CBSE Class 12 Physics Heat and Thermodynamics Exam Notes

CBSE Class 12 Physics Heat and Thermodynamics Exam Notes. Please refer to the examination notes which you can use for preparing and revising for exams. These notes will help you to revise the concepts quickly and get good marks.

HEAT AND THERMODYNAMICS

HEAT AND THERMODYNAMICS

Calorimetry

Unit of Heat

Heat energy flows from one body to another body due to their temperature difference. It is measured in units of calorie. A calorie is the amount of heat required to raise the temperature of 1 gm of water through 1°C.

\[1\ \text{kilocalorie}=10^3\ \text{calories},\qquad 1\ \text{calorie}=4.186\ \text{joule}\]

If the temperature of a body of mass \(m\) is raised through a temperature \(\Delta T\), then the heat \(\Delta Q\) given to the body is

\[\Delta Q=m\,S\,\Delta T\]

where \(S\) is the specific heat of the body, defined as the amount of heat required to raise the temperature of a unit mass of the body through 1°C. Its unit is cal/gm °C or J/kg·K. The thermal capacity of a body is the quantity of heat required to raise its temperature through 1°C and is equal to the product of mass and specific heat of the body.

Principle of Calorimetry

When two bodies at different temperatures are mixed, heat passes from the body at the higher temperature to the body at the lower temperature until the temperature of the mixture becomes constant. The principle of calorimetry implies that the heat lost by the body at the higher temperature is equal to the heat gained by the body at the lower temperature, assuming that there is no loss of heat to the surroundings.

Latent Heat

It is the amount of heat absorbed or given out by a body during the change of state, when its temperature remains constant.

• Latent heat of fusion: the quantity of heat required to change unit mass of a solid into its liquid state at a constant temperature, its melting point. The latent heat of fusion of ice is 80 calories per gm.
• Latent heat of vaporisation: the quantity of heat required to convert unit mass of a liquid to the gaseous state at its constant temperature of vaporisation. The latent heat of vaporisation of water is 540 calories/gm at 100°C under normal atmospheric pressure, when 1 gram of water gets converted into steam.

Specific Heat of Gases

In the case of solids and liquids, the change of volume or pressure, and the external work done during the change of temperature, is negligible. There is only one value of specific heat for solids or liquids. However, when the temperature of a gas is changed, there is a large change in its volume or pressure. The heat supplied thus depends upon the condition under which the temperature changes. It implies that the specific heat has to be specified separately for constant volume and constant pressure:

1. Specific heat at constant volume, \(c_v\): the amount of heat required to raise the temperature of one gram of a gas through 1°C when the volume of the gas is constant. \(\therefore\ \Delta Q=mc_v\Delta T\).
2. Specific heat at constant pressure, \(c_p\): the amount of heat required to raise the temperature of one gram of a gas through 1°C when the pressure of the gas is kept constant. \(\therefore\ \Delta Q=mc_p\Delta T\).

Molar Heat Capacity

The amount of heat required to increase the temperature of 1 mole of a gas through 1°C is called the molar heat capacity. The number of moles \(n\) in a mass \(m\) of the gas is given by \(n=\dfrac{\text{mass of gas}}{\text{molecular weight}}\).

Molar specific heat at constant volume, \(C_v\): If \(\Delta Q\) is 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 \(\Delta T\), then

\[(\Delta Q)_v=mc_v\Delta T=nMc_v\Delta T=nC_v\Delta T\]

where \(C_v\) is the molar specific heat at constant volume and is equal to \(Mc_v\).

Molar specific heat at constant pressure, \(C_p\): similarly,

\[(\Delta Q)_p=mc_p\Delta T=nMc_p\Delta T=nC_p\Delta T\]

where \(C_p\) is the molar specific heat at constant pressure and is equal to \(Mc_p\).

\[\text{Monoatomic gases: }C_p=\frac{5R}{2},\ C_v=\frac{3R}{2},\ \gamma=\frac{C_p}{C_v}=\frac53=1.67\]

\[\text{Diatomic gases: }C_p=\frac{7R}{2},\ C_v=\frac{5R}{2},\ \gamma=\frac{C_p}{C_v}=\frac75=1.4\]

