Consider a large number of hydrogen atoms with electron randomly distributed in the n=1,2, 3 and 4 orbits. How many different wavelengths of light are emitted by these atoms as the electrons fall into lower energy orbits

Options

A. 1
B. 3
C. 6
D. 9

Answer

To determine how many different wavelengths of light are emitted by hydrogen atoms as electrons transition from higher energy orbits (n=4, 3, 2) to lower energy orbits (n=1), we can follow these steps:

### Step 1: Identify the energy levels
The hydrogen atom has energy levels denoted by the principal quantum number \( n \). In this case, we have \( n = 1, 2, 3, 4 \).

### Step 2: Determine possible transitions
Electrons can transition from a higher energy level to a lower one, emitting light of a specific wavelength in the process. We need to consider all possible transitions between these levels.

### Step 3: List all transitions
1. From \( n = 4 \):
- \( 4 \to 3 \)
- \( 4 \to 2 \)
- \( 4 \to 1 \)

2. From \( n = 3 \):
- \( 3 \to 2 \)
- \( 3 \to 1 \)

3. From \( n = 2 \):
- \( 2 \to 1 \)

### Step 4: Count the transitions
Now, we will count the unique transitions:
- From \( n = 4 \): 3 transitions
- From \( n = 3 \): 2 transitions
- From \( n = 2 \): 1 transition

### Step 5: Total transitions
Adding these up:
- Total transitions = \( 3 + 2 + 1 = 6 \)

### Conclusion
Thus, the total number of different wavelengths of light emitted by the hydrogen atoms as the electrons fall into lower energy orbits is **6**.