A given mass of a gas at $0^{o} C$ is compressed reversibly and adiabatically to a pressure $20$ times the initial value. Calculate the final temperature of the gas. $[\frac{C_{P}}{C_{V}} = 1.42]$

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Solution

Adiabatic process $⟹ P^{1 - γ} T^{γ} = c o n s t a n t$
$V = \frac{C_{P}}{C_{V}} = 1.42,$ given....
$T_{1} = 0 + 273.15 = 273.15 K$
Let the initial pressure, $P_{1}^{1 - γ} T_{1}^{γ} = P_{2}^{1 - γ} T_{2}^{γ}$
$\frac{P^{- 0.42}}{{(20 P)}^{- 0.42}} = \frac{{(T_{2})}^{γ}}{{(273.15)}^{1.42}}$
$\frac{1}{{(20)}^{- 0.42}} = \frac{{(T_{2})}^{γ}}{2881.972} {T_{2}}^{1.42} = 10141.96971$
$∴$ Final temperature of the gas $= 662.54 K .$

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A given mass of a gas at 0oC is compressed reversibly and adiabatically to a pressure 20 times the initial value. Calculate the final temperature of the gas. [CPCV=1.42]

Adiabatic process ⟹P1−γTγ=constantV=CPCV=1.42, given....T1=0+273.15=273.15KLet the initial pressure, P1−γ1Tγ1=P1−γ2Tγ2P−0.42(20P)−0.42=(T2)γ(273.15)1.421(20)−0.42=(T2)γ2881.972T21.42=10141.96971∴ Final temperature of the gas =662.54K.

A given mass of a gas at $0^{o} C$ is compressed reversibly and adiabatically to a pressure $20$ times the initial value. Calculate the final temperature of the gas. $[\frac{C_{P}}{C_{V}} = 1.42]$