1. For an ideal gas in an adiabatic process,

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

where

$\gamma=\frac{C_p}{C_V}.$

2. Using the ideal gas law,

$PV=nRT \implies V=\frac{nRT}{P}.$

Substitute this into the adiabatic relation:

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

3. Simplify:

$P\cdot (nR)^{\gamma} T^{\gamma} P^{-\gamma}=\text{constant}$

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

Ignoring constants,

$T^{\gamma}P^{1-\gamma}=\text{constant}.$

So,

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

Hence,

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

4. Given in the question:

$P\propto T^3.$

Therefore,

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

5. Solve for $\gamma$:

$\gamma=3(\gamma-1)$

$\gamma=3\gamma-3$

$2\gamma=3$

$\gamma=\frac{3}{2}.$

6. Since

$\gamma=\frac{C_p}{C_V},$

we get

$\frac{C_p}{C_V}=\frac{3}{2}.$

7. Option check:

• A: $\frac{4}{3}$ — incorrect
• B: $2$ — incorrect
• C: $\frac{5}{3}$ — incorrect
• D: $\frac{3}{2}$ — correct

Therefore, the correct answer is Option D.