Final answer:
The shortest wavelength in the Paschen series is calculated using the Rydberg formula with nf=3 and ni=4, resulting in approximately 1875 nm, which falls in the infrared spectrum. Therefore, it can be concluded that the entire Paschen series consists of infrared light.
Step-by-step explanation:
To show that the entire Paschen series is in the infrared part of the spectrum, we need to calculate the shortest wavelength in the series. The Paschen series corresponds to electronic transitions in the hydrogen atom that terminate at the third energy level (n=3). According to the Rydberg formula, the wavelength (λ) of light emitted for any transition is given by:
1/λ = R * (1/nf² - 1/ni²)
where R is the Rydberg constant (1.097 x 10⁷ m⁻¹), nf is the final energy level, and ni is the initial energy level which is greater than nf.
For the Paschen series, nf=3. The shortest wavelength in this series occurs when the transition is from the next higher energy level (ni=4) to nf. By substituting nf=3 and ni=4, we obtain the shortest wavelength in the Paschen series. Since this shortest wavelength is in the infrared region, all longer wavelengths arising from transitions to the same final state (nf=3) will also be in the infrared.
1/λ = 1.097 x 10⁷ m⁻¹ * (1/3² - 1/4²)
1/λ = 1.097 x 10⁷ m⁻¹ * (1/9 - 1/16)
1/λ = 1.097 x 10⁷ m⁻¹ * (7/144)
λ ≈ 1.875 x 10⁻⁶ m or 1875 nm
Therefore, since the shortest wavelength of the Paschen series is around 1875 nm, which falls into the infrared part of the electromagnetic spectrum, we can conclude that all the transitions in the Paschen series will produce infrared light.