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Jet aircrafts fly at altitudes above $30,000 \mathrm{ft}$ where the air is very cold at $-40^{\circ} \mathrm{C}$ and the pressure is $0.28 \mathrm{~atm}$. The cabin is maintained at 1 atm pressure by means of a compressor which exchanges air from outside adiabatically. In order to have a comfortable cabin temperature of $25^{\circ} \mathrm{C}$, we will require in addition -
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an air-conditioner to cool the air injected into the cabin
$\mathrm{PV}^{\gamma}=\mathrm{C}$
$\begin{array}{l}
\mathrm{P}^{1-\gamma} \mathrm{T}^{\gamma}=\mathrm{C} \\
(0.28)^{1-\gamma} \times(233)^{\gamma}=1^{1-\gamma} \times \mathrm{T}^{\gamma} \\
\gamma=\frac{7}{5} \\
(0.28)^{1-7 / 5} \times(233)^{7 / 5}=1^{1-7 / 5} \times \mathrm{T}^{7 / 5} \\
\mathrm{~T}^{7 / 5}=233^{7 / 5} \times(0.28)^{-2 / 5} \\
\mathrm{~T}=233(0.28)^{-2 / 7} \\
\mathrm{~T}=\frac{233}{(0.28)^{2 / 7}}
\end{array}$
$\mathrm{T}$ is coming
more than $298 \mathrm{~K}$ or $25^{\circ} \mathrm{C}$
$\therefore \mathrm{T}$ is more than $25^{\circ} \mathrm{C}$
so to cool it an extra ac is required.
$\begin{array}{l}
\mathrm{P}^{1-\gamma} \mathrm{T}^{\gamma}=\mathrm{C} \\
(0.28)^{1-\gamma} \times(233)^{\gamma}=1^{1-\gamma} \times \mathrm{T}^{\gamma} \\
\gamma=\frac{7}{5} \\
(0.28)^{1-7 / 5} \times(233)^{7 / 5}=1^{1-7 / 5} \times \mathrm{T}^{7 / 5} \\
\mathrm{~T}^{7 / 5}=233^{7 / 5} \times(0.28)^{-2 / 5} \\
\mathrm{~T}=233(0.28)^{-2 / 7} \\
\mathrm{~T}=\frac{233}{(0.28)^{2 / 7}}
\end{array}$
$\mathrm{T}$ is coming
more than $298 \mathrm{~K}$ or $25^{\circ} \mathrm{C}$
$\therefore \mathrm{T}$ is more than $25^{\circ} \mathrm{C}$
so to cool it an extra ac is required.
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