Mayer’s relation:

\[C_p-C_v=R,\qquad\text{where }C_v=\frac{R}{\gamma-1},\ C_p=\frac{\gamma R}{\gamma-1}\]

Thermal Expansion

Thermal Expansion of Solids

Linear expansion: Almost all substances expand on heating. If a solid of very small area of cross-section has a length \(L_0\) at a temperature \(T_0\), and its length becomes \(L_T\) when its temperature is increased by \(\Delta T\), the increase in length \(\Delta L\) is given by

\[\Delta L=L_T-L_0=L_0\,\alpha\,\Delta T\]

where \(\alpha\) is the coefficient of linear expansion:

\[\alpha=\frac{L_T-L_0}{L_0\Delta T},\qquad L_T=L_0(1+\alpha\Delta T)\]

The coefficient of linear expansion is equal to the increase in length per unit length per degree rise of temperature.

Superficial expansion: If a solid plate of area \(A_0\) and very small thickness is heated through a temperature \(\Delta T\) so that its area increases to \(A_T\), the increase in area is

\[\Delta A=A_T-A_0=\beta A_0\Delta T\quad\text{or}\quad\beta=\frac{A_T-A_0}{A_0\Delta T}\]

where \(\beta\) is the coefficient of superficial expansion, and \(\beta=2\alpha\).

Volume expansion: If a solid of initial volume \(V_0\) is heated so that its volume increases to \(V_T\) with an increase of temperature \(\Delta T\), the increase in volume is

\[\Delta V=V_T-V_0=\gamma V_0\Delta T,\qquad\gamma=\frac{V_T-V_0}{V_0\Delta T}=\frac{\Delta V}{V_0\Delta T}\]

where \(\gamma\) is the coefficient of volume (cubical) expansion, and \(\gamma=3\alpha\).

As the temperature of a solid increases, the amplitude of oscillations of its atoms increases, which results in an increase of the average distance between atoms, due to which the volume increases.

If \(\rho_0\) is the density of a solid at 0°C and \(\rho_T\) its density at \(T\)°C, then for a constant mass \(m\), \(\rho_0=\dfrac{m}{V_0}\) and \(\rho_T=\dfrac{m}{V_T}\), where \(V_0\) and \(V_T\) are the respective volumes at 0°C and \(T\)°C. Therefore

\[\frac{\rho_0}{\rho_T}=\frac{V_T}{V_0}=\frac{V_0(1+\gamma T)}{V_0}=1+\gamma T\quad\therefore\quad\rho_T=\frac{\rho_0}{1+\gamma T}\]

Thermal Stress

If the ends of a rod of length \(L_0\) are rigidly fixed and it is heated, its length tends to increase due to the rise in temperature \(\Delta T\), but it is prevented from expanding. This sets up a compressive or tensile stress in the rod, called the thermal stress. As \(Y=\dfrac{\text{Stress}}{\text{Strain}}\),

\[\text{Stress}=Y\times\text{Strain}=\frac{Y\Delta L}{L_0}=\frac{Y\alpha L_0\Delta T}{L_0}=Y\alpha\Delta T\]

The force \(F\) on the rigid supports is

\[F=\text{Stress}\times A=YA\alpha\Delta T\]

where \(A\) is the area of cross-section of the rod.

Thermal Expansion of Liquids

As a liquid in a vessel acquires the shape of the vessel, heating first increases the volume of the vessel, which initially decreases the level of the liquid. When the temperature of the liquid increases further, the volume of the liquid increases. Thus the observed (apparent) level of the liquid rises less than the real level when the temperature increases. Thus the apparent expansion of the liquid is less than the real expansion, which gives a value of the coefficient of real expansion more than that of the coefficient of apparent expansion.

