The third longest wavelength in the Paschen series in hydrogen corresponds to what transitions?
To determine the transitions corresponding to the third longest wavelength in the Paschen series in hydrogen, we need to understand the Paschen series and the energy levels involved.
The Paschen series is a series of spectral lines in the emission spectrum of hydrogen. It occurs when electrons transition from higher energy levels to the third energy level (n = 3) in the hydrogen atom. These transitions result in the emission of photons with specific wavelengths.
The energy of an electron in a hydrogen atom can be described by the equation:
Eₙ = -13.6 eV / n²
Where Eₙ is the energy of the electron, -13.6 eV is the ionization energy of the hydrogen atom, and n is the principal quantum number (indicating the energy level).
The wavelengths of the spectral lines emitted can be calculated using the Rydberg formula:
1/λ = R_H (1/n₁² - 1/n₂²)
Where λ is the wavelength, R_H is the Rydberg constant for hydrogen (approximately 1.097 × 10⁷ m⁻¹), n₁ and n₂ are the initial and final energy levels respectively.
For the third longest wavelength (λ₃), we need to find the transition where the difference between the initial and final energy levels (1/n₁² - 1/n₂²) is the smallest while still being larger than the transitions with shorter wavelengths.
The formula for the difference in energy levels can be rearranged as:
1/λ₃ = R_H (1/n₁² - 1/n₂²)
To find the third longest wavelength, we can start by considering transitions from higher energy levels to the third energy level (n = 3). We calculate the differences (1/n₁² - 1/3²) for different initial energy levels (n₁) greater than 3 and find the smallest difference that is larger than the shorter wavelength transitions.
For example, let's consider the transition from the 7th energy level (n₁ = 7) to the 3rd energy level (n₂ = 3):
1/λ₃ = R_H (1/7² - 1/3²)
1/λ₃ = R_H (1/49 - 1/9)
1/λ₃ = R_H (9/441 - 49/441)
1/λ₃ = R_H (-40/441)
λ₃ = -441/40R_H
By calculating this expression, we can determine the value of the third longest wavelength in the Paschen series in hydrogen.