To solve the problem of a hydrogen atom being raised from the ground state to the third state, we will analyze the changes in kinetic energy (KE) and potential energy (PE) during this transition.
### Step-by-Step Solution:
1. **Understand the Energy Levels of Hydrogen Atom:**
The energy levels of a hydrogen atom can be calculated using the formula:
\[
E_n = -\frac{13.6 \, \text{eV}}{n^2}
\]
where \( n \) is the principal quantum number.
2. **Calculate the Energy in the Ground State (n=1):**
For the ground state (n=1):
\[
E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV}
\]
3. **Calculate the Energy in the Third State (n=3):**
For the third state (n=3):
\[
E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -\frac{13.6 \, \text{eV}}{9} \approx -1.51 \, \text{eV}
\]
4. **Determine the Change in Total Energy:**
The change in total energy when moving from the ground state to the third state is:
\[
\Delta E = E_3 - E_1 = -1.51 \, \text{eV} - (-13.6 \, \text{eV}) = 12.09 \, \text{eV}
\]
5. **Calculate Potential Energy (PE) and Kinetic Energy (KE):**
The potential energy (PE) and kinetic energy (KE) can be derived from the total mechanical energy:
- The potential energy is given by:
\[
PE = 2 \times E_n
\]
- The kinetic energy is given by:
\[
KE = -\frac{1}{2} \times E_n
\]
6. **Calculate PE and KE for Ground State:**
For the ground state (n=1):
- Total energy \( E_1 = -13.6 \, \text{eV} \)
- Potential Energy \( PE_1 = 2 \times (-13.6) = -27.2 \, \text{eV} \)
- Kinetic Energy \( KE_1 = -\frac{1}{2} \times (-13.6) = 6.8 \, \text{eV} \)
7. **Calculate PE and KE for Third State:**
For the third state (n=3):
- Total energy \( E_3 \approx -1.51 \, \text{eV} \)
- Potential Energy \( PE_3 = 2 \times (-1.51) \approx -3.02 \, \text{eV} \)
- Kinetic Energy \( KE_3 = -\frac{1}{2} \times (-1.51) \approx 0.755 \, \text{eV} \)
8. **Analyze the Changes in KE and PE:**
- From ground state to third state:
- Potential Energy increases from \( -27.2 \, \text{eV} \) to \( -3.02 \, \text{eV} \)
- Kinetic Energy decreases from \( 6.8 \, \text{eV} \) to \( 0.755 \, \text{eV} \)
### Conclusion:
When a hydrogen atom is raised from the ground state to the third state, the potential energy increases while the kinetic energy decreases.
