When a hydrogen atom emits a photon in going from n=5 to n=1, its recoil speed is almost

Options

A. 4 m/s
B. 800 m/s
C. 3mm/s
D. `01. mm//s`

Answer

To find the recoil speed of a hydrogen atom when it emits a photon while transitioning from n=5 to n=1, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Wavelength of the Emitted Photon:**
We use the Rydberg formula for hydrogen:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
Here, \( R \) is the Rydberg constant, \( n_1 = 1 \), and \( n_2 = 5 \).
\[ \frac{1}{\lambda} = R \left( \frac{1}{1^2} - \frac{1}{5^2} \right) = R \left( 1 - \frac{1}{25} \right) = R \left( \frac{24}{25} \right) \]
Therefore,
\[ \lambda = \frac{25}{24R} \]

2. **Use Conservation of Momentum:**
Before the emission of the photon, the total momentum is zero. After the emission, the momentum of the hydrogen atom (mass \( m \) and recoil speed \( v \)) and the momentum of the emitted photon must balance:
\[ 0 = -mv + p_{\text{photon}} \]
The momentum of the photon is given by:
\[ p_{\text{photon}} = \frac{h}{\lambda} \]
Thus, we can write:
\[ mv = \frac{h}{\lambda} \]

3. **Substitute for Wavelength:**
Substitute \( \lambda \) from step 1 into the momentum equation:
\[ mv = h \cdot \frac{24R}{25} \]
Rearranging gives:
\[ v = \frac{h \cdot 24R}{25m} \]

4. **Substitute Known Values:**
Now, we substitute the known constants:
- Planck's constant \( h = 6.626 \times 10^{-34} \, \text{Js} \)
- Rydberg constant \( R = 1.097 \times 10^7 \, \text{m}^{-1} \)
- Mass of the hydrogen atom (approximately the mass of a proton) \( m = 1.67 \times 10^{-27} \, \text{kg} \)

Plugging these values into the equation:
\[ v = \frac{(6.626 \times 10^{-34}) \cdot (24 \cdot 1.097 \times 10^7)}{25 \cdot (1.67 \times 10^{-27})} \]

5. **Calculate the Recoil Speed:**
Performing the calculation:
\[ v \approx \frac{(6.626 \times 10^{-34}) \cdot (26.328 \times 10^7)}{(4.175 \times 10^{-26})} \]
\[ v \approx \frac{1.747 \times 10^{-26}}{4.175 \times 10^{-26}} \approx 0.418 \, \text{m/s} \approx 4.16 \, \text{m/s} \]

6. **Final Result:**
The recoil speed of the hydrogen atom when it emits a photon transitioning from n=5 to n=1 is approximately \( 4 \, \text{m/s} \).