# During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio $\dpi{100} C_{P}/C_{V}$ for the gas is Option 1) $4/3\;$ Option 2) $\; 2\;$ Option 3) $\; 5/3\;$ Option 4) $\; 3/2$

As we learnt in

Equation of state -

$dQ= 0$

$n\, C_{V}\, dT+PdV= 0$

- wherein

On solving

$\gamma \frac{dV}{V}+\frac{dP}{P}= 0$

$\Rightarrow PV^{\gamma }= constant$

In the given question

$P \propto T^{3}$

or $P T^{-3}$ = constant ......(1)

$P^{1-\gamma } . T^{\gamma} = constant$.....(2)

From equation 1 and 2

$\frac{\gamma}{1-\gamma }= -3 \\ or 3\gamma -3=\gamma \\ or \ \gamma =3/2$

$\therefore \frac{C_p}{C_v}= \gamma = 3/2$

Option 1)

$4/3\;$

The option is incorrect.

Option 2)

$\; 2\;$

The option is incorrect.

Option 3)

$\; 5/3\;$

The option is incorrect.

Option 4)

$\; 3/2$

The option is correct.

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