Question

A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. The number of spectral lines emitted will be:

• A. \( 2 \)
• B. \( 1 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

For hydrogen spectral line calculations: - Determine the highest energy level an electron can reach. - Use the formula: \[ {Number of spectral lines} = \frac{n(n-1)}{2} \] where \( n \) is the highest excited state.

Answer 1

The Correct Option is C

Step 1: Understanding energy levels of hydrogen. The energy levels in hydrogen are given by the Bohr equation: \[ E_n = - \frac{13.6}{n^2} { eV} \] where \( n \) is the principal quantum number. 
Step 2: Effect of electron beam energy. - The incident energy is 12.5 eV. - The ground state energy (\( n = 1 \)) of hydrogen is \( -13.6 \) eV. - The energy required to excite an electron to \( n = 2 \) is: \[ E_2 = -\frac{13.6}{2^2} = -3.4 { eV} \] Energy difference: \[ \Delta E = E_2 - E_1 = (-3.4) - (-13.6) = 10.2 { eV} \] Since 12.5 eV is supplied, the electron can be excited up to \( n = 3 \). 
Step 3: Possible transitions and spectral lines. The number of spectral lines emitted follows: \[ {Number of spectral lines} = \frac{n(n-1)}{2} \] For \( n = 3 \), possible transitions are: - \( 3 \to 2 \) - \( 3 \to 1 \) - \( 2 \to 1 \) Thus, the total spectral lines: \[ \frac{3(3-1)}{2} = \frac{3 \times 2}{2} = 3 \] Final Answer: \[ \boxed{3} \]