A sample of gas contains `N_(1)` molecules and the total kinetic energy at `-123^(@)C is E_1` ergs. Another sample of gas at `27^@C` has total kinetic energy as `2E_(1)` ergs. Assuming gases to be ideal, the number of gas molecules in the second sample will be

Options

A. `N_(1)`
B. `N_(1)//2`
C. `2N_(1)`
D. `4N_(1)`

Answer

To solve the problem, we need to use the relationship between the kinetic energy of an ideal gas and the number of molecules it contains. The kinetic energy (E) of an ideal gas can be expressed as:

\[ E = \frac{3}{2} nRT \]

where:
- \( n \) is the number of moles of the gas,
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin.

### Step 1: Convert temperatures to Kelvin
- For the first sample at \(-123^\circ C\):
\[ T_1 = -123 + 273 = 150 \, K \]

- For the second sample at \(27^\circ C\):
\[ T_2 = 27 + 273 = 300 \, K \]

### Step 2: Write the kinetic energy expressions for both samples
- For the first sample:
\[ E_1 = \frac{3}{2} n_1 R T_1 \]

- For the second sample, given that its kinetic energy is \(2E_1\):
\[ E_2 = 2E_1 = \frac{3}{2} n_2 R T_2 \]

### Step 3: Set up the equation using kinetic energy expressions
From the expressions for \(E_1\) and \(E_2\):
\[ 2E_1 = \frac{3}{2} n_2 R T_2 \]
Substituting \(E_1\):
\[ 2 \left( \frac{3}{2} n_1 R T_1 \right) = \frac{3}{2} n_2 R T_2 \]

### Step 4: Simplify the equation
Cancelling \(\frac{3}{2} R\) from both sides:
\[ 2 n_1 T_1 = n_2 T_2 \]

### Step 5: Substitute the values of \(T_1\) and \(T_2\)
Substituting \(T_1 = 150 \, K\) and \(T_2 = 300 \, K\):
\[ 2 n_1 (150) = n_2 (300) \]

### Step 6: Solve for \(n_2\)
\[ 300 n_1 = 2 n_2 (150) \]
\[ 300 n_1 = 300 n_2 \]
Dividing both sides by 300:
\[ n_1 = n_2 \]

### Step 7: Relate number of molecules to moles
Since \(n = \frac{N}{N_A}\) (where \(N_A\) is Avogadro's number), we have:
\[ N_1 = N_2 \]

Thus, the number of gas molecules in the second sample will be equal to the number of gas molecules in the first sample.

### Final Answer
The number of gas molecules in the second sample is \(N_2 = N_1\).

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