Ratio of debroglie wavelengths of uncharged particle of mass m at `27^(0)C` to `127^(0)C` is nearly

Options

A. `1.16`
B. `0.16`
C. `1.33`
D. `0.8`

Explanation

To find the ratio of the de Broglie wavelengths of an uncharged particle of mass \( m \) at temperatures \( 27^{\circ}C \) and \( 127^{\circ}C \), we can follow these steps:

### Step 1: Convert Temperatures to Kelvin
First, we need to convert the given temperatures from Celsius to Kelvin.
- For \( 27^{\circ}C \):
\[ T_1 = 27 + 273 = 300 \, K \]
- For \( 127^{\circ}C \):
\[ T_2 = 127 + 273 = 400 \, K \]

### Step 2: Write the Expression for de Broglie Wavelength
The de Broglie wavelength \( \lambda \) of a particle is given by:
\[ \lambda = \frac{h}{p} \]
where \( p \) is the momentum of the particle. For an uncharged particle, we can express momentum in terms of its mass \( m \) and its root mean square speed \( v_{rms} \):
\[ p = mv_{rms} \]

### Step 3: Find the Expression for \( v_{rms} \)
The root mean square speed \( v_{rms} \) for a gas is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{m}} \]
where \( R \) is the universal gas constant.

### Step 4: Substitute \( v_{rms} \) into the Wavelength Formula
Substituting \( v_{rms} \) into the expression for momentum:
\[ p = m \cdot v_{rms} = m \cdot \sqrt{\frac{3RT}{m}} = \sqrt{3mRT} \]
Thus, the de Broglie wavelength becomes:
\[ \lambda = \frac{h}{\sqrt{3mRT}} \]

### Step 5: Determine the Ratio of Wavelengths
Since \( h \) and \( m \) are constants for the same particle, we can express the ratio of the wavelengths at the two temperatures:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\sqrt{3mRT_2}}{\sqrt{3mRT_1}} = \sqrt{\frac{T_2}{T_1}} \]

### Step 6: Substitute the Values of \( T_1 \) and \( T_2 \)
Now substituting \( T_1 = 300 \, K \) and \( T_2 = 400 \, K \):
\[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{400}{300}} = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} \approx 1.1547 \]

### Conclusion
Thus, the ratio of the de Broglie wavelengths of the uncharged particle at \( 27^{\circ}C \) to \( 127^{\circ}C \) is approximately:
\[ \frac{\lambda_1}{\lambda_2} \approx 1.1547 \]