A hydrogen atom emits a photon corresponding to an electron transition from `n = 5` to `n = 1`. The recoil speed of hydrogen atom is almost (mass of proton `~~1.6 xx 10^(-27) kg)`.

Options

A. `10ms^(-1)`
B. `2xx10^(-2) ms^(-1)
C. `4ms^(-1)`
D. `8xx10^(2)ms^(-1)`

Answer

To solve the problem of finding the recoil speed of a hydrogen atom after it emits a photon corresponding to an electron transition from \( n = 5 \) to \( n = 1 \), we can follow these steps:

### Step 1: Calculate the Energy Released During the Transition

The energy released when an electron transitions from a higher energy level \( n_2 \) to a lower energy level \( n_1 \) in a hydrogen atom is given by the formula:

\[ E = 13.6 \, \text{eV} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

For our case, \( n_1 = 1 \) and \( n_2 = 5 \):

\[ E = 13.6 \, \text{eV} \left( \frac{1}{1^2} - \frac{1}{5^2} \right) \]

Calculating this:

\[ E = 13.6 \, \text{eV} \left( 1 - \frac{1}{25} \right) = 13.6 \, \text{eV} \left( \frac{24}{25} \right) = 13.6 \times 0.96 = 13.056 \, \text{eV} \]

### Step 2: Convert Energy to Joules

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

\[ E = 13.056 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 2.089 \times 10^{-18} \, \text{J} \]

### Step 3: Calculate the Momentum of the Photon

The momentum \( p \) of the emitted photon can be calculated using the formula:

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

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

\[ p = \frac{2.089 \times 10^{-18} \, \text{J}}{3 \times 10^8 \, \text{m/s}} \approx 6.963 \times 10^{-27} \, \text{kg m/s} \]

### Step 4: Relate the Momentum of the Photon to the Recoil Momentum of the Hydrogen Atom

By conservation of momentum, the momentum of the hydrogen atom after emitting the photon will be equal in magnitude but opposite in direction to the momentum of the photon:

\[ p_{\text{atom}} = p_{\text{photon}} = 6.963 \times 10^{-27} \, \text{kg m/s} \]

### Step 5: Calculate the Recoil Speed of the Hydrogen Atom

The recoil speed \( v \) of the hydrogen atom can be calculated using the formula:

\[ p = m v \]

where \( m \) is the mass of the hydrogen atom (approximately equal to the mass of a proton, \( m \approx 1.6 \times 10^{-27} \, \text{kg} \)):

\[ v = \frac{p}{m} = \frac{6.963 \times 10^{-27} \, \text{kg m/s}}{1.6 \times 10^{-27} \, \text{kg}} \approx 4.35 \, \text{m/s} \]

### Final Answer

The recoil speed of the hydrogen atom is approximately \( 4.35 \, \text{m/s} \).

---