During an adiabatic process, the pressure $P$ of a gas is found to be proportional to the cube of its absolute temperature $T$, i.e., $P\propto T^3$. What is the adiabatic exponent $\gamma$ of the gas?

Solution

1. For an adiabatic process, the relationship between pressure $P$ and temperature $T$ is given by $P^{1-\gamma }{T}^{\gamma }=\text{constant}$, which can be rewritten as $P\propto T^{\frac{\gamma }{\gamma -1}}$.
2. The problem states that $P\propto T^3$, so we equate the exponents: $\frac{\gamma }{\gamma -1}=3$.
3. Solving for $\gamma$, we get $\gamma =3(\gamma -1)$, which simplifies to $\gamma =3\gamma -3$.
4. Rearranging the terms gives $2\gamma =3$, resulting in $\gamma =1.5$.
5. Hence the answer is (C).