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Specific Heat Capacity

Molar heat ca...

Question

Molar heat capacity of an ideal gas whose molar heat capacity at constant is $C_{v}$ for process $P = 2 e^{2 v}$ ( where P is pressure of gas and V is volume of gas)

C_{v} + \frac{R}{1 + 2 V}

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C_{v} + \frac{R}{2 V}

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C_{v} + \frac{R}{V}

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None of these

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Solution

The correct option is A $C_{v} + \frac{R}{1 + 2 V}$
$\begin{matrix} B y u sin g f i r s t l a w o f T h e r m o o d y n a m i c s : - d Q = d w + d U a n d, a l s o d Q = n C d T d w = P d v d U = n C_{v} d T N o w, s u b s t i t u t i n g t h e m i n t h e f i r s t l a w w e g e t \Rightarrow n C d T = P d V + n C_{n} d T C = \frac{P d V}{n d T} + C_{v} T o f i n d \frac{P d V}{n d T} w e w i l l u s e t h e i d e a l g a s e q u a t i o n P V = n R T 2 V e^{2 V} = n R T [R e p l a c e P = 2 e^{2 v}] D i f f e r e n t i a t i n g b o t h s i d e s w i t h r e s p e c t t o T 2 (e^{2 V} + 2 V e^{2 V}) \frac{d V}{d T} = n R N o w, f r o m t h i s w e h a v e \frac{P d V}{n d T} = \frac{R}{1 + 2 v} S o, w e g e t C = C_{v} + \frac{R}{1 + 2 v} H e n c e, t h e o p t i o n A i s t h e c o r r e c t a n s w e r . \end{matrix}$

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