Question

During an adiabatic process, the pressure of a gas is proportional to the cube of its absolute temperature. If the value of $\frac{C_p}{C_v}$ for that gas is $\frac{m}{n}$, where n is a prime number, then find the value of m+n.

Answer 1

In an adiabatic process,
$P{V}^{\gamma }=constant$ ......(1)
Given,
$P$ $\alpha T^3$
$\Rightarrow P=a{T}^3$ where a is a constant
$\therefore P=a\times (\frac{PV}{nR}{)}^3$
$\Rightarrow P{V}^{\frac{3}{2}}=constant$
Comparing with eqn(1)
we get $\gamma =\frac{3}{2}$
Given, $\gamma =\frac{m}{n}=\frac{3}{2}$ where m and n are prime numbers
$\therefore$ $m=3$ and $n=2$
$\Rightarrow m+n=3+2=5$