If a real gas obeys the Clausius equation of state P(v- b) = RT then

{(\frac{d u}{d v})}_{T} \neq 0

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{(\frac{d u}{d v})}_{T} = 0

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{(\frac{d u}{d v})}_{T} = 1

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{(\frac{d u}{d v})}_{T} = \frac{1}{P}

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Solution

The correct option is B ${(\frac{d u}{d v})}_{T} = 0$
For given real gas

$p (v - b) = R T$

$\Rightarrow P = \frac{R T}{v - b}$

$\Rightarrow {(\frac{d P}{d T})}_{v} = \frac{R}{v - b}$

using the first energy equation

${(\frac{d u}{d v})}_{T} = T {(\frac{δ P}{δ T})}_{v} - P$

$= \frac{T R}{v - b} - \frac{R T}{v - b} = 0$

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