1. For an adiabatic process of an ideal gas,

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

where

$\gamma = \frac{C_p}{C_v}$

2. Therefore, between initial and final states,

$P_i V_i^\gamma = P_f V_f^\gamma$

So,

$\frac{P_f}{P_i} = \left(\frac{V_i}{V_f}\right)^\gamma$

3. Substitute the given values:

$V_i = 8\text{ L}, \qquad V_f = 27\text{ L}, \qquad \frac{P_f}{P_i} = \frac{16}{81}$

Thus,

$\frac{16}{81} = \left(\frac{8}{27}\right)^\gamma$

4. Rewrite both fractions in powers:

$\frac{16}{81} = \frac{2^4}{3^4} = \left(\frac{2}{3}\right)^4$

and

$\frac{8}{27} = \frac{2^3}{3^3} = \left(\frac{2}{3}\right)^3$

So the equation becomes

$\left(\frac{2}{3}\right)^4 = \left[\left(\frac{2}{3}\right)^3\right]^\gamma = \left(\frac{2}{3}\right)^{3\gamma}$

5. Equate the powers:

$3\gamma = 4$

Hence,

$\gamma = \frac{4}{3}$

6. Since

$\gamma = \frac{C_p}{C_v}$

the required ratio is

$\boxed{\frac{4}{3}}$

7. Checking options:

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

So the correct option is B.