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Question

The equation of a certain gas can be written as $$\dfrac {T^{7/5}}{P^{2/5}}=constant$$. Its specific heat at constant volume will be :


A
32R
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B
52R
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C
72R
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D
2R
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Solution

The correct option is B $$\dfrac {5}{2}R$$
The equation of a certain gas can be written as

$$\dfrac{T^{7/5}}{P^{2/5}}=const$$

$$P^{-2/5}.T^{7/5}=const$$

$$P.T^{(\dfrac{7/5}{-2/5})}=const$$

$$PT^{-7/2}=const$$. . . . . .(1)

Equation for any polytropic process,

$$PT^{(\gamma/\gamma-1)}=const$$. . . . .(2)

Compare equation (1) and equation (2), we get, 

$$\dfrac{\gamma}{\gamma-1}=-\dfrac{7}{2}$$

$$2\gamma=-7\gamma +7$$

$$5\gamma=7$$

$$\gamma=\dfrac{7}{5}$$

The specific heat at constant volume will be given by

$$C_V=\dfrac{R}{\gamma-1}$$

$$C_V=\dfrac{5R}{2}$$

The correct option is B.

Physics

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