# During an adiabatic process , the pressure of a gas is formed to be proportional to the cube of its absolute temperature the value of r for the gas is

$(a)\;\large\frac{5}{3}\qquad(b)\;\large\frac{7}{5}\qquad(c)\;\large\frac{3}{2}\qquad(d)\;\large\frac{11}{9}$

$P \propto T^3$
$\implies P= KT^3$
For adiabatic process \begin{align*}\frac{T^x}{P^{x-1}} = C \end{align*}
where $C$ is a constant.
\implies \begin{align*} \frac{T^x}{(KT^3)^{x-1}} = C \end{align*}
\implies \begin{align*} \frac{T^x}{(T^3)^{x-1}} = k^{x-1}. C\end{align*}
\implies \begin{align*} \frac{T^x}{T^{3x-3}} = k^{x-1}. C\end{align*}
Let $K^{x-1}.C=A$
$\implies T^{3-2x } = A$
$\implies 3 - 2x = 0$
or $x = \frac{3}{2}$
answered Mar 21, 2014 by
edited Dec 17, 2017