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Heat Capacity

One mole of a...

Question

One mole of a mono-atomic ideal gas undergoes a quasi-static process, which is depicted by a straight line joining points $(V_{0}, T_{0})$ and $(2 V_{0}, 3 T_{0})$ in a V-T diagram. What is the value of the heat capacity of the gas at the point $(V_{0}, T_{0})$ ?

A

R

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B

\frac{3}{2} R

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C

2 R

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D

0

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Solution

The correct option is C $2 R$
$p d v + n c_{v} d T = n c d T$
$\Rightarrow C = C_{v} + \frac{p}{n} \frac{d v}{d T}$
at $V_{0}, T_{0}$
$\frac{d v}{d T} = \frac{V_{0}}{2 T_{0}}, P_{0} V_{0} = n R T_{0}$
$C = C_{v} + \frac{p_{0}}{n} \frac{d v}{d T}$ $\frac{p_{0}}{n} = \frac{R T_{0}}{v_{0}}$
$C = \frac{3}{2} R + \frac{R T + 0}{v_{0}} \times \frac{v_{0}}{2 T_{0}} = \frac{3}{2} R + \frac{R}{2} = 2 R$ .

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