A piston can freely move inside a horizontal cylinder closed from both ends. Initially, the piston separates the inside space of the cylinder into two equal parts each of volume `V_0` in which an ideal gas is contained under the same pressure `p_0` and at the same temperature. What work has to be performed in order to increase isothermally the volume of one part of gas `eta` times compared to that of the other by slowly moving the piston ?

Explanation

To solve the problem step-by-step, we will follow the process of determining the work done during the isothermal expansion of an ideal gas when the volume of one part is increased by a factor of \( \eta \) compared to the other part.

### Step 1: Understand the Initial Setup
Initially, the piston divides the cylinder into two equal volumes, each of volume \( V_0 \). The total volume of the gas in the cylinder is \( 2V_0 \).

### Step 2: Define the Final Volumes
Let the final volumes of the two parts after the piston has moved be \( V_1 \) and \( V_2 \). According to the problem, we have:
- \( V_1 = \eta V_2 \)

### Step 3: Use the Conservation of Volume
Since the total volume remains constant, we can write:
\[ V_1 + V_2 = 2V_0 \]
Substituting \( V_1 \) from the previous step:
\[ \eta V_2 + V_2 = 2V_0 \]
This simplifies to:
\[ V_2(\eta + 1) = 2V_0 \]
From this, we can solve for \( V_2 \):
\[ V_2 = \frac{2V_0}{\eta + 1} \]

### Step 4: Calculate \( V_1 \)
Now we can find \( V_1 \):
\[ V_1 = \eta V_2 = \eta \left(\frac{2V_0}{\eta + 1}\right) = \frac{2\eta V_0}{\eta + 1} \]

### Step 5: Work Done in Isothermal Process
The work done \( W \) during an isothermal expansion or compression of an ideal gas is given by:
\[ W = nRT \ln\left(\frac{V_f}{V_i}\right) \]
In our case, we can consider:
- \( V_f = V_1 = \frac{2\eta V_0}{\eta + 1} \)
- \( V_i = V_0 \)

Substituting these values into the work formula:
\[ W = nRT \ln\left(\frac{V_1}{V_0}\right) = nRT \ln\left(\frac{\frac{2\eta V_0}{\eta + 1}}{V_0}\right) \]
This simplifies to:
\[ W = nRT \ln\left(\frac{2\eta}{\eta + 1}\right) \]

### Final Expression for Work Done
Thus, the work done to increase the volume of one part of the gas \( \eta \) times compared to the other is:
\[ W = nRT \ln\left(\frac{2\eta}{\eta + 1}\right) \]