(a) Using Equation 6.5, calculate the energy of an electron...

(a) Using Equation $6.5,$ calculate the energy of an electron in the hydrogen atom when $n=2$ and when $n=6 .$ Calculate the wavelength of the radiation released when an electron moves from $n=6$ to $n=2 .$ (b) Is this line in the visible region of the electromagnetic spectrum? If so, what color is it?

Key Concepts

Visible Spectrum Characteristics

The visible region of the electromagnetic spectrum spans wavelengths approximately between 400 nm and 700 nm. Whether a spectral line falls within this range determines if it is visible to the human eye, and its exact wavelength can be associated with a particular color. This concept is important when analyzing spectral lines to infer the optical characteristics and colors associated with atomic transitions.

Energy Quantization in the Hydrogen Atom

This concept refers to the principle that an electron in a hydrogen atom can only occupy certain discrete energy levels, typically described by the quantum number n. The energy associated with each level is given by a formula (often involving a constant such as 13.6 eV for hydrogen) and inversely proportional to the square of n. This quantization is a fundamental result of the Bohr model and later quantum mechanics, demonstrating that electrons cannot exist at arbitrary energy levels but only at specific, allowed states.

Electron Transitions and Photon Emission

When an electron transitions between these quantized energy levels, the atom either absorbs or emits a photon with an energy exactly equal to the difference between the initial and final energy states. This process underlies the emission and absorption spectra of atoms. The energy released during a downward transition (from higher to lower energy states) manifests as electromagnetic radiation, with a wavelength that can be calculated if the energy difference is known.

Photon Energy-Wavelength Relationship

This key concept describes the fundamental relationship between the energy of a photon and its wavelength, typically given by the formula E = hc/?, where h is Planck’s constant and c is the speed of light. This equation allows one to convert the energy difference between quantized states into the corresponding wavelength of the emitted or absorbed radiation, linking quantum transitions to observable spectral lines.

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