When an electron jumps from a level `n = 4` to` n = 1`, the momentum of the recoiled hydrogen atom will be

Options

A. (a)`6.5xx10^(-27) kG - ms ^(-1)`
B. (b)`12.75xx10^(-19) kG - ms ^(-1)`
C. (c)`136xx10^(-19) kG - ms ^(-1)`
D. (d)Zero

Answer

To solve the problem of calculating the momentum of the recoiled hydrogen atom when an electron jumps from the energy level \( n = 4 \) to \( n = 1 \), we can follow these steps:

### Step 1: Understand the Energy Levels
The energy of an electron in a hydrogen atom at level \( n \) is given by the formula:
\[ E_n = -\frac{13.6 \text{ eV}}{n^2} \]
We need to calculate the energies for \( n = 4 \) and \( n = 1 \).

### Step 2: Calculate the Energy at \( n = 4 \)
Using the formula:
\[ E_4 = -\frac{13.6 \text{ eV}}{4^2} = -\frac{13.6 \text{ eV}}{16} = -0.85 \text{ eV} \]

### Step 3: Calculate the Energy at \( n = 1 \)
Using the formula:
\[ E_1 = -\frac{13.6 \text{ eV}}{1^2} = -13.6 \text{ eV} \]

### Step 4: Calculate the Energy Difference
The energy difference \( \Delta E \) when the electron transitions from \( n = 4 \) to \( n = 1 \) is:
\[ \Delta E = E_1 - E_4 = (-13.6 \text{ eV}) - (-0.85 \text{ eV}) = -13.6 + 0.85 = -12.75 \text{ eV} \]

### Step 5: Convert Energy to Joules
To convert electron volts to joules, we use the conversion factor \( 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \):
\[ \Delta E = -12.75 \text{ eV} \times 1.6 \times 10^{-19} \text{ J/eV} = -2.04 \times 10^{-18} \text{ J} \]

### Step 6: Calculate the Momentum of the Photon
The momentum \( p \) of the emitted photon can be calculated using the relation:
\[ p = \frac{\Delta E}{c} \]
where \( c \) is the speed of light \( (3 \times 10^8 \text{ m/s}) \).

### Step 7: Substitute the Values
Substituting the values we have:
\[ p = \frac{-2.04 \times 10^{-18} \text{ J}}{3 \times 10^8 \text{ m/s}} = -6.8 \times 10^{-27} \text{ kg m/s} \]

### Step 8: Determine the Recoil Momentum of the Hydrogen Atom
By conservation of momentum, the momentum of the recoiled hydrogen atom will be equal in magnitude but opposite in direction to the momentum of the emitted photon:
\[ p_{\text{atom}} = 6.8 \times 10^{-27} \text{ kg m/s} \]

### Final Answer
The momentum of the recoiled hydrogen atom is:
\[ \boxed{6.8 \times 10^{-27} \text{ kg m/s}} \]