The relation between `lambda_(1)=` wavelength of series limit of Lyman series ,`lambda_(2)=`the wavelength of the series limit of Balmer sereis & `lambda_(3)=` the wavelength of first line of Lyman series:

Options

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

Explanation

To derive the relationship between the wavelengths \( \lambda_1 \), \( \lambda_2 \), and \( \lambda_3 \), we need to understand the transitions of electrons in hydrogen-like atoms.

### Step 1: Identify the transitions
- The Lyman series corresponds to transitions where the electron falls to the ground state (\( n = 1 \)).
- The Balmer series corresponds to transitions where the electron falls to the first excited state (\( n = 2 \)).
- The series limit of the Lyman series (\( \lambda_1 \)) occurs when the electron transitions from \( n = \infty \) to \( n = 1 \).
- The series limit of the Balmer series (\( \lambda_2 \)) occurs when the electron transitions from \( n = \infty \) to \( n = 2 \).
- The first line of the Lyman series (\( \lambda_3 \)) corresponds to the transition from \( n = 2 \) to \( n = 1 \).

### Step 2: Write the energy equations
The energy of a photon emitted during these transitions can be expressed as:
- For Lyman series limit:
\[ E_1 = \frac{hc}{\lambda_1} \]
- For Balmer series limit:
\[ E_2 = \frac{hc}{\lambda_2} \]
- For the first line of Lyman series:
\[ E_3 = \frac{hc}{\lambda_3} \]

### Step 3: Set up the energy equality
At the series limit of Lyman and Balmer series, the energy released is equal:
\[ E_1 = E_2 \]
Thus:
\[ \frac{hc}{\lambda_1} = \frac{hc}{\lambda_2} + \frac{hc}{\lambda_3} \]

### Step 4: Simplify the equation
Cancel \( hc \) from both sides:
\[ \frac{1}{\lambda_1} = \frac{1}{\lambda_2} + \frac{1}{\lambda_3} \]

### Step 5: Rearranging the equation
Rearranging gives us the final relationship:
\[ \frac{1}{\lambda_1} - \frac{1}{\lambda_2} = \frac{1}{\lambda_3} \]

### Final Result
The relationship between the wavelengths is:
\[ \frac{1}{\lambda_1} - \frac{1}{\lambda_2} = \frac{1}{\lambda_3} \]