The Paschen series refers to a series of spectral lines in the emission spectrum of hydrogen. The spectral lines in the Paschen series are produced when an electron transitions from higher energy levels to the third energy level (n = 3) in the hydrogen atom.
The formula to calculate the wavelength of the spectral lines in the hydrogen atom is given by the Rydberg formula:
1/λ = R_H * (1/n_f^2 - 1/n_i^2)
Where:
- λ represents the wavelength of the spectral line.
- R_H is the Rydberg constant for hydrogen, approximately 1.097 × 10^7 per meter.
- n_i is the initial energy level.
- n_f is the final energy level.
For the 3rd spectral line in the Paschen series, n_i = ∞ (infinity) because the electron transitions from an energy level very far away to the n = 3 energy level. So the formula becomes:
1/λ = R_H * (1/3^2 - 1/∞^2)
Since 1/∞^2 is negligible, the equation simplifies to:
1/λ = R_H * (1/9)
Now we can solve for λ:
λ = 1 / (R_H * (1/9))
Plugging in the value of the Rydberg constant (R_H), we can calculate the wavelength. However, it seems you have requested the wavelength of the 3rd spectral line in the Paschen series specifically. The Paschen series typically refers to the transitions to the n = 3 energy level, but within that series, there are multiple spectral lines. To provide the exact wavelength for the 3rd spectral line, it would be helpful to know which transition specifically you are referring to (e.g., n = 4 to n = 3, n = 5 to n = 3, etc.).