The correct order of wavelength of Hydrogen `(._(1)H^(1))`, Deuterium `(._(1)H^(2))` and Tritium `(._(1)H^(3))` moving with same kinetic energy is

Options

A. `lambda_(H) gt lambda_(D) gt lambda_(r)`
B. `lambda_(H)=lambda_(D)=lambda_(r)`
C. `lambda_(H) lt lambda_(D) lt lambda_(r)`
D. `lambda_(H) lt lambda_(D) gt lambda_(r)`

Answer

To solve the problem of determining the correct order of wavelengths of Hydrogen (H), Deuterium (D), and Tritium (T) moving with the same kinetic energy, we can use the de Broglie wavelength formula:

### Step-by-Step Solution:

1. **Understand the de Broglie Wavelength Formula**:
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. The momentum \( p \) can also be expressed as:
\[ p = mv \]
where \( m \) is the mass and \( v \) is the velocity of the particle.

2. **Relate Kinetic Energy to Velocity**:
The kinetic energy (KE) of a particle is given by:
\[ KE = \frac{1}{2} mv^2 \]
Since we are given that all three isotopes have the same kinetic energy, we can express the velocity in terms of kinetic energy:
\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]

3. **Substituting Velocity into the Wavelength Formula**:
Substituting \( v \) into the de Broglie wavelength formula gives:
\[ \lambda = \frac{h}{mv} = \frac{h}{m \sqrt{\frac{2 \cdot KE}{m}}} = \frac{h \sqrt{m}}{\sqrt{2 \cdot KE}} \]
This shows that the wavelength is inversely proportional to the square root of the mass.

4. **Calculate the Wavelengths for Each Isotope**:
- For Hydrogen (H, mass = \( m \)):
\[ \lambda_H = \frac{h \sqrt{m}}{\sqrt{2 \cdot KE}} \]
- For Deuterium (D, mass = \( 2m \)):
\[ \lambda_D = \frac{h \sqrt{2m}}{\sqrt{2 \cdot KE}} = \frac{h \sqrt{2} \sqrt{m}}{\sqrt{2 \cdot KE}} = \frac{h \sqrt{m}}{\sqrt{KE}} = \frac{\lambda_H}{\sqrt{2}} \]
- For Tritium (T, mass = \( 3m \)):
\[ \lambda_T = \frac{h \sqrt{3m}}{\sqrt{2 \cdot KE}} = \frac{h \sqrt{3} \sqrt{m}}{\sqrt{2 \cdot KE}} = \frac{\lambda_H \sqrt{3}}{\sqrt{2}} \]

5. **Comparing the Wavelengths**:
Since the wavelengths are inversely proportional to the square root of their respective masses:
- \( \lambda_H \) is the largest,
- \( \lambda_D \) is smaller than \( \lambda_H \),
- \( \lambda_T \) is the smallest.

Therefore, the order of wavelengths is:
\[ \lambda_H > \lambda_D > \lambda_T \]

### Conclusion:
The correct order of wavelengths of Hydrogen, Deuterium, and Tritium moving with the same kinetic energy is:
\[ \lambda_H > \lambda_D > \lambda_T \]