If `r_n` is the radius of `n^(th)` orbit of hydrogen atom , then the relative change in the radius, when an electron jumps from `n^(th)` orbit to `(n-1)^(th) ` orbit is `(ngtgt1)`

Options

A. Directly proportional to n
B. Inversely proportional to n
C. Inversely proportional to `n ^2`
D. Independent of n

Answer

To solve the problem of finding the relative change in the radius of the hydrogen atom when an electron jumps from the \( n^{th} \) orbit to the \( (n-1)^{th} \) orbit, we can follow these steps:

### Step 1: Write the formula for the radius of the \( n^{th} \) orbit
The radius \( r_n \) of the \( n^{th} \) orbit in a hydrogen atom is given by the formula:
\[ r_n = a_0 n^2 \]
where \( a_0 \) is the Bohr radius (the radius of the first orbit) and \( n \) is the principal quantum number.

### Step 2: Write the formula for the radius of the \( (n-1)^{th} \) orbit
Using the same formula, the radius \( r_{n-1} \) of the \( (n-1)^{th} \) orbit is:
\[ r_{n-1} = a_0 (n-1)^2 \]

### Step 3: Calculate the change in radius
The change in radius \( \Delta r \) when the electron jumps from the \( n^{th} \) orbit to the \( (n-1)^{th} \) orbit is:
\[ \Delta r = r_n - r_{n-1} = a_0 n^2 - a_0 (n-1)^2 \]
Expanding \( (n-1)^2 \):
\[ \Delta r = a_0 n^2 - a_0 (n^2 - 2n + 1) = a_0 n^2 - a_0 n^2 + 2a_0 n - a_0 = 2a_0 n - a_0 \]
Thus,
\[ \Delta r = a_0 (2n - 1) \]

### Step 4: Calculate the relative change in radius
The relative change in radius is given by:
\[ \text{Relative Change} = \frac{\Delta r}{r_n} = \frac{a_0 (2n - 1)}{a_0 n^2} \]
The \( a_0 \) cancels out:
\[ \text{Relative Change} = \frac{2n - 1}{n^2} \]

### Step 5: Simplify the expression
For large \( n \) (i.e., \( n \gg 1 \)), the term \( -1 \) becomes negligible compared to \( 2n \):
\[ \text{Relative Change} \approx \frac{2n}{n^2} = \frac{2}{n} \]

### Conclusion
Thus, the relative change in the radius when an electron jumps from the \( n^{th} \) orbit to the \( (n-1)^{th} \) orbit is approximately:
\[ \frac{2}{n} \]