The coefficient of real expansion, \(\gamma_r\), of a liquid is defined as the real increase in volume per degree rise of temperature per unit original volume of the liquid.

\[\gamma_r=\frac{\text{real increase in volume}}{\text{original volume}\times\text{rise in temperature}}\]

The coefficient of apparent expansion, \(\gamma_a\), of 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.

\[\gamma_a=\frac{\text{apparent increase in volume}}{\text{original volume}\times\text{rise in temperature}}\]

If \(\gamma_g\) is the coefficient of cubical expansion of the material of the vessel, then

\[\gamma_r=\gamma_a+\gamma_g,\qquad\gamma_g=3\alpha\]

The density of the liquid \(\rho_T\) at temperature \(T\) is related to the density \(\rho_0\) at 0°C as \(\rho_T=\dfrac{\rho_0}{1+\gamma T}\), where \(\gamma\) is the coefficient of real expansion of the liquid and \(T\) is the rise in temperature.

Thermal Expansion of Gases

The molecules in an ideal gas have only kinetic energy due to forces of collision but do not possess any potential energy. The thermodynamic state of any gas is defined in terms of its pressure, volume and temperature, denoted by \(P\), \(V\) and \(T\) respectively. A change in one of these quantities produces a corresponding change in the others, depending upon the condition under which the transformation takes place. Such changes are governed by the following gas laws.

Boyle’s law: The pressure of a given mass of a gas is inversely proportional to its volume if the temperature \(T\) is kept constant.

\[P\propto\frac1V\quad\text{or}\quad PV=\text{constant}\]

If the pressure \(P_1\) and volume \(V_1\) change to \(P_2\), \(V_2\) at constant temperature, then \(P_1V_1=P_2V_2\).

Charles’s law of volume: The volume \(V\) of a given mass of a gas is directly proportional to its absolute temperature \(T\) when its pressure remains constant.

\[V\propto T\quad\text{or}\quad\frac VT=\text{constant}\]

If the volume \(V_1\) and temperature \(T_1\) change to \(V_2\), \(T_2\) at constant pressure, then \(\dfrac{V_1}{T_1}=\dfrac{V_2}{T_2}\), where the temperatures are in the Kelvin scale. If \(V_0\) and \(V_t\) are the volumes of the gas at 0°C and \(t\)°C respectively,

\[\frac{V_t}{273+t}=\frac{V_0}{273},\qquad V_t=V_0\left(1+\frac{t}{273}\right)=V_0[1+\alpha_vt]\]

where \(\alpha_v\) is the volume coefficient of a gas and is equal to \(1/273\).

Gay-Lussac’s law of pressure: The pressure of a given mass of a gas is directly proportional to its absolute temperature, provided the volume of the gas is kept constant.

\[P\propto T\quad\text{or}\quad\frac PT=\text{constant}\]

If the pressure and temperature \(P_1\), \(T_1\) change to \(P_2\), \(T_2\) at constant volume, then \(\dfrac{P_1}{T_1}=\dfrac{P_2}{T_2}=\text{constant}\). If \(P_t\) and \(P_0\) are the pressures of the gas at \(t\)°C and 0°C respectively, then

\[P_t=P_0(1+\alpha_pt)=P_0\left(1+\frac{t}{273}\right)\]

where \(\alpha_p\) is the pressure coefficient of the gas, which is also equal to \(1/273\).

Gas equation: If the above three laws are combined, then

\[\frac{PV}{T}=\text{constant},\qquad\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}=\text{constant}\]

The value of the constant depends on the mass of the gas. If the gas has \(n\) moles, \(PV=nRT\), which is called the equation of state of an ideal gas. \(R\) is called the universal or molar gas constant, and its value in SI units is \(8.314\ \text{J mol}^{-1}\text{K}^{-1}\).

Kinetic Theory of Gases

The macroscopic behaviour of gases is understood in terms of the motion of their molecules, based on the following assumptions:

1. All gases are made of molecules which are identical in all respects and are assumed to be rigid and perfectly elastic spheres, such that the size of the molecules is negligible as compared to their intermolecular separation.
2. All the molecules are in constant random motion and move in all directions with various speeds.
3. The molecules collide continuously with each other and with the walls of the container, such that the collisions are perfectly elastic and the duration of a collision is negligible as compared to the time spent in independent motion.
4. The volume of the gas molecules is negligible as compared to the volume occupied by the gas.
5. A molecule travels in a straight line with uniform velocity between two successive collisions. The average distance through which a molecule moves freely between successive collisions is called the mean free path.

