Step 1: Understanding the Concept:
An adiabatic process is a thermodynamic process in which there is no heat transfer into or out of the system ($dQ = 0$).
Step 2: Key Formula or Approach:
The approach starts with the First Law of Thermodynamics ($dQ = dU + dW$) and the ideal gas equation ($PV = nRT$).
Setting $dQ = 0$ gives $0 = nC_v dT + P dV$.
Using the differential form of the ideal gas law ($P dV + V dP = nR dT$) and combining these equations yields a separable differential equation.
Step 3: Detailed Explanation:
When integrating the combined differential equation, we use the fact that the specific heat ratio is defined as $\gamma = \frac{C_p}{C_v}$ and $R = C_p - C_v$.
The integration leads directly to Poisson's equation for a reversible adiabatic process of an ideal gas:
\[ P V^\gamma = \text{constant} \]
This relationship signifies that during an adiabatic expansion or compression, the pressure and volume change such that their product, with the volume raised to the power of $\gamma$, remains constant.
Option (A) $PV = \text{constant}$ is Boyle's law, which applies to isothermal processes, not adiabatic.
Step 4: Final Answer:
The correct option is (C).