The initial pressure and volume of a given mass of a gas `(C_p / C_v = gamma)` are `(p_0)` and `(V_0)`. The gas can exchange heat with surrounding. (a) It is slowly compressed to a volume `(V_0 / 2 )` and then suddenly compressed to `(V_0 /4)` . Find the final pressure. (b) If the gas is suddenly cpmpressed from the volume `(V_0 to V_0 / 2 )` and then slowly compressed to `(V_0 / 4 )`, What will be the final pressure?.

Explanation

To solve the problem, we will break it down into two parts as per the question.

### Part (a): Slowly compressing to \( V_0/2 \) and then suddenly to \( V_0/4 \)

1. **Initial Conditions**: The initial pressure and volume of the gas are \( P_0 \) and \( V_0 \).

2. **First Step (Slow Compression)**: The gas is slowly compressed to \( V_0/2 \). Since this is a slow process, it is isothermal (constant temperature).
- Using the ideal gas law, we can express the relationship between pressure and volume:
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