A vessel of volume `V` is evacuated by means of a piston air pump. One piston stroke captures the volume `Delta V`. How many strokes are needed to reduce the pressure in the vessel `eta` times ? The process is assumed to be isothermal, and the gas ideal.

Explanation

To solve the problem step by step, we will follow the reasoning outlined in the video transcript and derive the necessary equations.

### Step 1: Understand the Initial Conditions
We have a vessel of volume \( V \) that contains an ideal gas at an initial pressure \( P \). The density of the gas in the vessel is denoted as \( \rho \).

**Hint:** Remember that density is mass per unit volume, and for an ideal gas, pressure, volume, and temperature are related through the ideal gas law.

### Step 2: Relate Pressure and Density
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