Question

Calculate the wavelength of light emitted when each of the following transitions occur in the hydrogen atom. What type of electromagnetic radiation is emitted in each transition? a. \(n=3 \rightarrow n=2\) b. \(n=4 \rightarrow n=2\) c. \(n=2 \rightarrow n=1\)

Short answer

a. The transition from \(n=3\) to \(n=2\) results in the emission of visible light with a wavelength of approximately \(656.1\) nm. b. The transition from \(n=4\) to \(n=2\) results in the emission of visible light with a wavelength of approximately \(486.1\) nm. c. The transition from \(n=2\) to \(n=1\) results in the emission of ultraviolet light with a wavelength of approximately \(121.5\) nm.

Step-by-step solution

Apply the Rydberg Formula

Calculate the wavelength using the Rydberg formula: \[\frac{1}{\lambda} = R_H\left(\frac{1}{2^2} - \frac{1}{3^2}\right)\]

Solve for the Wavelength

Obtain the wavelength (\(\lambda\)) and convert it to nanometers (nm): \[\lambda = \frac{1}{R_H\left(\frac{1}{2^2} - \frac{1}{3^2}\right)} \approx 656.1\, nm\]

Determine Electromagnetic Radiation Type

Since \(390 \leq \lambda \leq 750\) nm, the electromagnetic radiation emitted is in the visible light range. b. \(n = 4 \rightarrow n = 2\)

Apply the Rydberg Formula

Calculate the wavelength using the Rydberg formula: \[\frac{1}{\lambda} = R_H\left(\frac{1}{2^2} - \frac{1}{4^2}\right)\]

Solve for the Wavelength

Obtain the wavelength (\(\lambda\)) and convert it to nanometers (nm): \[\lambda = \frac{1}{R_H\left(\frac{1}{2^2} - \frac{1}{4^2}\right)} \approx 486.1\, nm\]

Determine Electromagnetic Radiation Type

Since \(390 \leq \lambda \leq 750\) nm, the electromagnetic radiation emitted is in the visible light range. c. \(n = 2 \rightarrow n = 1\)

Apply the Rydberg Formula

Calculate the wavelength using the Rydberg formula: \[\frac{1}{\lambda} = R_H\left(\frac{1}{1^2} - \frac{1}{2^2}\right)\]

Solve for the Wavelength

Obtain the wavelength (\(\lambda\)) and convert it to nanometers (nm): \[\lambda = \frac{1}{R_H\left(\frac{1}{1^2} - \frac{1}{2^2}\right)} \approx 121.5\, nm\]

Determine Electromagnetic Radiation Type

Since \(\lambda < 390\) nm, the electromagnetic radiation emitted is in the ultraviolet range.