An electrons of a stationary hydrogen aton passes form the fifth enegry level to the ground level. The velocity that the atom acquired as a result of photon emission will be
`(m` is the mass of the electron, `R`, Rydberg constanrt and `h`, Planck's constant)

Options

A. `(25m)/(24 hR)`
B. `(24)/(25h R)`
C. `(24 h R) /(25 m)`
D. `(25 h R)/(24 m)`

Answer

To solve the problem of determining the velocity acquired by a hydrogen atom's electron when it transitions from the fifth energy level to the ground state, we can follow these steps:

### Step 1: Identify the Energy Levels
The electron transitions from the fifth energy level (n2 = 5) to the ground state (n1 = 1).

### Step 2: Calculate the Wavelength of the Emitted Photon
The wavelength (λ) of the emitted photon can be calculated using the Rydberg formula:
\[ \frac{1}{\lambda} = R \cdot (1/n_1^2 - 1/n_2^2) \]
Substituting the values:
\[ \frac{1}{\lambda} = R \cdot \left(1/1^2 - 1/5^2\right) = R \cdot \left(1 - \frac{1}{25}\right) = R \cdot \left(\frac{24}{25}\right) \]
Thus, we find:
\[ \lambda = \frac{25}{24R} \]

### Step 3: Calculate the Momentum of the Emitted Photon
The momentum (p) of a photon is given by:
\[ p = \frac{h}{\lambda} \]
Substituting the expression for λ:
\[ p = \frac{h}{\frac{25}{24R}} = \frac{24hR}{25} \]

### Step 4: Relate Momentum to the Velocity of the Electron
Since the hydrogen atom is stationary before the photon is emitted, the momentum acquired by the electron (which is equal to the momentum of the emitted photon) can be expressed as:
\[ mv = p \]
Substituting the expression for p:
\[ mv = \frac{24hR}{25} \]

### Step 5: Solve for Velocity (v)
To find the velocity (v) of the electron, we rearrange the equation:
\[ v = \frac{24hR}{25m} \]

### Final Answer
The velocity that the atom acquired as a result of photon emission is:
\[ v = \frac{24hR}{25m} \]