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

The correct option is B $-1$

It is given that $P{V}^n=constant$

And $C=\frac{C_p+C_v}{2}........\left(1\right)$

The expression for specific heat capacity is

$C=\frac{R}{\gamma -1}+\frac{R}{1-n}..........\left(2\right),\gamma =adiabaticindex$

$\because \gamma =\frac{C_P}{C_V}\&R=C_P-C_V$

So, equation (2) becomes

$\frac{C_p+C_v}{2}=\frac{C_P-C_V}{\frac{C_p}{C_V}-1}+\frac{C_P-C_V}{1-n}$

$\frac{C_P-C_V-2C_V}{2(C_P-C_V)}=\frac{1}{1-n}$

$\frac{1}{2}=\frac{1}{1-n}$

$n=-1$