n moles of gas is kept in a large tube of length L and cross-sectional area A. The tube is placed in a horizontal plane. If temperature of gas varies linearly from $T_{1}$ to $T_{2} (T_{2} > T_{1})$ . Then find the pressure of gas in tube.

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$p = \frac{n R (T_{1} + T_{2})}{2 A L}$

$p = \frac{n R (T_{2} - T_{1})}{2 A L \ln \frac{T_{2}}{T_{1}}}$

$p = \frac{n R (T_{2} + T_{1})}{A L \ln \frac{T_{2}}{T_{1}}}$

$p = \frac{n R (T_{2} - T_{1})}{A L \ln \frac{T_{2}}{T_{1}}}$

answer is C.

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Detailed Solution

We have,
$T (X) = \frac{T_{2} - T_{1}}{L} x + T_{1}$
We have $P V = n R T$
$\Rightarrow P A d x = n R [T_{1} + \frac{(T_{2} - T_{1})}{L} x]$
$\Rightarrow \frac{1}{P} = \int_{0}^{L} \frac{A d x}{n R [T_{1} + \frac{T_{2} - T_{1}}{L} x]}$
$\Rightarrow \frac{1}{P} = \frac{A L}{n R (T_{2} - T_{1})} {| l n \frac{T_{1} + (T_{2} - T_{1}) x}{L} |}_{0}^{L}$