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 of a hydrogen atom from the second excited state (n=3) and the first excited state (n=2) to the ground state (n=1). We will use the Rydberg formula for hydrogen:

\[ \frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]

where:
- \( R \) is the Rydberg constant,
- \( n_f \) is the final energy level,
- \( n_i \) is the initial energy level.

### Step 1: Calculate \( \lambda_{3-1} \)

For the transition from n=3 to n=1:

\[ n_f = 1, \quad n_i = 3 \]

Using the Rydberg formula:

\[ \frac{1}{\lambda_{3-1}} = R \left( \frac{1}{1^2} - \frac{1}{3^2} \right) \]

Calculating the terms:

\[ \frac{1}{\lambda_{3-1}} = R \left( 1 - \frac{1}{9} \right) = R \left( \frac{8}{9} \right) \]

Thus, we have:

\[ \lambda_{3-1} = \frac{9}{8R} \]

### Step 2: Calculate \( \lambda_{2-1} \)

For the transition from n=2 to n=1:

\[ n_f = 1, \quad n_i = 2 \]

Using the Rydberg formula:

\[ \frac{1}{\lambda_{2-1}} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \]

Calculating the terms:

\[ \frac{1}{\lambda_{2-1}} = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \]

Thus, we have:

\[ \lambda_{2-1} = \frac{4}{3R} \]

### Step 3: Find the ratio \( \frac{\lambda_{3-1}}{\lambda_{2-1}} \)

Now we can find the ratio of the two wavelengths:

\[ \frac{\lambda_{3-1}}{\lambda_{2-1}} = \frac{\frac{9}{8R}}{\frac{4}{3R}} = \frac{9}{8} \cdot \frac{3}{4} = \frac{27}{32} \]

### Step 4: Identify \( p \) and \( q \)

From the ratio \( \frac{\lambda_{3-1}}{\lambda_{2-1}} = \frac{27}{32} \), we can see that:

- \( p = 27 \)
- \( q = 32 \)

### Step 5: Calculate \( p + q \)

Now, we find \( p + q \):

\[ p + q = 27 + 32 = 59 \]

Thus, the final answer is:

\[ \boxed{59} \]