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Chapter 6: Problem 89

Calculate the wavelength of the first 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 first line in the Lyman series is calculated using the Rydberg formula for hydrogen: \(\frac{1}{\lambda} = R_H \left(\frac{1}{n_{1}^2} - \frac{1}{n_{2}^2}\right)\), where \(R_H\) is the Rydberg constant, \(n_1 = 1\), and \(n_2 = 2\). Substituting the values, we get \(\lambda = 1.215\times 10^{-7} m = 121.5 \, nm\). Since the ultraviolet range is between 10 and 400 nm, the first line in the Lyman series lies in the ultraviolet part of the spectrum.

Step by step solution

01

Calculate the wavelength of the first line in the Lyman series

For the Lyman series transitions, we can use the following Rydberg formula for hydrogen: \[\frac{1}{\lambda} = R_H \left(\frac{1}{n_{1}^2} - \frac{1}{n_{2}^2}\right)\] where: - \(\lambda\) is the wavelength of the spectral line, - \(R_H\) is the Rydberg constant for hydrogen, approximately 1.097 x 10^7 m−1, - \(n_1\) is the final energy level (for the Lyman series, n1 = 1), - \(n_2\) is the initial energy level (for the first line in the series, n2 = 2). We will now solve for the wavelength of the first line in the Lyman series.
02

Plug in the values for the first line in the Lyman series

We will plugin n1 = 1 and n2 = 2 into the Rydberg formula: \[\frac{1}{\lambda} = R_H \left(\frac{1}{1^2} - \frac{1}{2^2}\right)\]
03

Calculate the wavelength value

Now, let's solve for the wavelength, \(\lambda\): \[\frac{1}{\lambda} = 1.097\times 10^7 \left(\frac{1}{1} - \frac{1}{4}\right)\] \[\frac{1}{\lambda} = 1.097\times 10^7 \left(\frac{3}{4}\right)\] \[\frac{1}{\lambda} = 8.228\times 10^6 m^{-1}\] To get the wavelength (\(\lambda\)), we take the reciprocal of the value above: \[\lambda = \frac{1}{8.228\times 10^6}\]
04

Determine the wavelength

So, the wavelength of the first line in the Lyman series is: \[\lambda = 1.215\times 10^{-7} m = 121.5 \, nm\]
05

Checking if the first line lies in the ultraviolet (UV) part of the spectrum

Ultraviolet light has wavelengths between 10 and 400 nm. Since our calculated wavelength for the first line in the Lyman series (121.5 nm) falls within this range, we can conclude that the first line in the Lyman series lies in the ultraviolet part of the spectrum.

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