Consider an electron moving in the first Bohr orbit of a He$^+$ ion with velocity $v_1$. If it is allowed to move in the third Bohr orbit with velocity $v_3$, determine the correct $v_3 : v_1$ ratio.\\

Question

The ratio of the wavelengths of the light absorbed by a Hydrogen atom when it undergoes n = 2 $\rightarrow$ n = 3 and n = 4 $\rightarrow$ n = 6 transitions, respectively, is:

Answer 1

An electron's velocity in a Bohr orbit is inversely proportional to the principal quantum number (n):

$ v \propto \frac{1}{n} $

If the velocity in the first Bohr orbit (n=1) is $v_1$, and in the third Bohr orbit (n=3) is $v_3$, the proportionality gives:

$ \frac{v_3}{v_1} = \frac{1/3}{1/1} = \frac{1}{3} $

Therefore, the ratio $v_3 : v_1$ is 1:3.

Consider an electron moving in the first Bohr orbit of a He\(^+\) ion with velocity \(v_1\). If it is allowed to move in the third Bohr orbit with velocity \(v_3\), determine the correct \(v_3 : v_1\) ratio.\\

The Correct Option is C

Solution and Explanation

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