The wavelength of the spectral line corresponding to a transition in hydrogen from the n = 10 state to the lowest ground state (n = 1) can be determined using the Rydberg formula. The Rydberg formula provides a relationship between the wavelength (λ) of the emitted or absorbed photon and the principal quantum numbers (n) involved in the transition.
The formula for the calculation of the wavelength is:
1/λ = R_H * (1/n₁² - 1/n₂²)
Where: λ is the wavelength of the emitted or absorbed light, R_H is the Rydberg constant for hydrogen (approximately 1.097 x 10^7 per meter), n₁ is the initial principal quantum number (n = 10), n₂ is the final principal quantum number (n = 1).
Substituting the values into the formula:
1/λ = (1.097 x 10^7 m⁻¹) * (1/1² - 1/10²) = (1.097 x 10^7 m⁻¹) * (1 - 1/100) = (1.097 x 10^7 m⁻¹) * (99/100) ≈ 1.087 x 10^7 m⁻¹
Taking the reciprocal of both sides, we find:
λ ≈ 9.200 x 10^(-8) meters
So, the wavelength of the spectral line for the transition from the n = 10 state to the n = 1 state in hydrogen is approximately 9.200 x 10^(-8) meters, or 92.00 nanometers (nm).
In terms of the electromagnetic spectrum, this wavelength falls in the ultraviolet (UV) region. Specifically, it lies in the far ultraviolet range, which is beyond the range of visible light.