Question

In Bohr's theory of hydrogen atom, 'r' is the radius of the orbit, 'V' is the speed of electron and E is the total energy of the electron. Out of the following which physical quantity is inversely proportional to principal quantum number?

• A. \(\frac{1}{Vr}\)
• B. \(Vr\)
• C. \(V^2r\)
• D. \(Vr^2\)

Hint:

Use $r\propto n^2$ and $V\propto 1/n$, then combine them for each option.

Answer 1

The Correct Option is A

Step 1: Recall the Bohr model results
In the Bohr model of hydrogen, the radius of the \(n\)th orbit grows as the square of the principal quantum number \(n\).
The speed of the electron falls as \(n\) grows.
\[ r_n\propto n^2,\qquad V_n\propto \frac{1}{n} \]

Step 2: Test option (A)
Multiply the two proportionalities: \(Vr\propto \frac{1}{n}\times n^2 = n\).
So \(Vr\) is directly proportional to \(n\), and its reciprocal \(\frac{1}{Vr}\) is proportional to \(\frac{1}{n}\).
That is exactly "inversely proportional to \(n\)", so (A) is correct.

Step 3: Why the other options are wrong
(B) \(Vr\propto n\), so it is directly proportional to \(n\), not inversely.
(C) \(V^2r\propto \frac{1}{n^2}\times n^2 = n^0\), a constant that does not depend on \(n\).
(D) \(Vr^2\propto \frac{1}{n}\times n^4 = n^3\), which grows with \(n\).

Final Answer:
The reciprocal \(\frac{1}{Vr}\) is proportional to \(\frac{1}{n}\), which is option (A). \[ \boxed{\text{(A) } \frac{1}{Vr}} \]