Medium
Obtain the value of $\frac{C_{P}}{C_{V}}$ for a polyatomic gas.
(D) polyatomic gas has $3$ translational,$3$ rotational,and $f$ vibrational degrees of freedom. Thus,the total degree of freedom is $f_{total} = (3 + 3 + f) = (6 + f)$.
The internal energy $U$ is given by the law of equipartition of energy:
$U = \frac{f_{total}}{2} RT = \frac{6+f}{2} RT$.
The molar specific heat at constant volume is $C_{V} = \frac{dU}{dT} = \frac{6+f}{2} R$.
Using Mayer's relation,$C_{P} = C_{V} + R = \left(\frac{6+f}{2} + 1\right) R = \left(\frac{8+f}{2}\right) R$.
Therefore,the ratio $\gamma = \frac{C_{P}}{C_{V}} = \frac{\frac{8+f}{2} R}{\frac{6+f}{2} R} = \frac{8+f}{6+f}$.
The internal energy $U$ is given by the law of equipartition of energy:
$U = \frac{f_{total}}{2} RT = \frac{6+f}{2} RT$.
The molar specific heat at constant volume is $C_{V} = \frac{dU}{dT} = \frac{6+f}{2} R$.
Using Mayer's relation,$C_{P} = C_{V} + R = \left(\frac{6+f}{2} + 1\right) R = \left(\frac{8+f}{2}\right) R$.
Therefore,the ratio $\gamma = \frac{C_{P}}{C_{V}} = \frac{\frac{8+f}{2} R}{\frac{6+f}{2} R} = \frac{8+f}{6+f}$.
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Similar Questions
Which one of the following is $NOT$ a correct expression for an ideal gas?
[$C_{P}=$ Molar specific heat of a gas at constant pressure,
$C_{V}=$ Molar specific heat of a gas at constant volume,
$\gamma=$ Ratio of two specific heats of a gas,$R=$ Universal gas constant]
[$C_{P}=$ Molar specific heat of a gas at constant pressure,
$C_{V}=$ Molar specific heat of a gas at constant volume,
$\gamma=$ Ratio of two specific heats of a gas,$R=$ Universal gas constant]
$C_{v}$ and $C_{p}$ denote the molar specific heat capacities of a gas at constant volume and constant pressure,respectively. Then
$(A)$ $C_{p}-C_{v}$ is larger for a diatomic ideal gas than for a monoatomic ideal gas
$(B)$ $C_{p}+C_{v}$ is larger for a diatomic ideal gas than for a monoatomic ideal gas
$(C)$ $C_{p} / C_{v}$ is larger for a diatomic ideal gas than for a monoatomic ideal gas
$(D)$ $C_{p} \cdot C_v$ is larger for a diatomic ideal gas than for a monoatomic ideal gas
$(A)$ $C_{p}-C_{v}$ is larger for a diatomic ideal gas than for a monoatomic ideal gas
$(B)$ $C_{p}+C_{v}$ is larger for a diatomic ideal gas than for a monoatomic ideal gas
$(C)$ $C_{p} / C_{v}$ is larger for a diatomic ideal gas than for a monoatomic ideal gas
$(D)$ $C_{p} \cdot C_v$ is larger for a diatomic ideal gas than for a monoatomic ideal gas
DifficultIIT 2009
View SolutionAn ideal diatomic gas is heated at constant pressure. What is the fraction of total energy applied,which increases the internal energy of the gas?
The specific heat of an ideal gas is
Easy
View SolutionOne mole of an ideal monoatomic gas is heated at a constant pressure from $0^{\circ} C$ to $100^{\circ} C$. The change in the internal energy of the gas is (Given, $R = 8.32 \text{ J mol}^{-1} \text{ K}^{-1}$):
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