1. Identify the process

The container is thermally insulated, and the compression is sudden. Hence, there is no heat exchange:

$Q = 0$

So the process is treated as adiabatic.

For an adiabatic process of an ideal gas:

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

Given:

$\gamma = \frac{5}{3}$

and the final volume is

$V_2 = \frac{V_1}{8}$

2. Apply the adiabatic relation

$P_1 V_1^{\gamma} = P_2 V_2^{\gamma}$

Therefore,

$\frac{P_2}{P_1} = \left(\frac{V_1}{V_2}\right)^{\gamma}$

Substitute $V_2 = \frac{V_1}{8}$:

$\frac{V_1}{V_2} = \frac{V_1}{V_1/8} = 8$

So,

$\frac{P_2}{P_1} = 8^{5/3}$

3. Simplify

Since $8 = 2^3$,

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

Thus,

$\boxed{\frac{P_2}{P_1} = 32}$

4. Check options

• A: $16$ ❌
• B: $32$ ✅
• C: $28$ ❌
• D: $40$ ❌

So the correct option is B.