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Q.

One mole of an ideal gas (C_p /C_v = $γ$ ) at absolute temperature T₁ is diabatically 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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a

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

b

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

c

$T_{2} = T_{1} {(P_{2} / P_{1})}^{γ}$

d

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

answer is B.

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Detailed Solution

For adiabatic process,
${PV}^{γ} = constant$
For an ideal gas PV = RT or V = (RT/ P)
$∴ P (RT / P)^{γ} = constant$
or $T^{γ} / P^{γ - 1} = constant$ $(∵ R = constant)$
$∴ \frac{T_{2}^{γ}}{P_{2}^{γ - 1}} = \frac{T_{1}^{γ}}{P_{1}^{γ - 1}}$
or $T_{2} = T_{1} {(P_{2} / P_{1})}^{(γ - 1) / γ}$

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One mole of an ideal gas (Cp /Cv = γ) at absolute temperature T1 is diabatically compressed from an initial pressure P1 to a final pressure P2. The resulting temperature T2 of the gas is given by