An electron of kinetic energy K collides elastically with a stationary hydrogen atom in the ground state. Then,

Options

A. `K gt 13.6 eV`
B. `K gt 10.2 eV`
C. `K lt 10.2 eV`
D. data insufficient

Explanation

To solve the problem, we need to analyze the elastic collision between an electron with kinetic energy \( K \) and a stationary hydrogen atom in the ground state. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Energy Levels of Hydrogen
The energy levels of a hydrogen atom are given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
For the ground state (\( n = 1 \)):
\[ E_1 = -13.6 \, \text{eV} \]
For the first excited state (\( n = 2 \)):
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -3.4 \, \text{eV} \]

### Step 2: Calculate the Energy Difference
To transition from the ground state to the first excited state, the energy difference \( \Delta E \) is:
\[ \Delta E = E_2 - E_1 = (-3.4 \, \text{eV}) - (-13.6 \, \text{eV}) = 10.2 \, \text{eV} \]
This means that an electron must gain at least 10.2 eV of energy to excite the hydrogen atom from the ground state to the first excited state.

### Step 3: Analyze the Collision
In an elastic collision, the total kinetic energy before the collision must equal the total kinetic energy after the collision. If the kinetic energy \( K \) of the incoming electron is greater than 10.2 eV, it will provide enough energy to excite the hydrogen atom. This would result in a loss of kinetic energy for the incoming electron, making the collision inelastic.

### Step 4: Determine the Condition for Elastic Collision
For the collision to remain elastic, the kinetic energy \( K \) of the incoming electron must be less than 10.2 eV:
\[ K < 10.2 \, \text{eV} \]
This ensures that the incoming electron does not have enough energy to excite the hydrogen atom, thus conserving the total kinetic energy.

### Conclusion
The correct condition for the kinetic energy of the incoming electron in order to maintain an elastic collision with the hydrogen atom in the ground state is:
\[ K < 10.2 \, \text{eV} \]

### Final Answer
The answer is that the kinetic energy \( K \) of the incoming electron must be less than 10.2 eV.

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