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Question: A polytropic process for an ideal gas is represented by the equation \(P{V^n}\) constant. If gamma i...

A polytropic process for an ideal gas is represented by the equation PVnP{V^n} constant. If gamma is the ratio of specific heat CP/CV{{\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. γ  > n\gamma \; > \,n
B. γ  > n >1\gamma \; > \,n\, > 1
C.  n > γ\,n\, > \,\gamma
D. none as it is not possible

Explanation

Solution

The expression for the molar specific heat in a polytropic process is given by
C  =  Rγ  −1  =Rn−1{\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 PVn  =  {\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
C  =  Rγ  −1  −Rn−1{\rm{C}}\;{\rm{ = }}\;\dfrac{{\rm{R}}}{{\gamma \; - 1}}\; - \dfrac{{\rm{R}}}{{n - 1}}
Here R is the universal gas constant and γ\gamma is the ratio of specific heat of gases.
Upon simplifying the above relation we get,
C=(n−1)R− (γ−1)R(γ−1)(n−1)  =nR−R−γR  + R(γ−1)(n−1){\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)}}
⇒  (n−γ)R(γ−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−γ),(n−1),(γ−1)(n - \gamma ),(n - 1),(\gamma - 1)
We have two relations,
γ  =CPCV  \gamma \; = \dfrac{{{{\rm{C}}_{\rm{P}}}}}{{{{\rm{C}}_{\rm{V}}}}}\; And CP  −CV  = R{{\rm{C}}_P}\; - {{\rm{C}}_{\rm{V}}}\; = \,R
These expressions suggest that,
CP>CV  {{\rm{C}}_P}{\rm{ > }}{{\rm{C}}_V}\;and hence γ  >1\gamma \; > 1
So, γ  −1\gamma \; - 1 it will always be positive.
We can conclude that for C to be negative,(n−γ)(n - \gamma ) will be negative. That is
n−γ<0n - \gamma < 0
⇒n  <γ\Rightarrow n\; < \gamma ………… (1)
And (n-1) will be positive i.e.
n  −1>  0n\; - 1 > \;0
n  >  1n\; > \;1 …….. (2)
From I and 2 equations we conclude that,
γ  > n >1\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.