In hydrogen atom, electron makes transition from `n = 4` to `n = 1` level. Recoil momentum of the `H` atom will be
a.`13.6xx10^(-19) kg ms^(-1)` b.`6.8xx10^(-27) kg ms^(-1)`
c.`12.75xx10^(-24) kg ms^(-1)` d.None of these

Options

A. `13.6xx10^(-19) kg ms^(-1)`
B. `6.8xx10^(-27) kg ms^(-1)`
C. `12.75xx10^(-24) kg ms^(-1)`
D. None of these

Explanation

To find the recoil momentum of the hydrogen atom when an electron transitions from the n = 4 level to the n = 1 level, we can follow these steps:

### Step 1: Identify the Transition Levels
The electron is transitioning from the initial level \( n_2 = 4 \) to the final level \( n_1 = 1 \).

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wavelength \( \lambda \) of the emitted photon during this transition is given by:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R \) is the Rydberg constant, approximately \( 1.097 \times 10^7 \, \text{m}^{-1} \).

### Step 3: Calculate \( \frac{1}{\lambda} \)
Substituting the values of \( n_1 \) and \( n_2 \):
\[ \frac{1}{\lambda} = R \left( \frac{1}{1^2} - \frac{1}{4^2} \right) = R \left( 1 - \frac{1}{16} \right) = R \left( \frac{15}{16} \right) \]

### Step 4: Substitute Rydberg Constant
Now substituting the value of \( R \):
\[ \frac{1}{\lambda} = \frac{15}{16} R = \frac{15}{16} \times 1.097 \times 10^7 \, \text{m}^{-1} \]

### Step 5: Calculate \( \lambda \)
Now, we can find \( \lambda \):
\[ \lambda = \frac{16}{15R} = \frac{16}{15 \times 1.097 \times 10^7} \]

### Step 6: Calculate the Momentum of the Photon
The momentum \( p \) of the emitted photon can be calculated using the formula:
\[ p = \frac{h}{\lambda} \]
where \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \, \text{Js} \).

### Step 7: Substitute \( \lambda \) into the Momentum Formula
Substituting \( \lambda \) into the momentum formula:
\[ p = h \cdot \frac{15 \times 1.097 \times 10^7}{16} \]

### Step 8: Calculate the Recoil Momentum of the Hydrogen Atom
Since the momentum of the photon is equal and opposite to the recoil momentum of the hydrogen atom, the recoil momentum \( P_H \) of the hydrogen atom is:
\[ P_H = p \]

### Step 9: Final Calculation
After performing the calculations, we find that:
\[ P_H \approx 6.8 \times 10^{-27} \, \text{kg m/s} \]

### Conclusion
Thus, the recoil momentum of the hydrogen atom is approximately \( 6.8 \times 10^{-27} \, \text{kg m/s} \), which corresponds to option **b**.

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