Pressure Exerted by an Ideal Gas

If \(n\) molecules of a gas have different speeds \(v_1,v_2,v_3,\ldots,v_n\), then their mean square velocity \(\overline{v^2}\) is

\[\overline{v^2}=\frac{v_1^2+v_2^2+v_3^2+\cdots+v_n^2}{n}\]

It can be shown from the kinetic theory that the pressure \(P\) exerted by a mass \(M\) of a perfect gas of volume \(V\) is

\[P=\frac13\times\frac MV\times\overline{v^2}\]

where \(\overline{v^2}\) is the mean square velocity of the gas molecules. The density \(\rho\) of the gas is \(\rho=\dfrac MV=\dfrac{nM_m}{V}\), where \(n\) is the number of moles and \(M_m\) the molecular weight. Therefore

\[P=\frac13\rho\overline{v^2}\]

If the root mean square velocity is \(v_{rms}=\sqrt{\overline{v^2}}=\sqrt{\dfrac{v_1^2+v_2^2+\cdots+v_n^2}{n}}\), then

\[v_{rms}=\sqrt{\frac{3P}{\rho}},\qquad P=\frac13\rho v_{rms}^2=\frac13\frac MVv_{rms}^2\]

\[\therefore\ PV=\frac13nM_mv_{rms}^2=nRT\ \Rightarrow\ \frac13M_mv_{rms}^2=RT\ \Rightarrow\ v_{rms}=\sqrt{\frac{3RT}{M_m}}\]

where \(M_m\) is the molecular weight of the gas. The root mean square velocity of a gas molecule is proportional to the square root of the absolute temperature.

\[v_{rms}=\sqrt{\frac{3PV}{M}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3P}{\rho}}\]

Apart from the root mean square speed, molecular speeds can be measured as average speed and most probable speed.

Average speed: If \(v_1,v_2,\ldots,v_n\) are the velocities of \(n\) molecules of the gas, their average speed is the arithmetic mean of the various speeds:

\[v_{av}=\frac{v_1+v_2+v_3+\cdots+v_n}{n}=\sqrt{\frac{8RT}{\pi M}}\]

\(\therefore\ v_{av}=0.921\,v_{rms}\) and \(v_{rms}=1.085\,v_{av}\).

Most probable speed: It is the speed possessed by the maximum number of molecules of a gas at a given temperature \(T\).

\[v_{mps}=\sqrt{\frac{2RT}{M}}=\sqrt{\frac23}\,v_{rms}\]

\(v_{mps}=0.816\,v_{rms}\) and \(v_{rms}=1.224\,v_{mps}\). The ratio of these speeds is

\[v_{rms}:v_{av}:v_{mps}::\sqrt3:\sqrt{\frac8\pi}:\sqrt2,\qquad v_{mps}:v_{av}:v_{rms}::1:1.28:1.224\]

Kinetic Interpretation of Temperature

As \(PV=\dfrac13nM_mv_{rms}^2=nRT\), we get \(\dfrac12M_mv_{rms}^2=\dfrac32RT\). Therefore

\[\text{Kinetic energy per mole of gas}=E=\frac12M_mv_{rms}^2=\frac32RT\]

Thus the total translational kinetic energy per mole of molecules of an ideal gas is directly proportional to the absolute temperature. As \(R=\) Boltzmann’s constant \(\times\) Avogadro number, i.e. \(R=kN_A\),

\[E=\frac32kN_AT,\quad\text{where }k=1.38\times10^{-23}\ \text{J/K}\]

\[\text{Kinetic energy per molecule}=\frac32kT\]

Degrees of Freedom and Law of Equipartition of Energy

Each independent mode by which a system can absorb energy is called a degree of freedom. It is the number of independent motions possible, or the number of independent coordinates needed to specify the dynamic position of a body.

In a monoatomic gas, the molecules have three translational degrees of freedom along the three coordinate axes. The total kinetic energy \(\tfrac32kT\) is equally divided amongst its three degrees of freedom, so that each has kinetic energy \(\tfrac12kT\). In diatomic molecules like H₂, O₂ etc., there are three translational and two rotational degrees of freedom, i.e. five degrees of freedom. In triatomic or higher polyatomic gases, like H₂O, NH₃ etc., there are six degrees of freedom, due to three translational and three rotational degrees of freedom.

