A closed hollow insulated cylinder is filled with gas at `0^(@)C` and also contains an insulated piston of negligible weight and negligible thickness at the middle point. The gas on one side of the piston is heated to `100^(@)C`. If the piston moves `5 cm` the length of the hollow cylinder is

Options

A. `13.65 cm`
B. `27.3 cm`
C. `38.6 cm`
D. `64.6 cm`

Explanation

To solve the problem, we need to analyze the situation involving the closed hollow insulated cylinder with an insulated piston. We will use the ideal gas law and the concept of thermal equilibrium to find the length of the cylinder.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions:**
- The gas in the cylinder is initially at \(0^\circ C\), which is \(273 K\).
- The piston divides the cylinder into two equal parts, each with volume \(V = A \cdot \frac{L}{2}\), where \(A\) is the cross-sectional area and \(L\) is the total length of the cylinder.

2. **Heating One Side of the Cylinder:**
- The gas on one side of the piston is heated to \(100^\circ C\), which is \(373 K\).
- The other side remains at \(0^\circ C\) (\(273 K\)) because the walls are insulated.

3. **Movement of the Piston:**
- The piston moves \(5 cm\) (or \(0.05 m\)) towards the side that was heated.
- After the piston moves, the volume on the heated side becomes \(V_1 = A \left(\frac{L}{2} + 0.05\right)\).
- The volume on the cooler side becomes \(V_2 = A \left(\frac{L}{2} - 0.05\right)\).

4. **Applying the Ideal Gas Law:**
- According to the ideal gas law, for both sides of the piston, we have:
\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
- Here, \(T_1 = 373 K\) and \(T_2 = 273 K\).

5. **Substituting the Volumes:**
- Substitute the expressions for \(V_1\) and \(V_2\):
\[ \frac{A \left(\frac{L}{2} + 0.05\right)}{373} = \frac{A \left(\frac{L}{2} - 0.05\right)}{273} \]
- The area \(A\) cancels out:
\[ \frac{\frac{L}{2} + 0.05}{373} = \frac{\frac{L}{2} - 0.05}{273} \]

6. **Cross-Multiplying to Solve for \(L\):**
- Cross-multiply to eliminate the fractions:
\[ 273 \left(\frac{L}{2} + 0.05\right) = 373 \left(\frac{L}{2} - 0.05\right) \]
- Expanding both sides:
\[ 273 \cdot \frac{L}{2} + 13.65 = 373 \cdot \frac{L}{2} - 18.65 \]

7. **Rearranging the Equation:**
- Rearranging gives:
\[ 273 \cdot \frac{L}{2} - 373 \cdot \frac{L}{2} = -18.65 - 13.65 \]
- Simplifying:
\[ -100 \cdot \frac{L}{2} = -32.3 \]
- Thus:
\[ \frac{L}{2} = \frac{32.3}{100} \implies L = \frac{32.3 \cdot 2}{100} = 64.6 \text{ cm} \]

### Final Answer:
The length of the hollow cylinder is \(64.6 \text{ cm}\).