In a stationary hydrogen atom, an electron jumps from n = 3 ot n =1. The recoil speed of the hydrogen atom is about

Options

A. `4 m//s`
B. `4 cm//s`
C. `4 mm//s`
D. `4xx10^(-4) m//s`

Explanation

To find the recoil speed of a hydrogen atom when an electron jumps from the n=3 to n=1 energy level, we can follow these steps:

### Step 1: Calculate the energy of the electron in the n=3 and n=1 states.
The energy levels of a hydrogen atom are given by the formula:

\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]

For n=3:

\[ E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -\frac{13.6 \, \text{eV}}{9} \approx -1.51 \, \text{eV} \]

For n=1:

\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

### Step 2: Calculate the change in energy (ΔE) when the electron jumps from n=3 to n=1.
The change in energy is given by:

\[ \Delta E = E_1 - E_3 \]

Substituting the values:

\[ \Delta E = (-13.6 \, \text{eV}) - (-1.51 \, \text{eV}) = -13.6 + 1.51 = -12.09 \, \text{eV} \]

### Step 3: Convert the energy released into joules.
To convert electron volts to joules, we use the conversion factor \(1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}\):

\[ \Delta E = -12.09 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} \approx -1.9344 \times 10^{-18} \, \text{J} \]

### Step 4: Calculate the recoil speed of the hydrogen atom.
The momentum conservation principle states that the momentum gained by the hydrogen atom is equal to the momentum lost by the electron. The momentum change of the atom can be expressed as:

\[ MV = \Delta E / c \]

Where \(M\) is the mass of the hydrogen atom (approximately \(1.67 \times 10^{-27} \, \text{kg}\)) and \(c\) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)).

Rearranging gives us:

\[ V = \frac{\Delta E}{Mc} \]

Substituting the values:

\[ V = \frac{1.9344 \times 10^{-18} \, \text{J}}{(1.67 \times 10^{-27} \, \text{kg})(3 \times 10^8 \, \text{m/s})} \]

Calculating this gives:

\[ V \approx \frac{1.9344 \times 10^{-18}}{5.01 \times 10^{-19}} \approx 3.86 \, \text{m/s} \]

### Step 5: Round off the final answer.
The recoil speed of the hydrogen atom is approximately \(4 \, \text{m/s}\).

### Final Answer:
The recoil speed of the hydrogen atom is about **4 m/s**.

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