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 of how many strokes are needed to reduce the pressure in the vessel by a factor of `η`, we will follow these steps:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
For an isothermal process, the product \( PV \) remains constant.

### Step 2: Initial and Final Pressure
Let:
- \( P_1 \) be the initial pressure in the vessel.
- \( P_2 \) be the final pressure after \( n \) strokes.

According to the problem, we want to reduce the pressure by a factor of \( η \):
\[ P_2 = \frac{P_1}{η} \]

### Step 3: Volume Change with Each Stroke
Each stroke of the piston captures a volume \( \Delta V \). After \( n \) strokes, the total volume evacuated from the vessel is:
\[ V' = V + n \Delta V \]
where \( V \) is the original volume of the vessel.

### Step 4: Relate Initial and Final Pressures
Using the ideal gas law, we can express the initial and final pressures:
\[ P_1 V = nRT \]
\[ P_2 (V + n \Delta V) = nRT \]

### Step 5: Substitute for Final Pressure
Substituting \( P_2 \) into the equation gives:
\[ \frac{P_1}{η} (V + n \Delta V) = P_1 V \]

### Step 6: Simplify the Equation
Dividing both sides by \( P_1 \):
\[ \frac{1}{η} (V + n \Delta V) = V \]

### Step 7: Rearranging the Equation
Rearranging the equation leads to:
\[ V + n \Delta V = ηV \]
\[ n \Delta V = ηV - V \]
\[ n \Delta V = (η - 1)V \]

### Step 8: Solve for n
Now, we can solve for \( n \):
\[ n = \frac{(η - 1)V}{\Delta V} \]

### Final Answer
The number of strokes needed to reduce the pressure in the vessel by a factor of \( η \) is:
\[ n = \frac{(η - 1)V}{\Delta V} \]

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