Transition between three energy energy levels in a particular atom give rise to three Spectral line of wevelength , in increasing magnitudes. `lambda_(1), lambda_(2)` and `lambda_(3)`. Which one of the following equations correctly ralates `lambda_(1), lambda_(2)` and `lambda_(3)`?
`lambda_(1)=lambda_(2)-lambda_(3)`
`lambda_(1)=lambda_(3)-lambda_(2)`
`(1)/(lambda_(1))=(1)/(lambda_(2))+(1)/(lambda_(3))`
`(1)/(lambda_(2))=(1)/(lambda_(3))+(1)/(lambda_(1))`

Options

A. `lambda_(1)=lambda_(2)-lambda_(3)`
B. `lambda_(1)=lambda_(3)-lambda_(2)`
C. `(1)/(lambda_(1))=(1)/(lambda_(2))+(1)/(lambda_(3))`
D. `(1)/(lambda_(2))=(1)/(lambda_(3))+(1)/(lambda_(1))`

Answer

To solve the problem, we need to understand the relationships between the wavelengths of the spectral lines produced by transitions between three energy levels in an atom. Let's denote the wavelengths of the spectral lines as \( \lambda_1, \lambda_2, \) and \( \lambda_3 \) in increasing order of their magnitudes.

### Step-by-Step Solution:

1. **Understanding Energy Levels**:
- We have three energy levels in the atom, which we can denote as \( E_1, E_2, \) and \( E_3 \). The transitions between these energy levels result in the emission of photons with wavelengths \( \lambda_1, \lambda_2, \) and \( \lambda_3 \).

2. **Energy of Photons**:
- The energy of a photon is given by the equation:
\[ E = \frac{hc}{\lambda} \]
- Where \( h \) is Planck's constant and \( c \) is the speed of light.

3. **Energy Differences**:
- The energy differences corresponding to the transitions can be expressed as:
- For the transition from \( E_2 \) to \( E_1 \): \( E_1 = \frac{hc}{\lambda_1} \)
- For the transition from \( E_3 \) to \( E_2 \): \( E_2 = \frac{hc}{\lambda_2} \)
- For the transition from \( E_3 \) to \( E_1 \): \( E_3 = \frac{hc}{\lambda_3} \)

4. **Relating Energies**:
- According to the problem, the energy levels decrease as the wavelength increases. Thus, we can write:
\[ E_1 = E_2 + E_3 \]
- Substituting the expressions for energy, we have:
\[ \frac{hc}{\lambda_1} = \frac{hc}{\lambda_2} + \frac{hc}{\lambda_3} \]

5. **Canceling \( hc \)**:
- Since \( hc \) is a common factor, we can cancel it from both sides:
\[ \frac{1}{\lambda_1} = \frac{1}{\lambda_2} + \frac{1}{\lambda_3} \]

6. **Final Relation**:
- Rearranging gives us the final relation:
\[ \frac{1}{\lambda_1} = \frac{1}{\lambda_2} + \frac{1}{\lambda_3} \]

### Conclusion:
The correct equation that relates \( \lambda_1, \lambda_2, \) and \( \lambda_3 \) is:
\[ \frac{1}{\lambda_1} = \frac{1}{\lambda_2} + \frac{1}{\lambda_3} \]
This corresponds to the third option provided in the question.