An electron with kinetic energy =E eV collides with a hydrogen atom in the ground state. The collision will be elastic

Options

A. for all values of E
B. for `E lt 10.2 eV`
C. for `E lt 13.6 eV`
D. only for `E lt 3.4 eV`

Explanation

To determine whether the collision between an electron with kinetic energy \( E \) eV and a hydrogen atom in the ground state is elastic, we need to analyze the conditions under which elastic collisions occur and the energy levels of the hydrogen atom.

### Step-by-Step Solution:

1. **Understanding Elastic Collisions**:
- In an elastic collision, both momentum and kinetic energy are conserved. This means that the total kinetic energy before the collision is equal to the total kinetic energy after the collision.

2. **Kinetic Energy of the Electron**:
- The electron has a kinetic energy of \( E \) eV. This energy is the energy it possesses due to its motion.

3. **Energy Levels of Hydrogen Atom**:
- The ground state energy of a hydrogen atom is \( -13.6 \) eV. The energy required to transition from the ground state to the first excited state (n=2) is given by the difference in energy levels:
\[ \Delta E = E_2 - E_1 = -3.4 \, \text{eV} - (-13.6 \, \text{eV}) = 10.2 \, \text{eV} \]
- Therefore, an electron must have at least \( 10.2 \) eV of energy to excite the hydrogen atom from the ground state to the first excited state.

4. **Condition for Elastic Collision**:
- If the kinetic energy \( E \) of the electron is less than \( 10.2 \) eV, the electron does not have enough energy to cause an excitation in the hydrogen atom. Thus, the collision will be elastic, as all kinetic energy will be conserved.
- Conversely, if \( E \) is greater than \( 10.2 \) eV, some of the kinetic energy will be used to excite the hydrogen atom, leading to a loss of kinetic energy in the system. Therefore, the collision will not be elastic.

5. **Conclusion**:
- The collision will be elastic if \( E < 10.2 \) eV. If \( E \geq 10.2 \) eV, the collision will not be elastic due to energy loss in the excitation of the hydrogen atom.

### Final Answer:
The collision will be elastic if the kinetic energy \( E < 10.2 \) eV.

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