Certain transition of electron from excited state to ground state in one or more steps gives rise to a total of 10 lines find out the number of lines which are present in the visible range

Options

A. 3
B. 5
C. 6
D. 1

Answer

To solve the problem of finding the number of spectral lines present in the visible range from the given transitions of an electron from an excited state to the ground state, we will follow these steps:

### Step 1: Understand the Problem
The problem states that there are a total of 10 spectral lines resulting from transitions of an electron from an excited state to the ground state. We need to find out how many of these lines fall within the visible range.

### Step 2: Use the Formula for Spectral Lines
The formula for the number of spectral lines (N) when an electron transitions from a higher energy level (n2) to a lower energy level (n1) is given by:
\[ N = \frac{(n2 - n1)(n2 - n1 + 1)}{2} \]

### Step 3: Set Up the Equation
Since the total number of spectral lines is given as 10, we can set up the equation:
\[ 10 = \frac{(n2 - 1)(n2 - 1 + 1)}{2} \]
Here, n1 is equal to 1 (ground state).

### Step 4: Simplify the Equation
Multiplying both sides by 2 to eliminate the fraction:
\[ 20 = (n2 - 1)(n2) \]
Expanding this gives:
\[ 20 = n2^2 - n2 \]

### Step 5: Rearrange into Standard Quadratic Form
Rearranging the equation gives us:
\[ n2^2 - n2 - 20 = 0 \]

### Step 6: Factor the Quadratic Equation
To factor the quadratic equation:
\[ (n2 - 5)(n2 + 4) = 0 \]
This gives us two potential solutions for n2:
1. \( n2 = 5 \)
2. \( n2 = -4 \) (not valid since n2 cannot be negative)

### Step 7: Determine the Valid Excited State
The only valid solution is \( n2 = 5 \). This means the transitions are occurring from the energy level n2 = 5 to n1 = 1.

### Step 8: Identify the Visible Range Transitions
The visible range corresponds to the Balmer series, which includes transitions that end at n1 = 2. The possible transitions from n2 = 5 to n1 = 1 are:
- 5 to 2
- 4 to 2
- 3 to 2

### Step 9: Count the Visible Lines
Thus, the transitions that correspond to the visible lines are:
1. 5 to 2
2. 4 to 2
3. 3 to 2

This gives us a total of **3 lines** in the visible range.

### Conclusion
Therefore, the number of spectral lines present in the visible range is **3**.

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