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: Understand the Transition
The electron in the hydrogen atom is transitioning from the third energy level (n = 3) to the first energy level (n = 1). This transition involves the emission of a photon.

### Step 2: Calculate the Energy of the Photon
The energy of the photon emitted during the transition can be calculated using the formula:
\[ E = E_3 - E_1 \]
Where \( E_n \) is the energy of the electron at level n. For hydrogen, the energy levels are given by:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
Calculating for n = 3 and n = 1:
- \( E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -\frac{13.6 \, \text{eV}}{9} \approx -1.51 \, \text{eV} \)
- \( E_1 = -13.6 \, \text{eV} \)

Now, calculating the energy of the photon:
\[ E = E_3 - E_1 = (-1.51 \, \text{eV}) - (-13.6 \, \text{eV}) = 12.09 \, \text{eV} \]

### Step 3: Convert Energy to Joules
To find the momentum, we need the energy in joules. Using the conversion \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ E = 12.09 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 1.9344 \times 10^{-18} \, \text{J} \]

### Step 4: Calculate the Momentum of the Photon
The momentum \( P \) of the photon can be calculated using the formula:
\[ P = \frac{E}{c} \]
Where \( c \) is the speed of light (\( c \approx 3 \times 10^8 \, \text{m/s} \)):
\[ P = \frac{1.9344 \times 10^{-18} \, \text{J}}{3 \times 10^8 \, \text{m/s}} \]
\[ P \approx 6.448 \times 10^{-27} \, \text{kg m/s} \]

### Step 5: 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. Thus:
\[ P_{H} = -P_{photon} \]
\[ P_{H} \approx -6.448 \times 10^{-27} \, \text{kg m/s} \]

### Final Answer
The recoil momentum of the hydrogen atom is approximately:
\[ P_{H} \approx 6.45 \times 10^{-27} \, \text{kg m/s} \]