An atom emits a spectral line of wavelength `lambda` when an electron makes a transition between levels of energy `E_(1) and E_(2)`. Which expression correctly relates `lambda E_(1) and E_(2)` ?

Options

A. `lambda = (h c)/(E_(1) + E_(2))`
B. `lambda = (2 h c)/(E_(1) + E_(2))`
C. `lambda = (2 h c)/(E_(1) - E_(2))`
D. `lambda = (h c)/(E_(1) - E_(2))`

Explanation

To solve the problem, we need to relate the wavelength of the emitted spectral line (λ) to the energy levels (E₁ and E₂) of the atom.

### Step-by-Step Solution:

1. **Understand the Energy Transition**: When an electron transitions between two energy levels in an atom, it emits or absorbs energy in the form of a photon. The energy of the photon corresponds to the difference in energy between the two levels.

2. **Write the Energy Difference**: The energy difference (ΔE) between the two levels can be expressed as:
\[ \Delta E = E_2 - E_1 \]
(Note: If E₂ is higher than E₁, the transition is from E₂ to E₁, and the energy difference will be E₂ - E₁.)

3. **Relate Energy to Wavelength**: The energy of the emitted photon is also related to its wavelength (λ) by the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \(E\) is the energy of the photon,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \(c\) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)).

4. **Set the Two Expressions Equal**: Since the energy of the photon emitted during the transition is equal to the energy difference between the two levels, we can set the equations equal to each other:
\[ E_2 - E_1 = \frac{hc}{\lambda} \]

5. **Rearrange for Wavelength**: To find the expression for λ, rearrange the equation:
\[ \lambda = \frac{hc}{E_2 - E_1} \]

6. **Final Expression**: The final expression that relates λ, E₁, and E₂ is:
\[ \lambda = \frac{hc}{E_2 - E_1} \]

### Conclusion:
The correct expression relating λ, E₁, and E₂ is:
\[ \lambda = \frac{hc}{E_2 - E_1} \]