion; thermal expansion of solids, liquids and gases, anomalous expansion of water; specific heat capacity; Cp, Cv \u2013 calorimetry; change of state \u2013 latent heat capacity. Heat transfer-conduction, convection and radiation, thermal conductivity, qualitative ideas of Blackbody radiation, Wein\u2019s displacement…","_links":{"self":[{"href":"https:\/\/vartmaaninstitutesirsa.com\/index.php\/wp-json\/wp\/v2\/pages\/2282","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vartmaaninstitutesirsa.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/vartmaaninstitutesirsa.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/vartmaaninstitutesirsa.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/vartmaaninstitutesirsa.com\/index.php\/wp-json\/wp\/v2\/comments?post=2282"}],"version-history":[{"count":4,"href":"https:\/\/vartmaaninstitutesirsa.com\/index.php\/wp-json\/wp\/v2\/pages\/2282\/revisions"}],"predecessor-version":[{"id":4701,"href":"https:\/\/vartmaaninstitutesirsa.com\/index.php\/wp-json\/wp\/v2\/pages\/2282\/revisions\/4701"}],"wp:attachment":[{"href":"https:\/\/vartmaaninstitutesirsa.com\/index.php\/wp-json\/wp\/v2\/media?parent=2282"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}