Question

Which of the following is the incorrect statement about the molar heat capacity of an ideal gas system?

• A. Molar heat capacity at no exchange condition, $C_A=0$
• B. Molar heat capacity at constant temperature, $C_T\to \infty$
• C. Molar heat capacity at constant pressure, $C_p=\frac{\gamma R}{\gamma -1}$
• D. Molar heat capacity at constant volume, $C_v=\frac{R}{\gamma }$

Answer 1

The correct option is D Molar heat capacity at constant volume, $C_v=\frac{R}{\gamma }$

From the definition of specific heat capacity
$C=\frac{\Delta Q}{n\Delta T}......\left(1\right)$
where $n$ is the no. of moles.
Condition for no heat exchange is $\Delta Q=0$
So, from $\left(1\right)$, we can say that $\Delta Q=0\Rightarrow C=0$
At a constant temperature, $\Delta T=0$
So, at constant temperature, we can say that $C\to \infty$
At constant pressure, we know that
$C_P=\frac{\gamma R}{\gamma -1}$
and at constant volume, $C_V=\frac{R}{\gamma -1}$
Thus, the relation given in option (d) is incorrect.