According to the law of equipartition of energy, the total energy of a molecule is equally distributed on the average among all the degrees of freedom. For an ideal gas at temperature \(T\), the energy per molecule per degree of freedom is \(\tfrac12kT\), where \(k\) is Boltzmann’s constant, and the energy per mole per degree of freedom is \(\tfrac12NkT=\tfrac12RT\), where \(N\) is the number of molecules in one mole.

For \(n\) degrees of freedom, \(C_p\), \(C_v\) and \(\gamma\) for different types of gases are given as follows:

Thermodynamics

A thermodynamic system is one in which the state of a gas can be defined in terms of its pressure \(P\), volume \(V\) and temperature \(T\). If a thermodynamic system undergoes a change, these parameters change depending upon the condition or process under which the state of the gas changes. The changes taking place in various thermodynamic processes are as follows.

Isothermal process: A thermodynamic process in which the temperature remains constant is called an isothermal process. The changes in pressure and volume must be carried out very slowly, so that the heat produced due to compression (or the cooling during expansion) of the gas is conducted to the surroundings and the temperature of the gas does not change, due to thermal equilibrium. Thus the work done during an isothermal process from A to B is

\[W_{iso}=\int_{V_1}^{V_2}P\,dV=nRT\int_{V_1}^{V_2}\frac{dV}{V}=nRT\ln\left(\frac{V_2}{V_1}\right)\]

The isothermal process follows Boyle’s law, since the change takes place at a constant temperature.

Adiabatic process: It is a thermodynamic process in which there is no flow of heat into or out of the system. In an adiabatic process there is a change in temperature, and the heat developed or produced cannot be given up to or taken from the surroundings, as the system can be thermally insulated from them. Also, when a gas is compressed or expanded very rapidly, there is no time for exchange of heat with the surroundings. The pressure \(P\) and volume \(V\) for an adiabatic change follow the relation

\[PV^{\gamma}=\text{constant},\qquad\gamma=\frac{C_p}{C_v}\]

As the gas equation \(PV=nRT\) is followed, the process also follows the relations

\[TV^{\gamma-1}=\text{constant},\qquad T^{\gamma}P^{1-\gamma}=\text{constant}\]

The work done during an adiabatic change from A to B is \(W_{ad}=\displaystyle\int_{V_1}^{V_2}P\,dV\). As \(PV^\gamma=K\) (constant), \(P=\dfrac{K}{V^\gamma}\), so

\[W_{ad}=K\int_{V_1}^{V_2}V^{-\gamma}dV=K\left[\frac{V^{1-\gamma}}{1-\gamma}\right]_{V_1}^{V_2}=\frac{K}{1-\gamma}\left[V_2^{1-\gamma}-V_1^{1-\gamma}\right]=\frac{1}{1-\gamma}\left[P_2V_2^{\gamma}V_2^{1-\gamma}-P_1V_1^{\gamma}V_1^{1-\gamma}\right]\]

\[\therefore\ W_{ad}=\frac{1}{1-\gamma}\left[P_2V_2-P_1V_1\right]=\frac{nR}{1-\gamma}\left[T_2-T_1\right]\]

where \(T_1\) and \(T_2\) are the respective temperatures at A and B.

First Law of Thermodynamics

It states that when a certain quantity of heat \(Q\) is given to a thermodynamic system, a part of the heat \(\Delta U\) is used to change its internal energy and a part is used to do the external work \(W\):

\[Q=\Delta U+W\]

If the work is done by the gas itself in expanding, \(W\) is positive. In the case when no heat \(Q\) is given from outside, \(Q\) is zero, therefore \(\Delta U=-W\) and the body loses internal energy and cools. If the work is done on the gas while compressing it, \(W\) is negative and \(\Delta U=-(-W)=+W\); its internal energy increases and the gas is heated up and its temperature rises.

