An ideal gas undergoes a quasi-static, reversible process in which...

An ideal gas undergoes a quasi-static, reversible process in which its molar heat capacity C remains constant. If during this process the relation of pressure P and volume V is given by

P{V}^{n} =

constant, then n is given by (Here

{C}_{P}

and

{C}_{V}

are molar specific heat at constant pressure and constant volume, respectively) :

Options

A $n = \dfrac{{C}_{P} - C}{C - {C}_{V}}$
B $n = \dfrac{C - {C}_{V}}{C - {C}_{P}}$
C $n = \dfrac{{C}_{P}}{{C}_{V}}$
D $n = \dfrac{C - {C}_{P}}{C - {C}_{V}}$

Correct answer

D. $n = \dfrac{C - {C}_{P}}{C - {C}_{V}}$

Step-by-step solution

P{V}^{n} = k

C = {C}_{v} + \dfrac{R}{1 - n}

;

C - {C}_{v} = \dfrac{R}{1 - n}

1 - n = \dfrac{R}{C - {C}_{v}}

;

n = 1 - \dfrac{R}{C - {C}_{v}}

n = \dfrac{C - {C}_{v} - R}{C - {C}_{v}}

;

n = \dfrac{C - {C}_{v} - \left({C}_{p} - {C}_{v}\right)}{C - {C}_{v}}

n = \dfrac{C - {C}_{v} - {C}_{p} + {C}_{v}}{C - {C}_{v}}

;

n = \dfrac{C - {C}_{p}}{C - {C}_{v}}

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