The ratio of the velocity of electron in 3rd and 5th orbit of hydrogen atom is

Options

A. `5:3`
B. `1:3`
C. `5:1`
D. `1:1`

Answer

To find the ratio of the velocity of an electron in the 3rd and 5th orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the formula for velocity in orbits
The velocity \( V \) of an electron in the \( n^{th} \) orbit of a hydrogen-like atom is given by the formula:
\[ V_n = \frac{Z e^2}{2 \epsilon_0 h n} \]
where:
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)),
- \( e \) is the charge of the electron,
- \( \epsilon_0 \) is the permittivity of free space,
- \( h \) is the Planck constant,
- \( n \) is the principal quantum number (orbit number).

### Step 2: Identify the orbits
We need to find the velocities for the 3rd and 5th orbits:
- For the 3rd orbit, \( n_1 = 3 \)
- For the 5th orbit, \( n_2 = 5 \)

### Step 3: Write the velocity equations
Using the formula for each orbit:
- Velocity in the 3rd orbit:
\[ V_3 = \frac{Z e^2}{2 \epsilon_0 h \cdot 3} \]
- Velocity in the 5th orbit:
\[ V_5 = \frac{Z e^2}{2 \epsilon_0 h \cdot 5} \]

### Step 4: Set up the ratio of velocities
To find the ratio of the velocities \( \frac{V_3}{V_5} \):
\[ \frac{V_3}{V_5} = \frac{\frac{Z e^2}{2 \epsilon_0 h \cdot 3}}{\frac{Z e^2}{2 \epsilon_0 h \cdot 5}} \]

### Step 5: Simplify the ratio
Notice that \( Z, e^2, 2, \epsilon_0, \) and \( h \) cancel out:
\[ \frac{V_3}{V_5} = \frac{5}{3} \]

### Conclusion
Thus, the ratio of the velocity of the electron in the 3rd and 5th orbits of a hydrogen atom is:
\[ \frac{V_3}{V_5} = \frac{5}{3} \]