The de-broglie wavelength of neutron at `27^(@)C` is `lambda` . The wavelength at `927^(@)C` will be

Options

A. `(lambda)/(9)`
B. `(lambda)/(4)`
C. `(lambda)/(2)`
D. `(lambda)/(3)`

Explanation

To solve the problem of finding the de Broglie wavelength of a neutron at 927°C given its wavelength at 27°C is λ, we will follow these steps:

### Step-by-Step Solution:

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

2. **Relating Momentum to Kinetic Energy**:
The momentum \( p \) can be expressed in terms of kinetic energy (KE):
\[ p = \sqrt{2m \cdot KE} \]
where \( m \) is the mass of the neutron.

3. **Kinetic Energy of an Ideal Gas**:
For an ideal gas, the average kinetic energy can be expressed as:
\[ KE = \frac{3}{2} kT \]
where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature in Kelvin.

4. **Substituting Kinetic Energy into the Wavelength Formula**:
Substituting the expression for kinetic energy into the wavelength equation gives:
\[ \lambda = \frac{h}{\sqrt{2m \cdot \frac{3}{2} kT}} = \frac{h}{\sqrt{3mkT}} \]

5. **Proportionality to Temperature**:
From the above equation, we can see that:
\[ \lambda \propto \frac{1}{\sqrt{T}} \]
This means that the wavelength is inversely proportional to the square root of the temperature.

6. **Calculating Temperatures in Kelvin**:
- For 27°C:
\[ T_1 = 27 + 273 = 300 \, K \]
- For 927°C:
\[ T_2 = 927 + 273 = 1200 \, K \]

7. **Setting Up the Ratio of Wavelengths**:
Using the proportionality:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\sqrt{T_2}}{\sqrt{T_1}} \]
Substituting the temperatures:
\[ \frac{\lambda}{\lambda_2} = \frac{\sqrt{1200}}{\sqrt{300}} \]

8. **Simplifying the Ratio**:
\[ \frac{\lambda}{\lambda_2} = \frac{\sqrt{1200}}{\sqrt{300}} = \frac{\sqrt{1200/300}}{1} = \frac{\sqrt{4}}{1} = 2 \]
Thus, we have:
\[ \lambda_2 = \frac{\lambda}{2} \]

### Final Result:
The wavelength at 927°C is:
\[ \lambda_2 = \frac{\lambda}{2} \]