A vertical cylinder of cross-sectional area `0.1m^(2)` closed at both ends is fitted with a frictionless piston of mass M dividing the cylinder into two parts. Each part contains one mole of an ideal gas in equilibrium at `300K`. The volume of the upper part is `0.1m^(3)` and that of lower part is `0.05m^(3)` .What force must be applied to the piston so that the volumes of the two parts remain unchanged when the temperature is increased to `500K`?

Explanation

To solve the problem, we need to determine the force that must be applied to the piston to keep the volumes of the two parts of the cylinder unchanged when the temperature is increased from 300K to 500K. We will use the ideal gas law and the relationship between pressure, volume, and temperature.

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

2. **Calculate Initial Pressures**:
For the upper part of the cylinder (volume \( V_1 = 0.1 \, m^3 \)):
\[ P_1 = \frac{nRT_1}{V_1} = \frac{1 \times R \times 300}{0.1} \]
For the lower part of the cylinder (volume \( V_2 = 0.05 \, m^3 \)):
\[ P_2 = \frac{nRT_2}{V_2} = \frac{1 \times R \times 300}{0.05} \]

3. **Calculate the Change in Pressure**:
The net force on the piston is given by the difference in pressure multiplied by the area \( A \):
\[ F_{net} = (P_1 - P_2) \times A \]
The cross-sectional area \( A \) is given as \( 0.1 \, m^2 \).

4. **Calculate New Pressures at Higher Temperature**:
When the temperature is increased to \( 500K \):
\[ P_1' = \frac{nRT_1'}{V_1} = \frac{1 \times R \times 500}{0.1} \]
\[ P_2' = \frac{nRT_2'}{V_2} = \frac{1 \times R \times 500}{0.05} \]

5. **Calculate the New Net Force**:
The new net force on the piston when the temperature is increased:
\[ F_{net}' = (P_1' - P_2') \times A \]

6. **Determine the Force Required to Maintain Volume**:
The force that must be applied to the piston to keep the volumes unchanged is the difference in the forces due to the change in pressure:
\[ F_{applied} = F_{net}' - F_{net} \]

7. **Substituting Values**:
Substitute the values of \( P_1 \), \( P_2 \), \( P_1' \), and \( P_2' \) into the equations to find the required force.

8. **Final Calculation**:
After calculating the above values, we find that the force required to maintain the volume unchanged when the temperature is increased to \( 500K \) is:
\[ F_{applied} = \frac{nR \Delta T}{V} = \frac{1 \times \frac{25}{3} \times 200}{0.1} = \frac{5000}{3} \, N \]

### Final Answer:
The force that must be applied to the piston is \( \frac{5000}{3} \, N \).