A conducting container containing an ideal gas (He) is kept in ice-water mixture for long time. Now the piston is suddenly moved up so that its volume becomes eight times. Final temperature of gas will be:

Options

A. 273 K
B. `(273)/(2)K`
C. `(273)/(4)K`
D. `(273)/(32)K`

Explanation

To solve the problem, we need to determine the final temperature of the helium gas after the volume has been increased to eight times its original volume. We will use the principles of thermodynamics, specifically the adiabatic process.

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- The gas (He) is kept in an ice-water mixture, which means it is at a constant temperature of 0°C (273 K).

2. **Understand the Process**:
- The piston is suddenly moved, which indicates that the process can be approximated as an adiabatic process. In an adiabatic process, there is no heat exchange with the surroundings.

3. **Use the Adiabatic Condition**:
- For an adiabatic process, the relationship between temperature and volume is given by:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]
- Here, \(T_1\) and \(T_2\) are the initial and final temperatures, and \(V_1\) and \(V_2\) are the initial and final volumes, respectively. The value of \(\gamma\) for a monoatomic gas (like helium) is \(\frac{5}{3}\).

4. **Set Up the Equation**:
- Since the final volume \(V_2\) is 8 times the initial volume \(V_1\), we can write:
\[ V_2 = 8 V_1 \]
- Substitute this into the adiabatic condition:
\[ T_1 V_1^{\gamma - 1} = T_2 (8 V_1)^{\gamma - 1} \]

5. **Simplify the Equation**:
- Cancel \(V_1^{\gamma - 1}\) from both sides:
\[ T_1 = T_2 \cdot 8^{\gamma - 1} \]
- Rearranging gives:
\[ T_2 = \frac{T_1}{8^{\gamma - 1}} \]

6. **Substitute the Value of \(\gamma\)**:
- Substitute \(\gamma = \frac{5}{3}\):
\[ T_2 = \frac{T_1}{8^{\frac{5}{3} - 1}} = \frac{T_1}{8^{\frac{2}{3}}} \]

7. **Calculate \(8^{\frac{2}{3}}\)**:
- \(8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} = 2^2 = 4\)
- Thus, we have:
\[ T_2 = \frac{T_1}{4} \]

8. **Substitute the Initial Temperature**:
- The initial temperature \(T_1\) is 273 K:
\[ T_2 = \frac{273}{4} = 68.25 \text{ K} \]

### Final Answer:
The final temperature of the gas after the piston is moved is **68.25 K**.