Calculate the wavelength of the radiations in nanometers emitted when an electron in hydrogen atom jumps from third orbit to the ground state. `(R_(H)=109677 cm^(-1))`

Explanation

To calculate the wavelength of the radiation emitted when an electron in a hydrogen atom jumps from the third orbit to the ground state, we can follow these steps:

### Step 1: Identify the Initial and Final Energy Levels
- The initial energy level (N2) is 3 (third orbit).
- The final energy level (N1) is 1 (ground state).

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wavelength (λ) of emitted radiation is given by:

\[ \frac{1}{\lambda} = R_H \left( \frac{1}{N_1^2} - \frac{1}{N_2^2} \right) \]

Where:
- \( R_H \) is the Rydberg constant, given as \( R_H = 109677 \, \text{cm}^{-1} \).
- \( N_1 = 1 \)
- \( N_2 = 3 \)

### Step 3: Substitute the Values into the Formula
Substituting the values into the Rydberg formula:

\[ \frac{1}{\lambda} = 109677 \left( \frac{1}{1^2} - \frac{1}{3^2} \right) \]

Calculating the right side:

\[ \frac{1}{\lambda} = 109677 \left( 1 - \frac{1}{9} \right) = 109677 \left( \frac{9 - 1}{9} \right) = 109677 \left( \frac{8}{9} \right) \]

\[ \frac{1}{\lambda} = 109677 \times \frac{8}{9} = 97480.67 \, \text{cm}^{-1} \]

### Step 4: Calculate the Wavelength (λ)
Now, take the inverse to find λ:

\[ \lambda = \frac{1}{97480.67} \, \text{cm} \]

Calculating this gives:

\[ \lambda \approx 1.03 \times 10^{-5} \, \text{cm} \]

### Step 5: Convert to Nanometers
To convert from centimeters to nanometers, we use the conversion factor \( 1 \, \text{cm} = 10^7 \, \text{nm} \):

\[ \lambda \approx 1.03 \times 10^{-5} \, \text{cm} \times 10^7 \, \text{nm/cm} = 1.03 \times 10^2 \, \text{nm} = 103 \, \text{nm} \]

### Final Answer
The wavelength of the radiation emitted when an electron in a hydrogen atom jumps from the third orbit to the ground state is approximately **103 nm**.

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