1. Initial conditions

Both containers start with the same:

• Pressure $P$
• Volume $V$
• Temperature $T$

Final volume in both cases is: $V_f=\frac{V}{8}$

We need: $\frac{P_B}{P_A}$ where:

• $P_A$ = final pressure for isothermal compression
• $P_B$ = final pressure for adiabatic compression

2. Container A: Isothermal compression

For an isothermal process: $PV=\text{constant}$

So, $P_A V_f = PV$

Substitute $V_f=\frac{V}{8}$: $P_A\left(\frac{V}{8}\right)=PV$ $P_A=8P$

3. Container B: Adiabatic compression

For an adiabatic process: $PV^\gamma=\text{constant}$

For a monoatomic gas: $\gamma=\frac{5}{3}$

Thus, $P_B\left(\frac{V}{8}\right)^\gamma = PV^\gamma$

So, $P_B = P\left(\frac{V}{V/8}\right)^\gamma = P(8)^\gamma$

Hence, $P_B=P\cdot 8^{5/3}$

Now, $8^{5/3}=(2^3)^{5/3}=2^5=32$

Therefore, $P_B=32P$

4. Required ratio

$\frac{P_B}{P_A}=\frac{32P}{8P}=4$

5. Option check

• A: $\frac{1}{8}$ ❌
• B: $8^{3/2}$ ❌
• C: $4$ ✅
• D: $8$ ❌

So the correct option is: $\boxed{4}$ which is Option C.