During an adiabatic process, if the pressure of a gas is found to be proportional to the cube of its absolute temperature, then the ratio of for the gas is

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Solution

Q.
During an adiabatic process, if the pressure of a gas is found to be proportional to the cube of its absolute temperature, then the ratio of $\frac{C_{P}}{C_{V}}$ for the gas is [2024]

1 $\frac{3}{2}$

2 $\frac{7}{5}$

3 $\frac{5}{3}$

4 $\frac{9}{7}$

Ans.
(1)

$P = T^{3} \Rightarrow P T^{- 3} =$ constant

   $∵$ $\frac{P V}{T} = n R =$ constant from ideal gas equation $\Rightarrow T = \frac{P V}{n R}$

$P {(\frac{P V}{n R})}^{- 3} =$ constant $\Rightarrow P^{- 2} V^{- 3} =$ constant

   $\Rightarrow P V^{3 / 2} =$ constant ... (1)

$∴$ Process equation for adiabatic process is

   $P V^{γ} =$ constant ... (2)

Comparing equation (1) and (2)

$\Rightarrow \frac{C_{P}}{C_{V}} = γ = \frac{3}{2}$

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