What is the recoil velocity of Hydrogen atom when a photon is emitted due to corresponding transition from `n = 5` to `n = 1`. (`R_H` = Rydberg's constant, `m_H`, = mass of hydrogen)

Answer

To find the recoil velocity of a hydrogen atom when a photon is emitted due to a transition from \( n = 5 \) to \( n = 1 \), we can follow these steps:

### Step 1: Understand the Transition
When the hydrogen atom transitions from a higher energy level (\( n = 5 \)) to a lower energy level (\( n = 1 \)), it emits a photon. The energy of the emitted photon corresponds to the difference in energy between these two states.

### Step 2: Calculate the Wavelength of the Emitted Photon
The wavelength (\( \lambda \)) of the emitted photon can be calculated using the Rydberg formula:

\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

Where:
- \( R_H \) is the Rydberg constant.
- \( n_1 = 1 \) (final state).
- \( n_2 = 5 \) (initial state).

Substituting the values:

\[ \frac{1}{\lambda} = R_H \left( \frac{1}{1^2} - \frac{1}{5^2} \right) = R_H \left( 1 - \frac{1}{25} \right) = R_H \left( \frac{24}{25} \right) \]

Thus,

\[ \lambda = \frac{25}{24 R_H} \]

### Step 3: Calculate the Momentum of the Emitted Photon
The momentum (\( p_{photon} \)) of the emitted photon can be calculated using the formula:

\[ p_{photon} = \frac{h}{\lambda} \]

Substituting the value of \( \lambda \):

\[ p_{photon} = \frac{h}{\frac{25}{24 R_H}} = \frac{24 h R_H}{25} \]

### Step 4: Apply Conservation of Momentum
Initially, the hydrogen atom is at rest, so its initial momentum is zero. When the photon is emitted, the momentum of the hydrogen atom (\( p_{H} \)) will be equal in magnitude and opposite in direction to the momentum of the photon:

\[ p_{H} = -p_{photon} \]

Thus,

\[ m_H v = -\frac{24 h R_H}{25} \]

Where \( m_H \) is the mass of the hydrogen atom and \( v \) is the recoil velocity.

### Step 5: Solve for the Recoil Velocity
Rearranging the equation to find \( v \):

\[ v = -\frac{24 h R_H}{25 m_H} \]

The negative sign indicates that the direction of the recoil velocity is opposite to that of the emitted photon.

### Final Answer
The recoil velocity of the hydrogen atom when a photon is emitted during the transition from \( n = 5 \) to \( n = 1 \) is:

\[ v = \frac{24 h R_H}{25 m_H} \]

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