A vessel of volume V is evacuated by means of a piston air pump Upon cach double stroke a piston pump sucks in a volume V of air and expels it. The process is assumed to be isothermal, and the gas is ideal. When this pump is used to evacuate the air from a vessel with a volume V, it performs n double strokes. The initial pressure inside the vessel is `p_0` which is equal to the atmospheric pressure. (a) Calculate the final pressure inside the vessel.

Explanation

To solve the problem step-by-step, we will analyze the situation using the ideal gas law and the properties of isothermal processes.

### Step 1: Understand the Initial Conditions
The vessel has a volume \( V \) and is initially at a pressure \( p_0 \) (which is equal to atmospheric pressure). The piston pump, with each double stroke, removes a volume \( V \) of air from the vessel.

**Hint:** Remember that the initial state of the gas is defined by its pressure and volume.

### Step 2: Analyze the Isothermal Process
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