When an electron in hydrogen atom is taken from fourth excited state to ground state

Options

A. Both kinetic energy and potential energy
B. Both kinetic energy and potential energy decreases
C. Kinetic energy will increase while potential energy will decrease
D. Kinetic energy will decrease while potential energy will increase

Answer

To solve the question regarding the transition of an electron in a hydrogen atom from the fourth excited state to the ground state, we will analyze the changes in kinetic energy and potential energy during this transition.

### Step-by-Step Solution:

1. **Identify the Initial and Final States**:
- The electron is initially in the fourth excited state, which corresponds to the principal quantum number \( n = 4 \).
- The final state is the ground state, which corresponds to \( n = 1 \).

2. **Understanding the Radius of Orbit**:
- According to Bohr's model, the radius of the orbit for an electron in a hydrogen atom is given by the formula:
\[ r_n = a_0 n^2 \]
where \( a_0 \) is the Bohr radius and \( n \) is the principal quantum number.
- As \( n \) decreases from 4 to 1, the radius \( r \) will also decrease because it is proportional to \( n^2 \).

3. **Kinetic Energy Calculation**:
- The kinetic energy \( KE \) of the electron in a hydrogen atom is given by:
\[ KE = \frac{k e^2}{2r} \]
where \( k \) is Coulomb's constant and \( e \) is the charge of the electron.
- Since \( r \) decreases as \( n \) decreases, the kinetic energy \( KE \) will increase.

4. **Potential Energy Calculation**:
- The potential energy \( PE \) of the electron is given by:
\[ PE = -\frac{k e^2}{r} \]
- As \( r \) decreases, the absolute value of potential energy becomes larger (more negative), indicating that the potential energy decreases.

5. **Conclusion**:
- Therefore, during the transition from the fourth excited state to the ground state:
- The kinetic energy increases.
- The potential energy decreases (becomes more negative).

### Final Answer:
- The correct option is that the kinetic energy increases while the potential energy decreases.