To find the velocity of the recoiling hydrogen atom when an electron jumps from the 5th excited state to the ground state, we can follow these steps:
### Step 1: Identify the Energy Levels
The energy of the hydrogen atom in the nth orbit is given by the formula:
\[
E_n = -\frac{13.6 \, \text{eV}}{n^2}
\]
For the 5th excited state, the electron is in the 6th orbit (n=6), and for the ground state, it is in the 1st orbit (n=1).
### Step 2: Calculate the Energy of the States
Calculate the energy for n=6 and n=1:
\[
E_6 = -\frac{13.6 \, \text{eV}}{6^2} = -\frac{13.6 \, \text{eV}}{36} \approx -0.3778 \, \text{eV}
\]
\[
E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV}
\]
### Step 3: Find the Energy Difference
Now, calculate the energy difference when the electron transitions from the 6th orbit to the 1st orbit:
\[
\Delta E = E_1 - E_6 = -13.6 \, \text{eV} - (-0.3778 \, \text{eV}) = -13.6 + 0.3778 \approx -13.2222 \, \text{eV}
\]
### Step 4: Convert Energy to Joules
Convert the energy from electron volts to joules. The conversion factor is:
\[
1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}
\]
Thus,
\[
\Delta E \approx -13.2222 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} \approx -2.1155 \times 10^{-18} \, \text{J}
\]
### Step 5: Use Conservation of Momentum
When the electron transitions, it emits a photon, and by conservation of momentum, the momentum of the recoiling hydrogen atom must equal the momentum of the emitted photon:
\[
p_{\text{photon}} = p_{\text{atom}}
\]
The momentum of the photon can be expressed as:
\[
p = \frac{E}{c}
\]
where \(E\) is the energy of the photon and \(c\) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)).
### Step 6: Calculate the Velocity of the Recoiling Atom
The momentum of the recoiling hydrogen atom can be expressed as:
\[
p_{\text{atom}} = m_{\text{H}} \cdot v
\]
where \(m_{\text{H}} = 1.67 \times 10^{-27} \, \text{kg}\) (mass of hydrogen atom) and \(v\) is the velocity we want to find.
Setting the two momenta equal gives:
\[
\frac{\Delta E}{c} = m_{\text{H}} \cdot v
\]
Rearranging for \(v\):
\[
v = \frac{\Delta E}{m_{\text{H}} \cdot c}
\]
### Step 7: Substitute Values
Substituting the values:
\[
v = \frac{2.1155 \times 10^{-18} \, \text{J}}{(1.67 \times 10^{-27} \, \text{kg}) \cdot (3 \times 10^8 \, \text{m/s})}
\]
### Step 8: Calculate \(v\)
Calculating the above expression:
\[
v \approx \frac{2.1155 \times 10^{-18}}{5.01 \times 10^{-19}} \approx 4.22 \, \text{m/s}
\]
### Final Answer
The velocity of the recoiling hydrogen atom is approximately \(4.22 \, \text{m/s}\).
---
