What is the wave length of ${\mathrm{H}}_{\mathrm{\beta }}$ line in Balmer series of hydrogen spectrum? (R = Rydberg constant)

The wavelength of the Hβ line in the Balmer series of the hydrogen spectrum is approximately 486.1 nanometers (nm), and it is calculated using the Rydberg formula.

The Balmer series in the hydrogen spectrum includes lines that are visible to the human eye. The lines occur when an electron falls from a higher energy level (n) to the second energy level (n = 2) in a hydrogen atom. The Hβ line corresponds to a transition from the energy level n = 4 to n = 2.

Important Formula: Rydberg Formula

To calculate the wavelength of the Hβ line, we use the Rydberg formula for the hydrogen atom:

1/λ = R × (1/2² - 1/n²)

Where:

• λ = Wavelength of the line
• R = Rydberg constant = 1.097 × 10⁷ m^-1
• n = Higher energy level (n = 4 for Hβ line)

Step-by-Step Calculation

1. For the Hβ line, the transition is from n = 4 to n = 2.

2. Apply the values in the formula:

   1/λ = R × (1/2² - 1/4²)
   1/λ = R × (1/4 - 1/16)       

3. Calculate the fractions:

   1/λ = R × (4/16 - 1/16) = R × (3/16)       

4. Now, put the value of R = 1.097 × 10⁷ m^-1:

   1/λ = (1.097 × 107) × (3/16)
   1/λ = 2.056 × 106 m-1    

5. Find λ by taking reciprocal:

   λ = 1 / (2.056 × 106) m
   λ ≈ 4.861 × 10-7 m        

6. Convert meters to nanometers (1 m = 10⁹ nm):

   λ ≈ 486.1 nm   

Final Answer:

The wavelength of the Hβ line in the Balmer series is approximately 486.1 nanometers (nm).