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Chapter 6: Problem 90

Calculate the wavelength of the fifth line in the Lyman series and show that this line lies in the ultraviolet part of the spectrum.

Short Answer

Expert verified
The wavelength of the fifth line in the Lyman series is calculated using the Rydberg formula with electron transitions from n = 6 to n = 1. The resulting wavelength is found to be \(\lambda = 9.724 \times 10^{-8} m\), which lies within the ultraviolet part of the spectrum, typically in the range of \(10^{-9} m\) to \(4 \times 10^{-7} m\).

Step by step solution

01

Recall the Rydberg Formula

The Rydberg formula is used to calculate the wavelength of electromagnetic radiation emitted during electronic transitions in atoms, particularly hydrogen. The formula is given by: \( \frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \) where, - \(\lambda\) is the wavelength of the emitted light - \(R_H\) is the Rydberg constant for hydrogen (approximately \(1.097 \times 10^7 m^{-1}\)) - \(n_1\) and \(n_2\) are the initial and final energy levels, with \(n_1 < n_2\).
02

Plug in the energy levels for the fifth line of the Lyman series

We are given that the fifth line of the Lyman series corresponds to a transition from n = 6 to n = 1. Thus, we plug in these values for \(n_1\) and \(n_2\): \( \frac{1}{\lambda} = R_H \left(\frac{1}{1^2} - \frac{1}{6^2}\right) \)
03

Calculate the wavelength

Solve for \(\lambda\) by plugging the value of the Rydberg constant (\(R_H = 1.097 \times 10^7 m^{-1}\)): \( \frac{1}{\lambda} = 1.097 \times 10^7 \left(\frac{1}{1} - \frac{1}{36}\right) \) \( \frac{1}{\lambda} = 1.097 \times 10^7 \left(\frac{35}{36}\right) \) Dividing both sides by \(1.097 \times 10^7\) and then inverting yields the wavelength: \( \lambda = \frac{1}{1.097 \times 10^7 \times \frac{35}{36}} \) \( \lambda = 9.724 \times 10^{-8} m \)
04

Determine whether the wavelength is in the ultraviolet part of the spectrum

The ultraviolet (UV) part of the spectrum typically has wavelengths in the range of \(10^{-9} m\) to \(4 \times 10^{-7} m\). Since the calculated wavelength of the fifth line in the Lyman series is \(9.724 \times 10^{-8} m\), it falls within the UV range. Therefore, we have shown that the fifth line of the Lyman series lies in the ultraviolet part of the spectrum.

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