At what minimum kinetic energy must a hydrogen atom move for it's inelastic headon collision with another stationary hydrogen atom so that one of them emits a photon? Both atoms are supposed to be in the ground state prior to the collision.

Explanation

To solve the problem of determining the minimum kinetic energy required for a hydrogen atom to emit a photon during an inelastic head-on collision with another stationary hydrogen atom, we can follow these steps:

### Step 1: Understand the Excitation Process
In order for a hydrogen atom to emit a photon, it must first be excited to a higher energy state. For hydrogen, the ground state is n=1, and the first excited state is n=2. The energy difference between these states is what allows for photon emission.

### Step 2: Calculate the Energy Required for Excitation
The energy required to excite a hydrogen atom from the ground state (n=1) to the first excited state (n=2) can be calculated using the formula for the energy levels of hydrogen:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
Thus, the energy difference (ΔE) between n=2 and n=1 is:
\[ \Delta E = E_2 - E_1 = -\frac{13.6 \, \text{eV}}{2^2} - \left(-\frac{13.6 \, \text{eV}}{1^2}\right) \]
\[ \Delta E = -\frac{13.6 \, \text{eV}}{4} + 13.6 \, \text{eV} \]
\[ \Delta E = 13.6 \, \text{eV} \left(1 - \frac{1}{4}\right) = 13.6 \, \text{eV} \cdot \frac{3}{4} = 10.2 \, \text{eV} \]

### Step 3: Conservation of Momentum and Energy
In an inelastic collision, the total momentum before the collision must equal the total momentum after the collision. Let \( m \) be the mass of the hydrogen atom, \( V_1 \) be the velocity of the moving hydrogen atom, and \( V_2 \) be the velocity of the stationary hydrogen atom (which is 0). The conservation of momentum gives us:
\[ mV_1 + mV_2 = mV' \]
where \( V' \) is the final velocity of the system after the collision.

### Step 4: Relate Kinetic Energy to the Excitation Energy
The kinetic energy (T) of the moving hydrogen atom is given by:
\[ T = \frac{1}{2} m V_1^2 \]
For one of the atoms to emit a photon, the kinetic energy must be at least equal to the energy required for excitation:
\[ T \geq 10.2 \, \text{eV} \]

### Step 5: Find the Minimum Kinetic Energy
To find the minimum kinetic energy required for the collision, we can set:
\[ T = 10.2 \, \text{eV} \]

### Final Result
Thus, the minimum kinetic energy that the moving hydrogen atom must have for the inelastic collision to result in photon emission is:
\[ T = 10.2 \, \text{eV} \]