The ultraviolet (UV) series of the hydrogen atom corresponds to transitions of electrons from higher energy levels to the first energy level (n = 1). The least energetic spectral line in the UV series corresponds to the transition from the second energy level (n = 2) to the first energy level (n = 1).
The formula to calculate the wavelength of a spectral line in the hydrogen atom is given by:
1/λ = R_H * (1/n₁² - 1/n₂²)
Where:
- λ is the wavelength of the spectral line,
- R_H is the Rydberg constant for hydrogen (approximately 1.097 × 10^7 per meter),
- n₁ is the principal quantum number of the lower energy level, and
- n₂ is the principal quantum number of the higher energy level.
For the transition from n₂ = 2 to n₁ = 1, substituting these values into the formula:
1/λ = 1.097 × 10^7 m⁻¹ * (1/1² - 1/2²) = 1.097 × 10^7 m⁻¹ * (1 - 1/4) = 1.097 × 10^7 m⁻¹ * (3/4) = 8.2275 × 10^6 m⁻¹
To convert this to nanometers (nm), we can use the conversion factor: 1 nm = 10⁻⁹ m
λ = 1 / (8.2275 × 10^6 m⁻¹) = 1.214 nm
Therefore, the wavelength of the least energetic spectral line in the ultraviolet series of the hydrogen atom is approximately 1.214 nm.