In hydrogen atom, electron makes transition from `n = 4` to `n = 1` level. Recoil momentum of the `H` atom will be

Options

A. `3.4 xx 10^(-27)N-sec`
B. `6.8 xx 10^(-27)N-sec`
C. `3.4 xx 10^(-24)N-sec`
D. `6.8 xx 10^(-24)N-sec`

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: Understand the Transition
When an electron in a hydrogen atom transitions from a higher energy level (n = 4) to a lower energy level (n = 1), it emits a photon. The energy of this photon corresponds to the difference in energy between the two levels.

### Step 2: Calculate the Energy of the Photon
The energy of the photon emitted during the transition can be calculated using the formula:
\[ E = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R_H \) is the Rydberg constant, approximately \( 13.6 \, \text{eV} \), \( n_1 = 1 \), and \( n_2 = 4 \).

Substituting the values:
\[ E = 13.6 \left( \frac{1}{1^2} - \frac{1}{4^2} \right) = 13.6 \left( 1 - \frac{1}{16} \right) = 13.6 \left( \frac{15}{16} \right) \]

Calculating this gives:
\[ E = 13.6 \times \frac{15}{16} = 12.75 \, \text{eV} \]

### Step 3: Calculate the Momentum of the Photon
The momentum \( p \) of the photon can be calculated using the relation:
\[ p = \frac{E}{c} \]
where \( c \) is the speed of light, approximately \( 3 \times 10^8 \, \text{m/s} \).

First, convert the energy from eV to Joules (1 eV = \( 1.6 \times 10^{-19} \, \text{J} \)):
\[ E = 12.75 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 2.04 \times 10^{-18} \, \text{J} \]

Now calculate the momentum:
\[ p = \frac{2.04 \times 10^{-18} \, \text{J}}{3 \times 10^8 \, \text{m/s}} = 6.8 \times 10^{-27} \, \text{kg m/s} \]

### Step 4: Recoil Momentum of the Hydrogen Atom
By conservation of momentum, the recoil momentum of the hydrogen atom will be equal in magnitude and opposite in direction to the momentum of the emitted photon. Therefore, the recoil momentum of the hydrogen atom is:
\[ p_{\text{recoil}} = 6.8 \times 10^{-27} \, \text{kg m/s} \]

### Final Answer
The recoil momentum of the hydrogen atom when the electron makes a transition from \( n = 4 \) to \( n = 1 \) is:
\[ \boxed{6.8 \times 10^{-27} \, \text{kg m/s}} \]