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 step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the Problem
We are given that certain transitions of an electron from an excited state to the ground state result in a total of 10 spectral lines. We need to find out how many of these lines are in the visible range.

### Step 2: Use the Formula for Spectral Lines
The formula for the total number of spectral lines (N) when an electron transitions from a higher energy level (n2) to a lower energy level (n1) is given by:

\[ N = \frac{(n_2 - n_1 + 1)(n_2 - n_1)}{2} \]

### Step 3: Identify Ground and Excited States
In this case, the ground state (n1) is 1. We need to find the value of n2 that results in 10 spectral lines.

### Step 4: Set Up the Equation
Substituting n1 = 1 into the formula, we have:

\[ 10 = \frac{(n_2 - 1 + 1)(n_2 - 1)}{2} \]

This simplifies to:

\[ 10 = \frac{n_2(n_2 - 1)}{2} \]

### Step 5: Solve for n2
Multiplying both sides by 2 gives:

\[ 20 = n_2(n_2 - 1) \]

Rearranging this gives us a quadratic equation:

\[ n_2^2 - n_2 - 20 = 0 \]

### Step 6: Factor the Quadratic Equation
Factoring the quadratic equation, we get:

\[ (n_2 - 5)(n_2 + 4) = 0 \]

### Step 7: Find Possible Values for n2
Setting each factor to zero gives us:

1. \(n_2 - 5 = 0 \Rightarrow n_2 = 5\)
2. \(n_2 + 4 = 0 \Rightarrow n_2 = -4\) (not valid since n2 must be positive)

Thus, the only valid solution is \(n_2 = 5\).

### Step 8: Determine Visible Lines
Now, we need to find out how many of these transitions fall within the visible range, which corresponds to the Balmer series. The Balmer series involves transitions where the final state (n1) is 2, and the initial state (n2) can be 3, 4, 5, etc.

### Step 9: Identify Transitions from n2 = 5
From n2 = 5, the possible transitions to n1 = 2 are:

1. \(5 \to 2\)
2. \(4 \to 2\)
3. \(3 \to 2\)

### Step 10: Count the Lines
Thus, we have three transitions that correspond to three spectral lines in the visible range.

### Conclusion
The number of lines present in the visible range is **3**.

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