1. Use the adiabatic relation

For an ideal gas undergoing an adiabatic process, $PV^\gamma = \text{constant}$

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

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

2. Substitute the given values

Given:

• $\gamma = 1.5 = \frac{3}{2}$
• $V_i = 5$ L
• $V_f = 4$ L

Hence, $\frac{P_i}{P_f} = \left(\frac{4}{5}\right)^{3/2}$

3. Simplify

$\left(\frac{4}{5}\right)^{3/2} = \left(\sqrt{\frac{4}{5}}\right)^3 = \left(\frac{2}{\sqrt{5}}\right)^3$

Now, $\left(\frac{2}{\sqrt{5}}\right)^3 = \frac{8}{5\sqrt{5}}$

4. Match with the options

Thus, $\frac{P_i}{P_f} = \frac{8}{5\sqrt{5}}$

This matches Option B.