We have seen in Section 9.2.1 that the average kinetic energy of a molecule moving in x direction is \( \frac{1}{2} m v_{x}^{2} = \frac{1}{2} kT \).
Similarly, when the motion is in y direction, \( \frac{1}{2} m v_{y}^{2} = \frac{1}{2} kT \) and for the motion along z direction, \( \frac{1}{2} m v_{z}^{2} = \frac{1}{2} kT \).
According to kinetic theory, the average kinetic energy of system of molecules in thermal equilibrium at temperature \( T \) is uniformly distributed to all degrees of freedom (x or y or z directions of motion) so that each degree of freedom will get \( \frac{1}{2} kT \) of energy. This is called law of equipartition of energy.
Average kinetic energy of a monatomic molecule (with \( f = 3 \)) \( = 3 \times \frac{1}{2} kT = \frac{3}{2} kT \)
Average kinetic energy of diatomic molecule at low temperature (with \( f = 5 \)) \( = 5 \times \frac{1}{2} kT = \frac{5}{2} kT \)
Average kinetic energy of a diatomic molecule at high temperature (with \( f = 7 \)) \( = 7 \times \frac{1}{2} kT = \frac{7}{2} kT \)
Average kinetic energy of linear triatomic molecule (with \( f = 7 \)) \( = 7 \times \frac{1}{2} kT = \frac{7}{2} kT \)
Average kinetic energy of nonlinear triatomic molecule (with \( f = 6 \)) \( = 6 \times \frac{1}{2} kT = 3kT \)
9.4.1 Application of law of equipartition energy in specific heat of a gas#
Meyer’s relation \( C_{p} - C_{v} = R \) connects the two specific heats for one mole of an ideal gas.
Equipartition law of energy is used to calculate the value of \( C_{p} - C_{v} \) and the ratio between them \( \gamma = \frac{C_{p}}{C_{v}} \). Here \( \gamma \) is called adiabatic exponent.
i) Monatomic molecule
Average kinetic energy of a molecule \( = \left[ \frac{3}{2} kT \right] \)
Total energy of a mole of gas \( = \frac{3}{2} kT \times N_{A} = \frac{3}{2} RT \)
For one mole, the molar specific heat at constant volume \( C_{v} = \frac{dU}{dT} = \frac{d}{dT} \left[ \frac{3}{2} RT \right] \)
\[ C_{v} = \frac{3}{2} R \]\[ C_{p} = C_{v} + R = \frac{3}{2} R + R = \frac{5}{2} R \]The ratio of specific heats,
\[ \gamma = \frac{C_{p}}{C_{v}} = \frac{\frac{5}{2}R}{\frac{3}{2}R} = \frac{5}{3} = 1.67 \]ii) Diatomic molecule
Average kinetic energy of a diatomic molecule at low temperature \( = \frac{5}{2} kT \)
Total energy of one mole of gas \( = \frac{5}{2} kT \times N_{A} = \frac{5}{2} RT \)
(Here, the total energy is purely kinetic)
For one mole Specific heat at constant volume
\[ C_{v} = \frac{dU}{dT} = \frac{5}{2} R \]\[ C_{p} = C_{v} + R = \frac{5}{2} R + R = \frac{7}{2} R \]\[ \gamma = \frac{C_{p}}{C_{v}} = \frac{\frac{7}{2}R}{\frac{5}{2}R} = \frac{7}{5} = 1.40 \]Energy of a diatomic molecule at high temperature is equal to \( \frac{7}{2} RT \)
\[ C_{v} = \frac{dU}{dT} = \frac{7}{2} R \]\[ C_{p} = C_{v} + R = \frac{7}{2} R + R = \frac{9}{2} R \]\[ \gamma = \frac{C_{p}}{C_{v}} = \frac{\frac{9}{2}R}{\frac{7}{2}R} = \frac{9}{7} = 1.28 \]Note that the \( C_{v} \) and \( C_{p} \) are higher for diatomic molecules than the monoatomic molecules. It implies that to increase the temperature of diatomic gas molecules by \( 1^{\circ}C \) it requires more heat energy than monoatomic molecules.
iii) Triatomic molecule
a) Linear molecule
Energy of one mole \( = \frac{7}{2} kT \times N_{A} = \frac{7}{2} RT \)
\[ C_{v} = \frac{dU}{dT} = \frac{7}{2} R \]\[ C_{p} = C_{v} + R = \frac{7}{2} R + R = \frac{9}{2} R \]\[ \gamma = \frac{C_{p}}{C_{v}} = \frac{\frac{9}{2}R}{\frac{7}{2}R} = \frac{9}{7} = 1.28 \]b) Non-linear molecule
Energy of a mole \( = \frac{6}{2} kT \times N_{A} = 3RT \)
\[ C_{v} = \frac{dU}{dT} = 3R \]\[ C_{p} = C_{v} + R = 3R + R = 4R \]\[ \gamma = \frac{C_{p}}{C_{v}} = \frac{4R}{3R} = \frac{4}{3} = 1.33 \]Note that according to kinetic theory model of gases the specific heat capacity at constant volume and constant pressure are independent of temperature. But in reality it is not sure. The specific heat capacity varies with the temperature.
EXAMPLE 9.5
Find the adiabatic exponent \( \gamma \) for mixture of \( \mu_{1} \) moles of monoatomic gas and \( \mu_{2} \) moles of a diatomic gas at normal temperature \( (27^{\circ}C) \).
Solution
The specific heat of one mole of a monoatomic gas \( C_{V} = \frac{3}{2} R \)
For \( \mu_{1} \) mole, \( C_{V} = \frac{3}{2} \mu_{1} R \), \( C_{P} = \frac{5}{2} \mu_{1} R \)
The specific heat of one mole of a diatomic gas \( C_{V} = \frac{5}{2} R \)
For \( \mu_{2} \) mole, \( C_{V} = \frac{5}{2} \mu_{2} R \), \( C_{P} = \frac{7}{2} \mu_{2} R \)
The specific heat of the mixture at constant volume \( C_{V} = \frac{3}{2} \mu_{1} R + \frac{5}{2} \mu_{2} R \)
The specific heat of the mixture at constant pressure \( C_{P} = \frac{5}{2} \mu_{1} R + \frac{7}{2} \mu_{2} R \)
The adiabatic exponent \( \gamma = \frac{C_{P}}{C_{V}} = \frac{5\mu_{1} + 7\mu_{2}}{3\mu_{1} + 5\mu_{2}} \)