1. Use the adiabatic relation

For an ideal gas in an adiabatic process, $PV^\gamma = \text{constant}$ where $\gamma = \frac{C_p}{C_v}.$

Also, from the ideal gas law, $PV = nRT.$ So we can derive the relation between $P$ and $T$ during an adiabatic process.

2. Find the standard relation between $P$ and $T$ in adiabatic change

From $PV^\gamma = \text{constant}$ and $V = \frac{nRT}{P},$ substitute $V$ into the adiabatic equation:

$P\left(\frac{nRT}{P}\right)^\gamma = \text{constant}.$

Simplifying, $P \cdot \frac{(nR)^\gamma T^\gamma}{P^\gamma} = \text{constant}$ $(nR)^\gamma T^\gamma P^{1-\gamma} = \text{constant}.$

Since $(nR)^\gamma$ is constant, $T^\gamma P^{1-\gamma} = \text{constant}.$

Rearranging, $P^{\gamma-1} \propto T^\gamma.$

Hence, $P \propto T^{\frac{\gamma}{\gamma-1}}.$

3. Use the given condition

The question says: $P \propto T^3.$

So, $\frac{\gamma}{\gamma-1} = 3.$

4. Solve for $\gamma$

$\gamma = 3(\gamma-1)$ $\gamma = 3\gamma - 3$ $2\gamma = 3$ $\gamma = \frac{3}{2}.$

Therefore, $\frac{C_p}{C_v} = \frac{3}{2}.$

5. Check options

• A: $\frac{7}{5}$
• B: $\frac{3}{2}$
• C: $\frac{9}{7}$
• D: $\frac{5}{3}$

So the correct option is B.