One mole of an ideal gas with $\frac{C_{p}}{C_{v}} = γ$ at absolute temperature $T_{1}$ is adiabatically compressed from an initial pressure
$P_{1}$ to a final pressure $P_{2}$ . The resulting temperature $T_{2}$ of the gas is given by

$T_{2} = T_{1} {(\frac{P_{2}}{P_{1}})}^{\frac{γ}{γ - 1}}$

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$T_{2} = T_{1} {(\frac{P_{2}}{P_{1}})}^{\frac{γ - 1}{γ}}$

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$T_{2} = T_{1} {(\frac{P_{2}}{P_{1}})}^{γ}$

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$T_{2} = T_{1} {(\frac{P_{2}}{P_{1}})}^{γ - 1}$

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Solution

The correct option is B
$T_{2} = T_{1} {(\frac{P_{2}}{P_{1}})}^{\frac{γ - 1}{γ}}$

$F o r a n a d i a b a t i c p r o c e s s, P V^{γ} = c o n s t a n t . T h e i d e a l g a s e q u a t i o n i s P V = R T o r V = \frac{R T}{P} . T h e r e f o r e, w e h a v e P {(\frac{R T}{P})}^{γ} = c o n s t a n t o r \frac{T^{γ}}{P^{γ - 1}} = c o n s t a n t (∵ R i s a c o n s t a n t) ∴ \frac{T_{2}^{γ}}{P_{2}^{γ - 1}} = \frac{T_{1}^{γ}}{P_{1}^{γ - 1}} o r T_{2} = T_{1} {(\frac{P_{2}}{P_{1}})}^{\frac{γ - 1}{γ}}$

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One mole of an ideal gas with CpCv=γ at absolute temperature T1 is adiabatically compressed from an initial pressure P1 to a final pressure P2. The resulting temperature T2 of the gas is given by

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One mole of an ideal gas with $\frac{C_{p}}{C_{v}} = γ$ at absolute temperature $T_{1}$ is adiabatically compressed from an initial pressure
$P_{1}$ to a final pressure $P_{2}$ . The resulting temperature $T_{2}$ of the gas is given by