In the hydrogen atom spectrum, `lambda_(3-1) and lambda_(2-1)` represent wavelengths emitted due to transition from second and first excited states to the ground state respectively . The ratio `(lambda_(3-1))/(lambda_(2-1))=(p)/(q)`, where p and q are the smallest positive integers . What is the value of ( p + q ) ?

Answer

To solve the problem, we need to find the ratio of the wavelengths emitted during the transitions from the second and first excited states to the ground state in a hydrogen atom. The transitions are represented by \( \lambda_{3-1} \) and \( \lambda_{2-1} \).

### Step-by-step Solution:

1. **Understand the Transitions**:
- \( \lambda_{3-1} \): This corresponds to the transition from the third energy level (n=3) to the ground state (n=1).
- \( \lambda_{2-1} \): This corresponds to the transition from the second energy level (n=2) to the ground state (n=1).

2. **Use the Rydberg Formula**:
The wavelength of the emitted light during these transitions can be calculated using the Rydberg formula:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R \) is the Rydberg constant, \( n_1 \) is the lower energy level, and \( n_2 \) is the higher energy level.

3. **Calculate \( \lambda_{3-1} \)**:
For the transition from n=3 to n=1:
\[ \frac{1}{\lambda_{3-1}} = R \left( \frac{1}{1^2} - \frac{1}{3^2} \right) = R \left( 1 - \frac{1}{9} \right) = R \left( \frac{8}{9} \right) \]
Therefore,
\[ \lambda_{3-1} = \frac{9}{8R} \]

4. **Calculate \( \lambda_{2-1} \)**:
For the transition from n=2 to n=1:
\[ \frac{1}{\lambda_{2-1}} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \]
Therefore,
\[ \lambda_{2-1} = \frac{4}{3R} \]

5. **Find the Ratio \( \frac{\lambda_{3-1}}{\lambda_{2-1}} \)**:
Now, we can find the ratio of the two wavelengths:
\[ \frac{\lambda_{3-1}}{\lambda_{2-1}} = \frac{\frac{9}{8R}}{\frac{4}{3R}} = \frac{9}{8} \cdot \frac{3}{4} = \frac{27}{32} \]

6. **Identify \( p \) and \( q \)**:
From the ratio \( \frac{\lambda_{3-1}}{\lambda_{2-1}} = \frac{27}{32} \), we can identify \( p = 27 \) and \( q = 32 \).

7. **Calculate \( p + q \)**:
Finally, we calculate \( p + q \):
\[ p + q = 27 + 32 = 59 \]

### Final Answer:
The value of \( p + q \) is \( 59 \).