The electronic velocity in the fourth Bohr orbit of hydrogen is v. The velocity of the electron in the first orbit would be

Options

A. 4v
B. 16v
C. v/4
D. v/16

Answer

To find the velocity of the electron in the first Bohr orbit of hydrogen when the velocity in the fourth Bohr orbit is given as \( v \), we can use the formula for the velocity of an electron in a Bohr orbit:

\[ v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s} \]

where:
- \( v_n \) is the velocity of the electron in the nth orbit,
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)),
- \( n \) is the principal quantum number (orbit number).

### Step 1: Calculate the velocity in the fourth orbit
For the fourth orbit (\( n = 4 \)):
\[ v_4 = 2.18 \times 10^6 \frac{1}{4} = \frac{2.18 \times 10^6}{4} = 0.545 \times 10^6 \text{ m/s} \]
This is given as \( v \).

### Step 2: Calculate the velocity in the first orbit
Now, for the first orbit (\( n = 1 \)):
\[ v_1 = 2.18 \times 10^6 \frac{1}{1} = 2.18 \times 10^6 \text{ m/s} \]

### Step 3: Relate the velocities
We know that the velocity in the fourth orbit is \( v \), which we calculated as \( 0.545 \times 10^6 \text{ m/s} \). The velocity in the first orbit is \( 2.18 \times 10^6 \text{ m/s} \).

### Step 4: Express \( v_1 \) in terms of \( v \)
Since \( v_4 = \frac{v_1}{4} \) (as the speed increases by a factor of \( n \) when moving from the nth orbit to the first orbit):
\[ v_1 = 4 \times v_4 \]
Given \( v_4 = v \):
\[ v_1 = 4v \]

### Conclusion
Thus, the velocity of the electron in the first orbit is \( 4v \).