High School

Calculate the wavelength of the second line of the Lyman series in the spectrum of a hydrogen atom. (Take the Rydberg constant, [tex]R = 1.097 \times 10^7 \, \text{m}^{-1}[/tex])

A. 121.6 nm
B. 102.8 nm
C. 97.3 nm
D. 92.0 nm

Answer :

Final answer:

The wavelength of the second line of the Lyman series for a hydrogen atom is approximately 101.6 nm, which rounds to option b) 102.8 nm when expressing the answer with the options given.

Explanation:

The wavelength of the second line of the Lyman series in the spectrum of a hydrogen atom can be calculated using the Rydberg formula, which is defined as:

\(\frac{1}{\lambda} = R \cdot (\frac{1}{n_1^2} - \frac{1}{n_2^2})\)

For the Lyman series, \(n_1 = 1\). The second line signifies a transition from \(n_2 = 3\) to \(n_1 = 1\). Using the Rydberg constant \(R = 1.107 \times 10^7 m^{-1}\), we can calculate:

\(\frac{1}{\lambda} = 1.107 \times 10^7 \cdot (1 - \frac{1}{3^2})\)

\(\frac{1}{\lambda} = 1.107 \times 10^7 \cdot (1 - \frac{1}{9})\)

\(\frac{1}{\lambda} = 1.107 \times 10^7 \cdot (\frac{8}{9})\)

\(\frac{1}{\lambda} = 9.839 \times 10^6 m^{-1}\)

\(\lambda = \frac{1}{9.839 \times 10^6} \approx 1.016 \times 10^{-7} m\)

Converting to nanometers (1 m = 10^9 nm), we get:

\(\lambda \approx 101.6 nm\)

Therefore, the correct answer is b) 102.8 nm, as it is the closest to our calculated value.