Problem 6 - University Physics with Modern Physics

Question

(a) A molecule decreases its vibrational energy by 0.250 \(\mathrm{eV}\) by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electromagnetic spectrum does that wavelength of light lie? (b) An atom decreases its energy by 8.50 eV by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electromagnetic spectrum does that wavelength of light lie? (c) A molecule decreases its rotational energy by \(3.20 \times 10^{-3} \mathrm{eV}\) by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electromagnetic spectrum does that wavelength of light lie?

Step-by-Step Solution

Answer

(a) 4960 nm, infrared; (b) 91.3 nm, ultraviolet; (c) 3880 nm, infrared.

Step 1: Formula for Calculating Wavelength

To find the wavelength of light, use the formula \( \lambda = \frac{hc}{E} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J·s} \), \( c \) is the speed of light \( 3.00 \times 10^8 \text{ m/s} \), and \( E \) is the energy in joules. First, convert the energy from electronvolts (eV) to joules using the conversion factor: \( 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \).

Step 2: Calculate Wavelength for (a)

Convert \( 0.250 \text{ eV} \) to joules: \( E_a = 0.250 \times 1.602 \times 10^{-19} = 4.005 \times 10^{-20} \text{ J} \). Using the wavelength formula, \( \lambda_a = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{4.005 \times 10^{-20}} \approx 4.96 \times 10^{-6} \text{ m} \) or 4960 nm. This places it in the infrared part of the spectrum.

Step 3: Calculate Wavelength for (b)

Convert \( 8.50 \text{ eV} \) to joules: \( E_b = 8.50 \times 1.602 \times 10^{-19} = 1.3617 \times 10^{-18} \text{ J} \). Using the wavelength formula, \( \lambda_b = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{1.3617 \times 10^{-18}} \approx 9.13 \times 10^{-8} \text{ m} \) or 91.3 nm. This places it in the ultraviolet part of the spectrum.

Step 4: Calculate Wavelength for (c)

Convert \( 3.20 \times 10^{-3} \text{ eV} \) to joules: \( E_c = 3.20 \times 10^{-3} \times 1.602 \times 10^{-19} = 5.1264 \times 10^{-22} \text{ J} \). Using the wavelength formula, \( \lambda_c = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{5.1264 \times 10^{-22}} \approx 3.88 \times 10^{-3} \text{ m} \) or 3880 nm. This places it in the infrared part of the spectrum.

Key Concepts

Vibrational Energy Transition

When a molecule decreases its vibrational energy, it releases a photon. Vibrational energy transitions occur when molecules vibrate less energetically, transitioning from a higher to a lower vibrational state. This energy change is quite significant and involves interactions at the molecular level.

• Molecules are made up of atoms that are in constant motion. These movements include vibrations - think of bonds stretching and compressing like springs.
• A vibrational transition involves the release or absorption of a photon as the molecule switches between vibrational states.

Given an energy change of 0.250 eV, the decrease relates directly to the vibrational energy level change. By using the formula to convert this energy release to a wavelength, it becomes possible to identify where this light falls on the electromagnetic spectrum.
The resulting wavelength of 4960 nm indicates the release of a photon in the infrared region, typical for vibrational transitions. This is because infrared light corresponds precisely to the energy levels involved in molecular vibrations.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation. This spectrum ranges from very short wavelengths (gamma rays) to very long wavelengths (radio waves). Each type of radiation is characterized by its unique wavelength and energy.

• Visible light is just a small part of the spectrum, ranging from about 400 nm (violet) to 700 nm (red).
• Infrared light is on the spectrum just below visible light, from about 700 nm up to 1 mm. This range involves lower energy than visible light.
• Ultraviolet light has a shorter wavelength than visible light and can be divided into UV-A, UV-B, and UV-C, with increasing energy levels.

Understanding where a given wavelength fits in the electromagnetic spectrum allows us to deduce the energy changes occurring within photons. In various scientific applications, identifying and categorizing these wavelengths can provide insights into the processes happening at an atomic or molecular scale.

Rotational Energy Transition

Rotational transitions occur when molecules change their rotational energy levels. This energy change is typically much smaller than vibrational or electronic energy changes.

• Molecules have the ability to rotate around their bond axes, and changes in this rotation involve the release or absorption of very low energy photons.
• When a molecule gives up 3.20 x 10^{-3} eV to transition to a lower rotational state, the associated photon is of low energy, producing a longer wavelength.

With an energy change resulting in a wavelength of 3880 nm, the photon falls within the infrared region. Rotational transitions, like vibrational ones, often fall within the longer wavelengths of the electromagnetic spectrum due to their comparatively low energy requirements. This highlights the usefulness of infrared spectroscopy in studying the rotational transitions of molecules.