Question

A polytropic process for an ideal gas is represented by the equation $P{V^n}$ constant. If gamma is the ratio of specific heat ${{\rm{C}}_P}/{{\rm{C}}_V}$. Then the value of n for which molar heat capacity of the process is negative is given as
A. $\gamma \; > \,n$
B. $\gamma \; > \,n\, > 1$
C. $\,n\, > \,\gamma $
D. none as it is not possible

Answer 1

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

Complete step by step answer:
A polytropic process is mathematically expressed as ${\rm{P}}{{\rm{V}}^n}\; = \;$constant.
Where P is the pressure, V is the volume and 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., is C negative.
 Molar specific heat is given as
${\rm{C}}\;{\rm{ = }}\;\dfrac{{\rm{R}}}{{\gamma \; - 1}}\; - \dfrac{{\rm{R}}}{{n - 1}}$
Here R is the universal gas constant and $\gamma $ is the ratio of specific heat of gases.
Upon simplifying the above relation we get,
${\rm{C = }}\dfrac{{(n - 1){\rm{R - }}\,{\rm{(}}\gamma {\rm{ - 1)R}}}}{{{\rm{(}}\gamma {\rm{ - 1)}}(n - 1){\rm{ }}}}\; = \dfrac{{{\rm{nR - R - }}\gamma {\rm{R}}\;{\rm{ + }}\,{\rm{R}}}}{{{\rm{(}}\gamma {\rm{ - 1)}}(n - 1)}}$
$ \Rightarrow \;\dfrac{{({\rm{n - }}\gamma {\rm{) R}}}}{{(\gamma - 1)\;(n - 1)}}$
Now we will focus on three terms of the expression obtained and analyses their range,
$(n - \gamma ),(n - 1),(\gamma - 1)$
We have two relations,
$\gamma \; = \dfrac{{{{\rm{C}}_{\rm{P}}}}}{{{{\rm{C}}_{\rm{V}}}}}\;$ And ${{\rm{C}}_P}\; - {{\rm{C}}_{\rm{V}}}\; = \,R$
These expressions suggest that,
${{\rm{C}}_P}{\rm{ > }}{{\rm{C}}_V}\;$and hence $\gamma \; > 1$
So, $\gamma \; - 1$ it will always be positive.
We can conclude that for C to be negative,$(n - \gamma )$ will be negative. That is
$n - \gamma < 0$
$ \Rightarrow n\; < \gamma $ ………… (1)
And (n-1) will be positive i.e.
$n\; - 1 > \;0$
$n\; > \;1$ …….. (2)
From I and 2 equations we conclude that,
$\gamma \; > \,n\, > 1$

Therefore (B) option 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 factorize the expression and then one by one evaluate each factor to get the values. Good care must be taken by solving the inequalities.