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Equipartition Theorem

Select the in...

Question

Select the incorrect relation. (Where symbols have their usual meanings)

A

C_{p} = \frac{γ R}{γ - 1}

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B

C_{p} - C_{v} = R

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C

Δ U = \frac{P_{f} V_{f} - P_{i} V_{i}}{1 - γ}

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D

C_{v} = \frac{R}{γ - 1}

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Solution

The correct option is C $Δ U = \frac{P_{f} V_{f} - P_{i} V_{i}}{1 - γ}$
Solution:- (C) $Δ U = \frac{P_{f} V_{f} - P_{i} V_{i}}{1 - γ}$
As we know that,
$Δ U = n C_{v} Δ T$
Whereas,
$n =$ No. of moles
$C_{v} =$ Heat capacity at constant volume $= \frac{R}{γ - 1}$
$Δ T =$ Change in temperature $= T_{f} - T_{i}$
$∴ Δ U = n \frac{R}{γ - 1} (T_{f} - T_{i})$
$\Rightarrow Δ U = \frac{n R T_{f} - n R T_{i}}{γ - 1} . . . . . (1)$
Now from ideal gas equation,
$P V = n R T$
For the same no. of moles of a gas,
$n R T_{f} = P_{f} V_{f}$
$n R T_{i} = P_{i} V_{i}$
Substituting these values in ${e q}^{n} (1)$ , we have
$Δ U = \frac{P_{f} V_{f} - P_{i} V_{i}}{γ - 1}$
Hence the given option $(C) Δ U = \frac{P_{f} V_{f} - P_{i} V_{i}}{1 - γ}$ is incorrect.

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