Applications of the First Law of Thermodynamics

(1) Isothermal process: If the transformation takes place from A to B, the volume increases from \(V_i\) to \(V_f\). As the temperature in an isothermal process remains constant, \(\Delta T=0\), so the internal energy change \(\Delta U=nC_v\Delta T=0\). The work done in the isothermal process from A to B is

\[W_{iso}=\int_{V_i}^{V_f}P\,dV=nRT\ln\left(\frac{V_f}{V_i}\right),\qquad\therefore\ Q=\Delta U+W_{iso}=nRT\ln\left(\frac{V_f}{V_i}\right)\]

(2) Adiabatic process: If an adiabatic transformation takes place from the state A (\(P_i,V_i\)) to the state B (\(P_f,V_f\)), the system does not exchange heat with the surroundings. \(\therefore\ Q=0=\Delta U+W\), so \(\Delta U=-W\). The work done in the adiabatic transformation from A to B, with \(PV^\gamma=\) constant, is

\[W_{ad}=\int_{V_i}^{V_f}P\,dV=\frac{P_fV_f-P_iV_i}{1-\gamma}=\frac{nR(T_f-T_i)}{1-\gamma}\]

where \(T_i\) and \(T_f\) correspond to the temperatures at A and B respectively.

1. If the work is done by the gas in expanding, \(W\) is positive and \(\Delta U\) is negative; heat is given out and cooling takes place.
2. If the work is done on the gas by compressing it, \(W\) is negative and \(\Delta U\) is positive; heat is produced, which raises the temperature.

(3) Isochoric process: The volume of the system remains constant, such that heating takes place if the gas is compressed from C (\(P_2\)) to D (\(P_1\)) at constant volume, and cooling takes place if the pressure of the gas is reduced from A (\(P_1\)) to B (\(P_2\)) at constant volume. As \(\Delta V=0\), \(W=\int P\,dV=0\), so

\[Q=\Delta U+W=\Delta U=nC_v\Delta T=\frac{nR\,\Delta T}{\gamma-1}\]

(4) Isobaric process: The transformation of the thermodynamic system takes place at constant pressure, such that the gas expands from A (\(V_i\)) to B (\(V_f\)), or the volume of the gas is reduced from C (\(V_f\)) to D (\(V_i\)), at constant pressure.

\[W_{AB}=P(V_f-V_i),\qquad Q=\Delta U+W\]

If the temperature changes by \(\Delta T\), \(Q=nC_v\Delta T+P(V_f-V_i)\), i.e. \(Q=nC_p\Delta T=nC_p(T_f-T_i)\).

(5) Cyclic process: In a P-V diagram, the initial state A goes to state B via path I and returns to the initial state A via path II. The change in internal energy, \(\Delta U\), is zero in a complete cycle. The work done in the clockwise direction from A to B along path I is positive and is equal to the area \(\int_{V_i}^{V_f}P\,dV\) under curve I. The work done in the anticlockwise direction from B to A along path II is negative and is equal to the area \(\int_{V_i}^{V_f}P\,dV\) under path II. Therefore

\[Q=W=\text{area of the shaded portion ACBDA}\]

Second Law of Thermodynamics

It states that it is impossible to construct an engine, operating in a cycle, which will extract heat from a source and convert the whole of it into work. It implies that it is impossible to make heat flow from a colder body to a hotter body without the help of an external agency or an engine from outside.

A heat engine is a device which converts input heat into work. It absorbs a quantity of heat \(Q_1\) from a source, performs an amount of work \(W\), and returns to the initial state after rejecting some heat \(Q_2\) to a sink. The working substance, which is a gas or a liquid, undergoes a cyclic thermodynamic process. The source is at a higher temperature than the sink. The efficiency \(\eta\) of a heat engine is

\[\eta=\frac{W}{Q_1}=\frac{Q_1-Q_2}{Q_1}=1-\frac{Q_2}{Q_1}\]

The Carnot cycle is the most efficient heat engine and undergoes a cycle of two isothermal and two adiabatic processes. It absorbs heat \(Q_1\) by expanding isothermally at \(T_1\) along AB and then expands adiabatically along BC to temperature \(T_2\). It is compressed isothermally at \(T_2\), rejecting heat \(Q_2\) along CD, and is compressed adiabatically along DA. Its efficiency is

\[\eta=1-\frac{T_2}{T_1}\]

Propagation of Heat

Heat can be transferred from one place to another by the three processes of conduction, convection and radiation.

