Question

A certain ideal gas undergoes a polytropic process $P{V^n} =$constant such that the molar specific heat during the process is negative. If the ratio of the specific heats of the gas be $\gamma$, then the range of values of $n$ will be:
A)$0 < n < \gamma$
B)$1 < n < \gamma$
C)$n = \gamma$
D)$n > \gamma$

Answer 1

Hint The expression for the molar specific heat in a polytropic process is given by
$C = \dfrac{R}{{\gamma - 1}} - \dfrac{R}{{n - 1}}$
We are given the question that $C$ is negative. Hence, we are to find only those values of $n$ for which the entire expression becomes negative.

Complete step-by-step solution :
A polytropic process is mathematically expressed as $P{V^n} =$constant
Where
$P$ is the pressure
$V$is the volume
$n$ is the polytropic index
Value of $n$ ranges from $0$ to infinity but in the situation given above, we have to find only those values of $n$ for which molar specific heat, i.e., $C$ is negative.
Molar specific heat is given as
$C = \dfrac{R}{{\gamma - 1}} - \dfrac{R}{{n - 1}}$
Here , $R$ is universal gas constant and $\gamma$ is the ratio of specific heats of gas.
Upon simplifying the above expression, we get
$C = \dfrac{{\left( {n - 1} \right)R - \left( {\gamma - 1} \right)R}}{{\left( {\gamma - 1} \right)\left( {n - 1} \right)}} = \dfrac{{nR - R - \gamma R + R}}{{\left( {\gamma - 1} \right)\left( {n - 1} \right)}} \\\ = \dfrac{{\left( {n - \gamma } \right)R}}{{\left( {\gamma - 1} \right)\left( {n - 1} \right)}} \\\$
Now we will focus on three terms of the expression obtained and analyse their range, $\left( {n - \gamma } \right),\left( {n - 1} \right),\left( {\gamma - 1} \right)$
We have two relations
$\gamma = \dfrac{{{C_P}}}{{{C_V}}}$ and ${C_P} - {C_V} = R$
These expressions suggest that
${C_P} > {C_V}$ and hence, $\gamma > 1$
So, $\gamma - 1$ will always be positive.
We can conclude that for $C$ to be negative, $n - \gamma$ will be negative. That is
$n - \gamma < 0$
$n < \gamma$ (1)
And $n - 1$ will be positive. i.e.,
$n - 1 > 0$
$n > 1$ (2)
From (1) and (2), we get
$1 < n < \gamma$
Hence option B is correct.

Note:- For any question of the type where we are supposed to predict the values of a variable, it is a good practice to factorise the expression and then one by one evaluate each factor to get the values. Good care must be taken by solving the inequalities.