A moving hydrogen atom makes a head on collision with a stationary hydrogen atom. Before collision both atoms are in in ground state and after collision they move together. What is the minimum value of the kinetic energy of the moving hydrogen atom, such that one of the atoms reaches one of the excited state?

Answer

To solve the problem, we need to analyze the collision of two hydrogen atoms and determine the minimum kinetic energy required for one of the atoms to reach an excited state after the collision. Here’s a step-by-step solution:

### Step 1: Understand the Collision
We have two hydrogen atoms: one is moving with kinetic energy \( K \) and the other is stationary. After the collision, both atoms move together.

### Step 2: Apply Conservation of Momentum
Before the collision, the momentum of the moving hydrogen atom is given by:
\[ p = mv \]
where \( m \) is the mass of the hydrogen atom and \( v \) is its velocity.

After the collision, both atoms (with a combined mass of \( 2m \)) move together with a velocity \( v' \). By conservation of momentum:
\[ mv = (2m)v' \]
This simplifies to:
\[ v' = \frac{v}{2} \]

### Step 3: Calculate Kinetic Energy After Collision
The kinetic energy before the collision is:
\[ K = \frac{1}{2} mv^2 \]

After the collision, the kinetic energy \( K' \) of the two atoms moving together is:
\[ K' = \frac{1}{2} (2m) (v')^2 = \frac{1}{2} (2m) \left(\frac{v}{2}\right)^2 = \frac{1}{2} (2m) \frac{v^2}{4} = \frac{mv^2}{4} \]

### Step 4: Relate Kinetic Energies
From the above, we can relate the kinetic energy before and after the collision:
\[ K' = \frac{K}{2} \]

### Step 5: Apply Conservation of Energy
The energy difference required for one of the hydrogen atoms to transition from the ground state to the excited state is given as \( \Delta E = 10.2 \, \text{eV} \).

According to the conservation of energy:
\[ K = K' + \Delta E \]
Substituting \( K' = \frac{K}{2} \):
\[ K = \frac{K}{2} + 10.2 \]

### Step 6: Solve for Kinetic Energy K
Rearranging the equation:
\[ K - \frac{K}{2} = 10.2 \]
\[ \frac{K}{2} = 10.2 \]
\[ K = 2 \times 10.2 = 20.4 \, \text{eV} \]

### Conclusion
The minimum value of the kinetic energy of the moving hydrogen atom required for one of the atoms to reach an excited state is:
\[ \boxed{20.4 \, \text{eV}} \]

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