Let `v_(1)` be the frequency of series limit of Lyman series, `v_(2)` the frequency of the first line of Lyman series and `v_(3)` the frequency of series limit of Balmer series. Then which of the following is correct ?

Options

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

Answer

To solve the problem, we need to find the relationship between the frequencies \( v_1 \), \( v_2 \), and \( v_3 \) associated with the Lyman and Balmer series of hydrogen.

### Step-by-Step Solution:

1. **Understanding the Series:**
- The Lyman series corresponds to transitions where the electron falls to the first energy level (n=1).
- The Balmer series corresponds to transitions where the electron falls to the second energy level (n=2).

2. **Frequency of Series Limit:**
- The frequency of the series limit for the Lyman series (\( v_1 \)) occurs when the electron transitions from \( n = \infty \) to \( n = 1 \):
\[ v_1 = \frac{c}{\frac{1}{1^2} - \frac{1}{\infty^2}} = \frac{c}{1 - 0} = c \]

3. **Frequency of the First Line of Lyman Series:**
- The first line of the Lyman series (\( v_2 \)) occurs when the electron transitions from \( n = 2 \) to \( n = 1 \):
\[ v_2 = \frac{c}{\frac{1}{1^2} - \frac{1}{2^2}} = \frac{c}{1 - \frac{1}{4}} = \frac{c}{\frac{3}{4}} = \frac{4c}{3} \]

4. **Frequency of Series Limit of Balmer Series:**
- The frequency of the series limit for the Balmer series (\( v_3 \)) occurs when the electron transitions from \( n = \infty \) to \( n = 2 \):
\[ v_3 = \frac{c}{\frac{1}{2^2} - \frac{1}{\infty^2}} = \frac{c}{\frac{1}{4} - 0} = 4c \]

5. **Finding the Relationship:**
- Now we can find the relationship between \( v_1 \), \( v_2 \), and \( v_3 \):
- We know:
\[ v_1 = c, \quad v_2 = \frac{4c}{3}, \quad v_3 = 4c \]
- To find \( v_1 - v_2 \):
\[ v_1 - v_2 = c - \frac{4c}{3} = \frac{3c}{3} - \frac{4c}{3} = -\frac{c}{3} \]
- We can also express \( v_3 \) in terms of \( v_1 \):
\[ v_3 - v_1 = 4c - c = 3c \]
- Therefore, we can conclude:
\[ v_1 - v_2 = v_3 \]

### Conclusion:
The correct relationship is:
\[ v_1 - v_2 = v_3 \]