Question

Calculate the energy in joules and the wavelength in nano- meters of the spectral line produced in the hydrogen spectrum when an electron falls from the fourth Bohr orbit to the first. (Report your answer using three significant figures.) In which region of the electromagnetic spectrum (UV, visible, or infrared) is the line?

Short answer

Energy released is approximately 2.05 x \(10^{-18}\) joules, wavelength is approximately 96.9 nanometers, falling under the UV region of the spectrum.

Step-by-step solution

Calculate the Energy Released

Use the formula derived from the Rydberg equation for the energy difference between two orbits: \( E = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \), where \( R_H \) is the Rydberg constant for hydrogen (2.18 x \(10^{-18}\) joules), \( n_1 \) is the lower energy level, and \( n_2 \) is the higher energy level. In this case, \( n_1 = 1 \) for the first orbit and \( n_2 = 4 \) for the fourth orbit.

Calculate the Wavelength of the Photon

The energy of a photon (E) is related to its wavelength (\( \lambda \)) by the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant (6.626 x \(10^{-34}\) J·s) and \( c \) is the speed of light (3.00 x \(10^8\) m/s). Rearrange this equation to solve for \( \lambda \): \( \lambda = \frac{hc}{E} \).

Convert the Wavelength to Nanometers

Convert the wavelength from meters to nanometers by multiplying by \(10^9\) because there are \(10^9\) nanometers in a meter.

Determine the Region of the Electromagnetic Spectrum

Compare the calculated wavelength with the general ranges of the electromagnetic spectrum to determine if it is in the UV, visible, or infrared region.

Key concepts

Bohr Model of Hydrogen

The Bohr model of hydrogen, envisioned by Niels Bohr in 1913, was a pivotal step towards understanding atomic structure. This model introduced the concept of quantized electron orbits around the nucleus, meaning electrons can only inhabit certain regions of space with fixed energy levels.

Bohr depicted the hydrogen atom somewhat like a miniature solar system with the nucleus at the center and electrons circling around it in discrete orbits. When an electron jumps from a higher energy orbit to a lower one, it emits a photon whose energy equals the difference in the energy levels of the two orbits.

The lower the energy level's principal quantum number (\( n \)), the closer the orbit is to the nucleus. In exercises like calculating the spectral line produced when an electron falls from the fourth to the first orbit, we apply Bohr's postulates to determine the energy released, a cornerstone in quantum mechanics.

Rydberg Equation for Energy

The Rydberg equation is crucial for understanding atomic spectra. It calculates the energy differences between orbits in an atom based on their quantum numbers. For hydrogen, this formula looks like:
\[ E = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] where \( R_H \) is the Rydberg constant for hydrogen (2.18 x \(10^{-18}\) joules), \( n_1 \) is the principal quantum number of the lower energy orbit, and \( n_2 \) the higher one.

Students often utilize the Rydberg equation to compute the energy emitted or absorbed as an electron transitions between orbits in a hydrogen atom. Step by step, they plug in the designated quantum numbers to reveal the energy in joules, a process that aids them in further calculations, like finding the photon's wavelength.

Photon Wavelength Calculation

To bridge the gap between energy and the wavelength of the emitted or absorbed light, we use the relationship \( E = \frac{hc}{\lambda} \), where \( E \) is the energy of the photon, \( h \) is Planck's constant (6.626 x \(10^{-34}\) J·s), \( c \) is the speed of light in a vacuum (3.00 x \(10^8\) m/s), and \( \lambda \) is the photon's wavelength.

To calculate \( \lambda \), we rearrange the equation: \[ \lambda = \frac{hc}{E} \]. This gives students a clear pathway to deriving the wavelength once the energy is known. Converting between meters and nanometers is a simple multiplication or division by \(10^9\), making it a practical exercise for understanding the nuances of light associated with atomic transitions.

Electromagnetic Spectrum Regions

The electromagnetic spectrum encompasses all types of electromagnetic radiation, from radio waves to gamma rays. It is divided into regions based on wavelengths and frequencies, the common ones being radio, microwave, infrared, visible light, ultraviolet, X-rays, and gamma rays. Each type of electromagnetic radiation has a characteristic range of wavelengths.

For instance, the visible spectrum is roughly from 380 to 750 nanometers, what we perceive as colors from violet to red. Ultraviolet (UV) light has shorter wavelengths, while infrared (IR) lies just beyond the red end of the visible spectrum with longer wavelengths.

When solving for the spectral line of a photon emitted by an electron transition in hydrogen, pinpointing the wavelength is the key to determining if it falls within the UV, visible, or IR ranges. This classification helps in understanding both the theoretical aspect and practical applications of electromagnetic radiation in various fields.