Conider a given sample of an ideal gas `(C_p / C_v = gamma )` having initial pressure `(p_0) and volume (V_0)` . (a) The gas is isothermally taken to a pressure `(P_0 / 2 )` and from there adiabatically ta a pressure `(p_0 / 2)` and from there isothermally to a pressure `(p_0 / 4 )` . find the volume.

Explanation

To solve the problem step-by-step, we will analyze the process of the ideal gas as it undergoes different transformations: isothermal and adiabatic.

### Step 1: Initial Conditions
We start with an ideal gas at initial pressure \( P_0 \) and volume \( V_0 \).

### Step 2: Isothermal Process to Pressure \( \frac{P_0}{2} \)
In an isothermal process, the product of pressure and volume remains constant. Therefore, we have:

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