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. \(n=4 \rightarrow n=3\) b. \(n=5 \rightarrow n=4\) c. \(n=5 \rightarrow n=3\)

Short answer

The wavelengths of the light emitted during the transitions are: a. \(1.28 \times 10^{-6}\,\text{m}\) (Infrared) b. \(1.63 \times 10^{-6}\,\text{m}\) (Infrared) c. \(6.98 \times 10^{-7}\,\text{m}\) (Visible light, red)

Step-by-step solution

Recall the Rydberg formula for Hydrogen

The Rydberg formula for hydrogen 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, \(R_H\) is the Rydberg constant (approximately \(1.097 \times 10^7 m^{-1}\)), \(n_1\) is the initial energy level, and \(n_2\) is the final energy level of the transition.

Calculate the emitted wavelength for each transition

Use the Rydberg formula to calculate the wavelength of the light emitted during each transition. a. For \(n_1 = 4\) and \(n_2 = 3\), we plug in the values and solve for \(\lambda\): \[ \frac{1}{\lambda} = R_H \left( \frac{1}{4^2} - \frac{1}{3^2} \right) \] \[ \lambda = (\frac{1}{R_H} * \frac{1}{\frac{1}{4^2} - \frac{1}{3^2}})^{-1} \] \[ \lambda \approx 1.28 \times 10^{-6}\,\text{m} \] b. For \(n_1 = 5\) and \(n_2 = 4\), we plug in the values and solve for \(\lambda\): \[ \frac{1}{\lambda} = R_H \left( \frac{1}{5^2} - \frac{1}{4^2} \right) \] \[ \lambda = (\frac{1}{R_H} * \frac{1}{\frac{1}{5^2} - \frac{1}{4^2}})^{-1} \] \[ \lambda \approx 1.63 \times 10^{-6}\,\text{m} \] c. For \(n_1 = 5\) and \(n_2 = 3\), we plug in the values and solve for \(\lambda\): \[ \frac{1}{\lambda} = R_H \left( \frac{1}{5^2} - \frac{1}{3^2} \right) \] \[ \lambda = (\frac{1}{R_H} * \frac{1}{\frac{1}{5^2} - \frac{1}{3^2}})^{-1} \] \[ \lambda \approx 6.98 \times 10^{-7}\,\text{m} \]

Identify the type of electromagnetic radiation for each transition

Based on the calculated wavelengths, we can determine the type of electromagnetic radiation associated with each transition using the electromagnetic spectrum. a. The wavelength \(\lambda \approx 1.28 \times 10^{-6}\,\text{m}\) corresponds to infrared b. The wavelength \(\lambda \approx 1.63 \times 10^{-6}\,\text{m}\) corresponds to infrared c. The wavelength \(\lambda \approx 6.98 \times 10^{-7}\,\text{m}\) corresponds to visible light (red) So, the types of electromagnetic radiation emitted in each transition are: a. Infrared b. Infrared c. Visible light (red)