In conduction, heat flows from a place of higher temperature to a place of lower temperature with the medium remaining stationary. The molecules of the body oscillate about their equilibrium positions more violently at a place of higher temperature and collide with the molecules of the adjacent portion, thus transferring a part of their energy to these molecules, which now vibrate more violently. Thus heat can be transmitted by collision of molecules. In metals, the conduction of heat takes place by free electrons. In liquids and gases, heat is transferred not only by collisions but also by the motion of heated molecules, which carry the heat in such media.

When a liquid in a vessel is heated, the lighter molecules in the lower layer get heated and rise to the surface, and the cold molecules at the surface go towards the bottom of the vessel. Such a movement of molecules due to heating produces convection currents, due to which heat is propagated through a liquid or a gas. This is convection.

Radiation is a mode of transfer of heat in which the heat travels directly from one place to another without any intervening medium, or in vacuum. The heat from the sun propagates mostly through vacuum to reach the earth by radiation.

Conduction

Heat flows in unequally heated bodies like a rod or other solids from the place of higher temperature to the place of lower temperature due to conduction. After some time, the temperature of each section becomes constant; this is called the steady state, in which the temperature of a section remains independent of time because the heat gained by that section is conducted on to the next section.

The quantity of heat \(Q\) conducted in time \(t\) across a slab of length \(l\), area of cross-section \(A\), with steady-state temperatures \(\theta_1\) and \(\theta_2\) at the hot and cold ends respectively, is

\[Q=\frac{kA(\theta_1-\theta_2)t}{l}\]

where \(k\) is the coefficient of thermal conductivity, 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. Units of \(k\): kilocalorie/metre·second·°C or J m⁻¹ s⁻¹ K⁻¹. In CGS units, \(k\) is expressed in cal cm⁻¹ (°C)⁻¹.

The temperature gradient per unit distance is \(-\dfrac{d\theta}{dx}\), so \(Q=-kA\left(\dfrac{d\theta}{dx}\right)t\).

The rate of flow of heat \(\dfrac Qt\) for a compound bar made of a bar of length \(l_1\), area of cross-section \(A_1\) and thermal conductivity \(k_1\), and another bar of length \(l_2\), area of cross-section \(A_2\) and thermal conductivity \(k_2\), with temperatures \(\theta_1\) and \(\theta_2\) at the ends, is given in the steady state by

\[\frac Qt=\frac{\theta_1-\theta_2}{\dfrac{l_1}{k_1A_1}+\dfrac{l_2}{k_2A_2}}\]

\(\dfrac{l}{kA}\) is also called the thermal resistance.

Radial Flow of Heat Through a Cylindrical Tube

Consider a cylindrical tube of length \(l\) and inner and outer radii \(r_1\) and \(r_2\). If heat flows radially, i.e. perpendicular to the axis of the tube, from the steady-state temperature \(\theta_1\) at the inner surface to the temperature \(\theta_2\) at the outer surface, then the rate of heat flowing through an element of shell lying between radius \(r\) and \(r+dr\) is

\[\Delta Q=-k(2\pi rl)\frac{d\theta}{dr}\]

where \(d\theta\) is the temperature difference across the shell. It can be integrated for the total heat flow per second:

\[Q\int_{r_1}^{r_2}\frac{dr}{r}=2\pi kl(\theta_1-\theta_2)\ \Rightarrow\ Q=\frac{2\pi kl(\theta_1-\theta_2)}{\ln\left(\dfrac{r_2}{r_1}\right)}\]

Similarly, ice can be formed on a lake due to solidification of water when the temperature above the lake is negative and much below the temperature of solidification of water. If the temperature of water at the bottom of a layer of ice in the lake is \(\theta_1\)°C, the thickness of ice is \(x_1\), the temperature of the environment is \(\theta_2\), \(\rho\) is the density of ice and \(L\) the latent heat of fusion of ice, then the time \(t\) taken to solidify ice up to a thickness \(x_2\) is

\[t=\frac{\rho L\left(x_2^2-x_1^2\right)}{2k(\theta_1-\theta_2)}\]

where \(k\) is the coefficient of thermal conductivity of ice.

