A hypothetical gas expands adiabatically such that its volume changes from 08 litres to 27 litres. If the ratio of final pressure of the gas to initial pressure of the gas is $$\dfrac{16}{81}$$. Then the ratio of $$\dfrac{C_p}{C_v}$$ will be.

We need to find $$\gamma = \frac{C_p}{C_v}$$ for an adiabatic process.

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

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

Substitute given values: $$V_1 = 8$$ litres, $$V_2 = 27$$ litres, $$\frac{P_2}{P_1} = \frac{16}{81}$$

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

Express both sides as powers: $$\frac{8}{27} = \left(\frac{2}{3}\right)^3$$ and $$\frac{16}{81} = \left(\frac{2}{3}\right)^4$$

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

Equate exponents: $$3\gamma = 4 \implies \gamma = \frac{4}{3}$$

The correct answer is Option A: $$\frac{4}{3}$$.