Question

Relation between constant volume specific heat $\left(C_v\right)$ and degree of freedom $\left(f\right)$ for a gas is ($R$ is gas constant)

• A. $C_v=\frac{f}{2}R$
• B. $C_v=fR$
• C. $C_v=\frac{f}{R}$
• D. $C_v=\frac{R}{f}$

Answer 1

The correct option is A $C_v=\frac{f}{2}R$

The internal energy of 1 mole of an ideal gas at temperature $T$ , having $f$ degrees of freedom is given by ,

$U=N_A\times \left(1/2\right)f{k}_{B}T=\left(1/2\right)fRT$ ......................eq1

where $N_A=$Avogadro's number

$k_B=$ Boltzmann's constant

from first law of thermodynamics ,

$dU=dQ-dW$ ............eq2

At constant volume , $dV=0$

hence $dW=PdV=0$

eq2 becomes , $dU=dQ$

but $dQ=C_{V}dT$ (for 1 mole of gas)

therefore $dU=C_{V}dT$

or $C_V=dU/dT$

putting the value of $U$ in this equation, we get

$C_V=\frac{d\left(1/2\right)fRT}{dT}=\frac{f}{2}R$