An ideal gas undergoes an adiabatic process obeying the relation $PV^{4/3} =$ constant. If its initial temperature is $300\,\, K$ and then its pressure is increased upto four times its initial value, then the final temperature is (in Kelvin):
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$\mathrm{T}_{\mathrm{i}}=300 \mathrm{K} \quad \mathrm{T}_{\mathrm{f}}=?$
$\mathrm{P}_{\mathrm{i}}=\mathrm{P} \quad \mathrm{P}_{\mathrm{f}}=4 \mathrm{P}$
By gas equation we know: $-V=\frac{n R T}{P}$
$\therefore \mathrm{PV}^{4 / 3}=\mathrm{constant}$
$\Rightarrow P\left(\frac{n R T}{P}\right)^{\frac{4}{3}}=$ constant
$\Rightarrow \frac{T^{\frac{4}{3}}}{p^{\frac{1}{3}}}=$ constant
$\therefore T_{f}=\left(\frac{P_{t}}{P_{i}}\right)^{\frac{4}{3}-\frac{4}{3}} \times T_{i}=\left(\frac{4 P}{P}\right)^{\frac{1}{4}} \times 300 \mathrm{K}$
$\Rightarrow T_{f}=300 \sqrt{2} \mathrm{K}$
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