An electron in `H` atom makes a transition from `n = 3 to n = 1`. The recoil momentum of the `H` atom will be

Options

A. `6.45 xx 10^(-27) N s`
B. `6.8 xx 10^(-27) N s`
C. `6.45 xx 10^(-24) N s`
D. `6.8 xx 10^(-24) N s`

Explanation

To find the recoil momentum of the hydrogen atom when an electron transitions from \( n = 3 \) to \( n = 1 \), we can follow these steps:

### Step 1: Determine the energy of the emitted photon
The energy of the photon emitted during the transition can be calculated using the formula for the energy levels of a hydrogen atom:

\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]

For \( n = 3 \):
\[ E_3 = -\frac{13.6}{3^2} = -\frac{13.6}{9} \approx -1.51 \, \text{eV} \]

For \( n = 1 \):
\[ E_1 = -\frac{13.6}{1^2} = -13.6 \, \text{eV} \]

The energy of the emitted photon (the difference in energy between the two levels) is:

\[ E_{\text{photon}} = E_1 - E_3 = (-13.6) - (-1.51) = -13.6 + 1.51 = -12.09 \, \text{eV} \]

### Step 2: Convert the energy of the photon to joules
To convert the energy from electron volts to joules, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):

\[ E_{\text{photon}} = 12.09 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 1.9344 \times 10^{-18} \, \text{J} \]

### Step 3: Calculate the momentum of the emitted photon
The momentum \( p \) of a photon can be calculated using the formula:

\[ p = \frac{E}{c} \]

where \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)).

Substituting the values:

\[ p = \frac{1.9344 \times 10^{-18} \, \text{J}}{3 \times 10^8 \, \text{m/s}} \approx 6.448 \times 10^{-27} \, \text{kg m/s} \]

### Step 4: Determine the 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:

\[ p_{\text{recoil}} = -p_{\text{photon}} = -6.448 \times 10^{-27} \, \text{kg m/s} \]

Thus, the recoil momentum of the hydrogen atom is approximately:

\[ p_{\text{recoil}} \approx 6.45 \times 10^{-27} \, \text{kg m/s} \]

### Final Answer
The recoil momentum of the hydrogen atom is \( 6.45 \times 10^{-27} \, \text{kg m/s} \).

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