Question

The electron in hydrogen atom is moving in an orbit of radius 0.53 Å. It takes \( 1.571 \times 10^{-16} \, \text{s} \) to complete one revolution. The velocity of electron will be

• A. \( 5.3 \times 10^6 \, \text{m/s} \)
• B. \( 4 \times 10^6 \, \text{m/s} \)
• C. \( 3 \times 10^8 \, \text{m/s} \)
• D. \( 2.12 \times 10^6 \, \text{m/s} \)

Hint:

In problems involving circular motion, use the formula \( v = \frac{2 \pi r}{T} \) to find the velocity when the radius and time period are given.

Answer 1

The Correct Option is D

Step 1: Formula for velocity in hydrogen atom.
For an electron moving in a circular orbit in a hydrogen atom, the centripetal force is provided by the electrostatic force. Using Bohr's model of the hydrogen atom, the velocity of the electron \( v \) is given by: \[ v = \frac{2 \pi r}{T} \] Where: - \( r = 0.53 \, \text{Å} = 0.53 \times 10^{-10} \, \text{m} \) (radius of the orbit) - \( T = 1.571 \times 10^{-16} \, \text{s} \) (time for one revolution) Step 2: Calculating the velocity.
Substitute the values into the equation for \( v \): \[ v = \frac{2 \pi \times 0.53 \times 10^{-10}}{1.571 \times 10^{-16}} \] \[ v \approx 2.12 \times 10^6 \, \text{m/s} \] Step 3: Conclusion.
Thus, the velocity of the electron is \( 2.12 \times 10^6 \, \text{m/s} \), which corresponds to option (D).