Statement I: The de Broglie wavelength of a molecule (in a sample of ideal gas) varies inversely as the square root of absolute temperature.
Statement II: The de Broglie wavelength of a molecule (in sample of ideal gas) depends on temperature.

Options

A. Statement I is True, Statement II is True,Statement II is a correct explanation for Statement I
B. Statement I is True, Statement II is True, Statement II is NOT a correct explanation for Statement I
C. Statement I is True, Statement II is False .
D. Statement I is false, Statement II is True.

Explanation

To analyze the statements regarding the de Broglie wavelength of a molecule in an ideal gas, we can break down the problem step by step.

### Step 1: Understand the de Broglie Wavelength
The de Broglie wavelength (λ) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

### Step 2: Relate Momentum to Temperature
For a molecule in an ideal gas, the momentum \( p \) can be expressed in terms of the root mean square (RMS) speed (\( v_{RMS} \)):
\[ p = mv_{RMS} \]
where \( m \) is the mass of the molecule.

### Step 3: Determine the RMS Speed
The RMS speed of gas molecules is related to the absolute temperature \( T \) of the gas:
\[ v_{RMS} = \sqrt{\frac{3kT}{m}} \]
where \( k \) is the Boltzmann constant. This shows that \( v_{RMS} \) is proportional to the square root of the temperature \( T \):
\[ v_{RMS} \propto \sqrt{T} \]

### Step 4: Substitute into the de Broglie Wavelength Formula
Now substituting \( p \) in the de Broglie wavelength formula:
\[ \lambda = \frac{h}{mv_{RMS}} = \frac{h}{m \sqrt{\frac{3kT}{m}}} = \frac{h \sqrt{m}}{\sqrt{3kT}} \]
This indicates that:
\[ \lambda \propto \frac{1}{\sqrt{T}} \]
Thus, the de Broglie wavelength varies inversely with the square root of the absolute temperature.

### Step 5: Evaluate the Statements
- **Statement I**: The de Broglie wavelength of a molecule varies inversely as the square root of absolute temperature. **(True)**
- **Statement II**: The de Broglie wavelength of a molecule depends on temperature. **(True)**

### Step 6: Determine the Relationship Between the Statements
While both statements are true, Statement II does not provide a correct explanation for Statement I. Statement I is specifically about the inverse relationship with the square root of temperature, while Statement II is more general.

### Conclusion
- Both statements are true.
- Statement II is not the correct explanation for Statement I.

Thus, the correct answer is that both statements are true, but Statement II does not explain Statement I.

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