Question

Number of spectral lines obtained in He⁺ spectra, when an electron makes transition from fifth excited state to first excited state will be __________.

Answer 1

Correct Answer: 10

To calculate the number of spectral lines obtained during the transition of an electron from the fifth excited state to the first excited state in the He⁺ ion, we use the formula for the maximum number of spectral lines during a transition:

Number of spectral lines = \[ \frac{n_2(n_2 + 1)}{2} - \frac{n_1(n_1 + 1)}{2} \]

Where:
\(n_1\) is the final state (ground state or first excited state).
\(n_2\) is the initial state (fifth excited state).

Assign Values:

• For the first excited state: \(n_1 = 2\)
• For the fifth excited state: \(n_2 = 6\)

Calculate the Maximum Number of Lines:

\[ \Delta n = n_2 - n_1 = 6 - 2 = 4 \]

The maximum number of spectral lines is given by:

\[ \text{Number of lines} = \frac{n_2(n_2 + 1)}{2} - \frac{n_1(n_1 + 1)}{2} \] \[ = \frac{6(6 + 1)}{2} - \frac{2(2 + 1)}{2} \] \[ = \frac{6 \times 7}{2} - \frac{2 \times 3}{2} \] \[ = 21 - 3 = 18 \]

Since we are considering all possible transitions from \(n_2 = 6\) to \(n_1 = 2\), we have:

Final Calculation:

\[ \text{Maximum number of spectral lines} = \frac{\Delta n (\Delta n + 1)}{2} = \frac{4(4 + 1)}{2} = \frac{20}{2} = 10 \]

Thus, the number of spectral lines obtained in He⁺ spectra when an electron transitions from the fifth excited state to the first excited state is: 10

Answer 2

Calculate the number of spectral lines in the He⁺ ion spectrum when an electron transitions from the fifth excited state to the first excited state.

Concept Used:

The number of spectral lines produced when an electron transitions from a higher energy level \( n_2 \) to a lower energy level \( n_1 \) is given by the formula:

\[ \text{Number of spectral lines} = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2} \]

where \( n_1 \) and \( n_2 \) are the principal quantum numbers of the lower and higher states respectively. The ground state is \( n = 1 \), so the first excited state is \( n = 2 \) and the fifth excited state is \( n = 6 \).

Step-by-Step Solution:

Step 1: Identify the principal quantum numbers for the given states.

• Ground state: \( n = 1 \)
• First excited state: \( n = 2 \)
• Fifth excited state: \( n = 6 \)

Thus, the transition is from \( n_2 = 6 \) to \( n_1 = 2 \).

Step 2: Apply the formula for the number of spectral lines.

\[ \text{Number of spectral lines} = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2} = \frac{(6 - 2)(6 - 2 + 1)}{2} \]

Step 3: Simplify the expression.

\[ = \frac{(4)(5)}{2} = \frac{20}{2} = 10 \]

Therefore, the number of spectral lines obtained is 10.