Convection

The heat from a liquid can be lost either due to natural convection in the atmosphere or by forced convection, when a steady stream of air is passed over a hot solid body.

Newton’s Law of Cooling

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 \(\theta_1\) is placed in surroundings at a lower temperature \(\theta_2\), the rate of cooling is \(\dfrac{dQ}{dt}\propto(\theta_1-\theta_2)\), where \(dQ\) is the quantity of heat lost in time \(dt\). Newton’s law of cooling gives

\[\frac{dQ}{dt}=-k(\theta_1-\theta_2)\]

where \(k\) is a constant. If a body of mass \(m\) and specific heat \(s\) loses a temperature \(d\theta\) in time \(dt\), then

\[\frac{dQ}{dt}=ms\frac{d\theta}{dt}=-k(\theta_1-\theta_2)\]

Radiation

Thermal radiations are electromagnetic waves which are invisible. They travel with the velocity of light in a straight line and can undergo reflection, refraction, interference, diffraction and polarization. A perfectly black body can be considered as an ideal body which emits as well as absorbs all radiations.

Emissive power: The emissive power of the surface of a body at a certain temperature and for a particular wavelength is defined as the radiant energy emitted per second per unit area of the body per unit wavelength range. It can be compared to that of a black body under similar conditions, where a perfectly black body has emissive power equal to unity.

Absorptive power: It is the rate of absorption of thermal radiation per unit area of the surface as compared to a black body under the same conditions.

All bodies emit radiations at all temperatures, and the emission increases with temperature. There is continuous exchange of radiation between a body and its surroundings. The temperature of a body rises or falls as a result of this exchange of radiations.

Kirchhoff’s Law

It states that the ratio of the emissive power to the absorptive power for radiation of a given wavelength is equal for all substances at a given temperature, and is equal to the emissive power of a perfectly black body at the same temperature. Thus good emitters are good absorbers and bad emitters are bad absorbers.

Wien’s Displacement Law

If a curve is drawn for the emissive power \(E_\lambda\) versus wavelength \(\lambda\) at a certain temperature \(T\), the spectral distribution of radiation energy versus wavelength is as shown in the figure, with the maximum value of \(E_\lambda\) at wavelength \(\lambda_m\). Wien’s displacement law states that for a black body, the wavelength \(\lambda_m\) for which the emissive power \(E_\lambda\) has its maximum value is inversely proportional to its absolute temperature. If \(E_\lambda\) is maximum for \(\lambda_m\prime\) at temperature \(T\prime\), the curve is displaced as shown, and \(\lambda_mT=\lambda_m\prime T\prime\).

\[\lambda_mT=\text{constant}=2.93\times10^{-3}\ \text{m K}\]

Stefan’s Law

It states that the total amount of radiant energy emitted per second from unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature. Thus the rate of loss of heat radiation per unit area, \(E\), is

\[E=\sigma T^4\]

where \(\sigma\) is Stefan’s constant, \(\sigma=5.672\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}\). If a black body at temperature \(T_1\) K is surrounded by another black body at \(T_2\) K, the net loss of energy per second per unit area is

\[E=\sigma\left(T_1^4-T_2^4\right)\]

If the body is not a perfectly black body and the emissivity of the radiating surface is \(e\), then \(E=e\sigma\left(T_1^4-T_2^4\right)\). If \(A\) is the surface area of the body, the rate of energy radiated by the body is

\[E=eA\sigma\left(T_1^4-T_2^4\right)\]

Solar Constant and Temperature of the Sun

The solar constant is defined as the amount of radiation received from the sun at the earth per minute per cm² of a surface placed at right angles to the solar radiation, at the mean distance of the earth from the sun. Assuming that the absorption of solar radiation by the atmosphere near the earth is negligible, the value of the solar constant \(S\) is 1.94 cal cm⁻² min⁻¹. The temperature of the sun, \(T\), is given by

\[T^4=\frac S\sigma\left(\frac Rr\right)^2\]

where \(S\) is the solar constant, \(\sigma\) is Stefan’s constant, \(R\) is the mean distance of the earth from the sun and \(r\) is the radius of the sun.