Question

If \( v_n \) and \( v_p \) are orbital velocities in \( n^{th} \) and \( p^{th} \) orbit respectively, then the ratio \( \frac{v_p}{v_n} \) is

• A. \( \frac{p^2}{n^2} \)
• B. \( \frac{n}{p} \)
• C. \( \frac{n^2}{p^2} \)
• D. \( \frac{p}{n} \)

Hint:

The orbital velocity of an electron is inversely proportional to the square of the principal quantum number, so the ratio of velocities is simply the ratio of the quantum numbers.

Answer 1

The Correct Option is B

Step 1: Understanding the orbital velocity.
The orbital velocity \( v \) of an electron in an orbit is given by the equation: \[ v = \sqrt{\frac{KZ}{r}} \] where \( K \) is a constant, \( Z \) is the atomic number, and \( r \) is the radius of the orbit. The radius of the orbit is related to the principal quantum number \( n \) by: \[ r \propto n^2 \] Thus, the velocity is inversely proportional to the square of the principal quantum number: \[ v \propto \frac{1}{n} \] Step 2: Finding the ratio.
Therefore, the ratio \( \frac{v_p}{v_n} \) is given by: \[ \frac{v_p}{v_n} = \frac{n}{p} \] Step 3: Conclusion.
Thus, the ratio \( \frac{v_p}{v_n} \) is \( \frac{n}{p} \), corresponding to option (B).