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. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is C

To determine the number of spectral lines emitted when a 12.5 eV electron beam bombards gaseous hydrogen, we assess the electronic transitions within a hydrogen atom. This involves calculating how many possible transitions can occur given the excitation energy.

Step 1: Understanding Electron Excitation

Hydrogen atoms absorb energy, and electrons get excited to higher energy levels. The energy of an electron in a hydrogen atom is given by E_n = -\frac{13.6}{n^2} \text{ eV}, where \( n \) is the principal quantum number.

Step 2: Calculating Excitation

The electron beam has an energy of 12.5 eV. This energy will excite the electron from the ground state (n=1) to a higher energy level. The ground state energy is E_1 = -13.6 \text{ eV}. The energy difference between the levels is:

\Delta E = E_{n} - E_1 = -\frac{13.6}{n^2} + 13.6

Setting \Delta E = 12.5 \text{ eV} gives us the excitation condition:

13.6 \left(1 - \frac{1}{n^2}\right) = 12.5

Simplifying this equation for \( n \), we have:

\frac{1}{n^2} = \frac{1.1}{13.6}

Calculating further:

n^2 = \frac{13.6}{1.1} \approx 12.36

Therefore, \( n \approx 3.5 \). However, \( n \) must be an integer, so the closest possibility is \( n = 3 \). Thus the electron is excited to the third energy level (n=3).

Step 3: Determining Spectral Lines

When the electron transitions from higher to lower energy levels, it emits photons, producing spectral lines. For an electron excited to \( n = 3 \), it can transition to:

1. From n=3 to n=2
2. From n=3 to n=1
3. From n=2 to n=1

These transitions produce three distinct spectral lines.

Conclusion

The number of spectral lines emitted is 3. Thus, the correct answer is option 3.