An ideal gas filled in a cylinder occupies volume V. The gas is compressed isothermally to the volume V/3. Now the cylinder valve is opened and the gas is allowed to leak keeping temperature same. What percentage of the number of molecules escape to bring the pressure in the cylinder back to its original value.

Options

A. `66%`
B. `33%`
C. `0.33%`
D. `0.66%`

Explanation

To solve the problem step by step, we will use the ideal gas law and the concept of isothermal processes.

### Step 1: Understand the Initial Conditions
Initially, the gas occupies a volume \( V \) at pressure \( P \) and temperature \( T \). According to the ideal gas law:
\[ PV = n_1RT \quad \text{(1)} \]
where \( n_1 \) is the number of moles of gas.

### Step 2: Conditions After Compression
The gas is compressed isothermally to a volume of \( \frac{V}{3} \). The pressure after compression can be denoted as \( P_2 \). Using the ideal gas law again:
\[ P_2 \left(\frac{V}{3}\right) = n_1RT \quad \text{(2)} \]
Since the temperature remains constant, we can relate the pressures and volumes before and after compression:
\[ P_2 = \frac{3P}{1} = 3P \quad \text{(from (1) and (2))} \]

### Step 3: Conditions After Gas Leaks
After opening the valve, some gas escapes, and we denote the number of moles left as \( n_2 \). The pressure in the cylinder returns to its original value \( P \). The volume remains \( \frac{V}{3} \):
\[ P \left(\frac{V}{3}\right) = n_2RT \quad \text{(3)} \]

### Step 4: Relate \( n_2 \) to \( n_1 \)
We can now relate equations (2) and (3):
From equation (2):
\[ P_2 = 3P \implies 3P \left(\frac{V}{3}\right) = n_1RT \]
From equation (3):
\[ P \left(\frac{V}{3}\right) = n_2RT \]
Dividing equation (3) by equation (2):
\[ \frac{P \left(\frac{V}{3}\right)}{3P \left(\frac{V}{3}\right)} = \frac{n_2RT}{n_1RT} \]
This simplifies to:
\[ \frac{1}{3} = \frac{n_2}{n_1} \implies n_2 = \frac{n_1}{3} \]

### Step 5: Calculate the Number of Moles Escaped
The number of moles that escaped can be calculated as:
\[ \text{Moles escaped} = n_1 - n_2 = n_1 - \frac{n_1}{3} = \frac{2n_1}{3} \]

### Step 6: Calculate the Percentage of Moles Escaped
To find the percentage of the number of molecules that escaped:
\[ \text{Percentage escaped} = \left(\frac{\text{Moles escaped}}{n_1}\right) \times 100 = \left(\frac{\frac{2n_1}{3}}{n_1}\right) \times 100 = \frac{2}{3} \times 100 = 66.67\% \]

### Final Answer
Thus, the percentage of the number of molecules that escaped to bring the pressure in the cylinder back to its original value is \( 66.67\% \).

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