Class 11PhysicsKinetic Theory of GasesMolar Specific Heat of gas and relation between them (Mayer's formula)
MediumObtain 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 $\gamma = \frac{C_{P}}{C_{V}} = \frac{\frac{8+f}{2} R}{\frac{6+f}{2} R} = \frac{8+f}{6+f}$.
Explore More
Similar Questions
The figure shows the variation in heat supplied $(Q)$ with the change in temperature $(\Delta T)$ in an isobaric process for a monoatomic $(M)$,diatomic $(D)$,and a polyatomic $(P)$ gas. The initial state of all the gases is the same,and the scales for the two axes coincide. Ignoring vibrational degrees of freedom,the lines $a, b,$ and $c$ respectively correspond to
$40 \, \text{calories}$ of heat is needed to raise the temperature of $1 \, \text{mole}$ of an ideal monoatomic gas from $20^{\circ}C$ to $30^{\circ}C$ at a constant pressure. The amount of heat required to raise its temperature over the same interval at a constant volume $(R = 2 \, \text{cal} \, \text{mol}^{-1} \text{K}^{-1})$ is ..... $\text{calories}$.
Medium
View SolutionWhen some amount of heat energy is supplied to a monatomic gas, the percentage of heat energy used for increasing the internal energy of the gas $(\gamma = 5/3)$ is
The amount of heat energy required to raise the temperature of $1 \ g$ of helium from $T_1 \ K$ to $T_2 \ K$ at $NTP$ is ........
Difficult
View SolutionThe molar specific heats of an ideal gas at constant pressure and constant volume are denoted by $C_P$ and $C_V$ respectively. If $\gamma = C_P/C_V$ and $R$ is the universal gas constant,then $C_V = $ ........
Medium
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo