A monoatomic gas at pressure $$P$$ and volume $$V$$ is suddenly compressed to one eighth of its original volume. The final pressure at constant entropy will be

A monoatomic gas at pressure $$P$$ and volume $$V$$ is suddenly compressed to $$\dfrac{V}{8}$$ (one-eighth of its original volume). We need to find the final pressure given constant entropy (adiabatic process).

Constant entropy means the process is adiabatic (isentropic). For an adiabatic process:

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

For a monoatomic ideal gas:

$$\gamma = \frac{C_P}{C_V} = \frac{5/2 \cdot R}{3/2 \cdot R} = \frac{5}{3}$$

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

$$P \cdot V^{5/3} = P_2 \cdot \left(\frac{V}{8}\right)^{5/3}$$

$$P_2 = P \times \left(\frac{V}{V/8}\right)^{5/3} = P \times 8^{5/3}$$

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

$$P_2 = 32P$$

The correct answer is Option C: $$32P$$.