Question

The ratio of the wavelength of spectral lines \( H_\alpha \) and \( H_\beta \) in the Balmer series is \( \frac{x}{20} \). The value of \( x \) is _______.

Answer 1

The Balmer series involves transitions where an electron in a hydrogen atom drops to the n=2 energy level. The wavelengths of the emitted photons are described by the Rydberg formula:

1/λ = R (1/2² - 1/n²)

Where:

• λ is the wavelength of the emitted photon
• R is the Rydberg constant (approximately 1.097 x 10⁷ m^-1)
• n is the principal quantum number of the initial energy level (n > 2)

Hα and Hβ Transitions

• Hα: This corresponds to the transition from n=3 to n=2.
• Hβ: This corresponds to the transition from n=4 to n=2.

Calculating the Wavelengths

1. Wavelength of Hα (λ_α):
   1/λ_α = R (1/2² - 1/3²) = R (1/4 - 1/9) = R (5/36)
   λ_α = 36 / (5R)
2. Wavelength of Hβ (λ_β):
   1/λ_β = R (1/2² - 1/4²) = R (1/4 - 1/16) = R (3/16)
   λ_β = 16 / (3R)

Finding the Ratio λ_α/λ_β

(λ_α / λ_β) = (36 / 5R) / (16 / 3R) = (36 / 5R) * (3R / 16) = (36 * 3) / (5 * 16) = 108 / 80 = 27/20

Determining the Value of x

The problem states that the ratio λ_α/λ_β = x/20. We found that the ratio is 27/20. Therefore, x = 27.