Which of the following expressions represents the spectrum of Balmer series (If n is the principal quantum number of higher energy level) in Hydrogen atom ?
(a)`overset-v=(R(n-1)(n+1))/n^(2)`
(b)`overset-v=(R(n-2)(n+2))/(4n^(2))`
(c)`overset-v=(R(n-2)(n+2))/(n^(2))`
(d)`overset-v=(R(n-1)(n+1))/(4n^(2))`

Options

A. `overset-v=(R(n-1)(n+1))/n^(2)`
B. `overset-v=(R(n-2)(n+2))/(4n^(2))`
C. `overset-v=(R(n-2)(n+2))/(n^(2))`
D. `overset-v=(R(n-1)(n+1))/(4n^(2))`

Answer

To solve the problem of identifying the correct expression that represents the spectrum of the Balmer series in the hydrogen atom, we will follow these steps:

### Step 1: Understand the Balmer Series
The Balmer series corresponds to the transitions of electrons from higher energy levels (n ≥ 3) to the second energy level (n = 2) in a hydrogen atom.

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wave number (ν̅) of spectral lines in hydrogen is given by:
\[ \overset{v}{=}\ R \cdot Z^2 \left( \frac{1}{n_l^2} - \frac{1}{n_h^2} \right) \]
where:
- \( R \) is the Rydberg constant,
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)),
- \( n_l \) is the principal quantum number of the lower energy level (for Balmer series, \( n_l = 2 \)),
- \( n_h \) is the principal quantum number of the higher energy level (can be 3, 4, 5, ...).

### Step 3: Substitute Values into the Rydberg Formula
For the Balmer series:
- Set \( n_l = 2 \)
- Let \( n_h = n \) (where \( n \) is the principal quantum number of the higher energy level)

Substituting these values into the Rydberg formula gives:
\[ \overset{v}{=}\ R \cdot 1^2 \left( \frac{1}{2^2} - \frac{1}{n^2} \right) \]
This simplifies to:
\[ \overset{v}{=}\ R \left( \frac{1}{4} - \frac{1}{n^2} \right) \]

### Step 4: Simplify the Expression
To combine the terms, we can find a common denominator:
\[ \overset{v}{=}\ R \left( \frac{n^2 - 4}{4n^2} \right) \]
This can be factored as:
\[ \overset{v}{=}\ \frac{R(n-2)(n+2)}{4n^2} \]

### Step 5: Identify the Correct Option
Now we can compare our derived expression with the given options:
- (a) \(\overset{v}{=}\frac{R(n-1)(n+1)}{n^2}\)
- (b) \(\overset{v}{=}\frac{R(n-2)(n+2)}{4n^2}\)
- (c) \(\overset{v}{=}\frac{R(n-2)(n+2)}{n^2}\)
- (d) \(\overset{v}{=}\frac{R(n-1)(n+1)}{4n^2}\)

The expression we derived is:
\[ \overset{v}{=}\frac{R(n-2)(n+2)}{4n^2} \]
Thus, the correct answer is **(b)**.