A monoatomic ideal gas, initially at temperature `T_1,` is enclosed in a cylinder fitted with a friction less piston. The gas is allowed to expand adiabatically to a temperature `T_2` by releasing the piston suddenly. If `L_1 and L_2` are the length of the gas column before expansion respectively, then `(T_1)/(T_2)` is given by

Options

A. `((L_1)/(L_2))^(2//3)`
B. `L_1/L_2`
C. `L_2/L_1`
D. `((L_2)/(L_1))^(2//3)`

Explanation

To solve the problem, we need to find the ratio \( \frac{T_1}{T_2} \) for a monoatomic ideal gas that expands adiabatically. Here’s the step-by-step solution:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, the relationship between temperature and volume for an ideal gas is given by the equation:
\[ T V^{\gamma - 1} = \text{constant} \]
where \( \gamma \) is the adiabatic index.

### Step 2: Determine the Value of \( \gamma \)
For a monoatomic ideal gas, the degree of freedom \( f \) is 3. The adiabatic index \( \gamma \) is given by:
\[ \gamma = \frac{f + 2}{f} = \frac{3 + 2}{3} = \frac{5}{3} \]

### Step 3: Express Initial and Final States
Let:
- \( T_1 \) = initial temperature
- \( T_2 \) = final temperature
- \( L_1 \) = initial length of the gas column (related to initial volume)
- \( L_2 \) = final length of the gas column (related to final volume)

The volumes can be expressed in terms of the lengths:
\[ V_1 \propto L_1 \quad \text{and} \quad V_2 \propto L_2 \]

### Step 4: Apply the Adiabatic Condition
Using the adiabatic condition for the initial and final states:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]

Substituting the expressions for volume:
\[ T_1 (L_1)^{\gamma - 1} = T_2 (L_2)^{\gamma - 1} \]

### Step 5: Substitute \( \gamma \) Value
Substituting \( \gamma = \frac{5}{3} \):
\[ T_1 (L_1)^{\frac{5}{3} - 1} = T_2 (L_2)^{\frac{5}{3} - 1} \]
This simplifies to:
\[ T_1 (L_1)^{\frac{2}{3}} = T_2 (L_2)^{\frac{2}{3}} \]

### Step 6: Rearranging to Find the Ratio
Rearranging the equation to find \( \frac{T_1}{T_2} \):
\[ \frac{T_1}{T_2} = \frac{(L_2)^{\frac{2}{3}}}{(L_1)^{\frac{2}{3}}} \]
This can be rewritten as:
\[ \frac{T_1}{T_2} = \left(\frac{L_2}{L_1}\right)^{\frac{2}{3}} \]

### Final Answer
Thus, the ratio \( \frac{T_1}{T_2} \) is:
\[ \frac{T_1}{T_2} = \left(\frac{L_2}{L_1}\right)^{\frac{2}{3}} \]

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