Question

An ideal gas undergoes a polytropic given by equation $P V^{n}=$ constant. If molar heat capacity of gas during this process is arithmetic mean of its molar heat capacity at constant pressure and constant volume then value of $n$ is

• A. Zero
• B. $-1$
• C. $+1$
• D. $\gamma$

Answer 1

Answer: (B)

Polytropic process
$P V^{n}=$ constant
Given heat capacities is average of $C_{p}$ and $C_{V}$ So
$C=\frac{C_{P}+C_{V}}{2}$
or $C=\frac{2 C_{V}+R}{2}$
or $C=\frac{C_{V}+R}{2}$ ..... (i)
Now formula for specific heat of polytropic process is given by
$C=\frac{R}{y-1}+\frac{R}{1-n}$ ..... (ii)
or $\frac{R}{y-1}+\frac{R}{2}=\frac{R}{y-1}+\frac{R}{1-n}$ as $C_{V}=\frac{R}{y-1}$
$\frac{R}{2}=\frac{R}{1-n}$
or $n=-1$