The ratio of frequencies of the first line of the Lyman series and the first line of Balmer series.is `(p)/(q)`. Find (p+q).

Answer

To find the ratio of frequencies of the first line of the Lyman series and the first line of the Balmer series, we will follow these steps:

### Step 1: Understand the Lyman and Balmer Series
The Lyman series corresponds to transitions where the electron falls to the first energy level (n=1) from higher levels (n=2, 3, ...). The first line of the Lyman series is the transition from n=2 to n=1.

The Balmer series corresponds to transitions where the electron falls to the second energy level (n=2) from higher levels (n=3, 4, ...). The first line of the Balmer series is the transition from n=3 to n=2.

### Step 2: Use the Rydberg Formula
The frequency of the emitted light during these transitions can be calculated using the Rydberg formula:

\[ \mu = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

where:
- \( \mu \) is the frequency,
- \( R_H \) is the Rydberg constant,
- \( n_1 \) and \( n_2 \) are the principal quantum numbers of the lower and upper energy levels, respectively.

### Step 3: Calculate Frequency for the Lyman Series
For the first line of the Lyman series (n=2 to n=1):

\[ \mu_{Lyman} = R_H \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R_H \left( 1 - \frac{1}{4} \right) = R_H \left( \frac{3}{4} \right) \]

### Step 4: Calculate Frequency for the Balmer Series
For the first line of the Balmer series (n=3 to n=2):

\[ \mu_{Balmer} = R_H \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R_H \left( \frac{1}{4} - \frac{1}{9} \right) \]

To simplify:

\[ \frac{1}{4} - \frac{1}{9} = \frac{9 - 4}{36} = \frac{5}{36} \]

Thus,

\[ \mu_{Balmer} = R_H \left( \frac{5}{36} \right) \]

### Step 5: Calculate the Ratio of Frequencies
Now we can find the ratio of the frequencies:

\[ \frac{\mu_{Lyman}}{\mu_{Balmer}} = \frac{R_H \left( \frac{3}{4} \right)}{R_H \left( \frac{5}{36} \right)} = \frac{3}{4} \cdot \frac{36}{5} = \frac{3 \cdot 36}{4 \cdot 5} = \frac{108}{20} = \frac{27}{5} \]

### Step 6: Identify p and q
From the ratio \( \frac{27}{5} \), we can identify \( p = 27 \) and \( q = 5 \).

### Step 7: Calculate p + q
Finally, we find \( p + q \):

\[ p + q = 27 + 5 = 32 \]

### Final Answer
Thus, the answer is \( 32 \).

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