Question

(a) What is the relationship between the wavelength and the frequency of radiant energy? (b) Ozone in the upper atmosphere absorbs energy in the \(210-230-\mathrm{nm}\) range of the spectrum. In what region of the electromagnetic spectrum does this radiation occur?

Short answer

(a) The relationship between wavelength (\(\lambda\)) and frequency (\(\nu\)) of radiant energy is inversely proportional, given by the formula \(\nu = \frac{c}{\lambda}\) or \(\lambda = \frac{c}{\nu}\), where \(c\) is the speed of light in a vacuum (\(3.00 \times 10^8 \mathrm{m/s}\)). (b) The wavelength range \(210-230 \mathrm{nm}\) falls under the ultraviolet region of the electromagnetic spectrum, as it is between \(400 \mathrm{nm}\) and \(10 \mathrm{nm}\).

Step-by-step solution

Part (a): Relationship between wavelength and frequency

The relationship between wavelength (\(\lambda\)) and frequency (\(\nu\)) of radiant energy (electromagnetic waves) is given by the following formula: \[ c = \lambda \nu \] where `c` is the speed of light in a vacuum, approximately equal to \(3.00 \times 10^8 \mathrm{m/s}\). To express the relationship between the wavelength and frequency, you can rearrange the equation as follows: \[ \nu = \frac{c}{\lambda} \] or \[ \lambda = \frac{c}{\nu} \] This relationship shows that as the wavelength increases, the frequency decreases, and vice versa. They are inversely proportional.

Part (b): Region of the electromagnetic spectrum

The electromagnetic spectrum is divided into various regions, based on the wavelength or frequency of the radiation. Given the wavelength range \(210-230 \mathrm{nm}\), we want to determine the region it falls under in the electromagnetic spectrum. Below are the regions and their respective wavelength ranges: 1. Radio waves: \(\lambda > 1 \mathrm{m}\) 2. Microwaves: \(1 \mathrm{m} > \lambda > 1 \mathrm{mm}\) 3. Infrared: \(1 \mathrm{mm} > \lambda > 700 \mathrm{nm}\) 4. Visible light: \(700 \mathrm{nm} > \lambda > 400 \mathrm{nm}\) 5. Ultraviolet: \(400 \mathrm{nm} > \lambda > 10 \mathrm{nm}\) 6. X-rays: \(10 \mathrm{nm} > \lambda > 0.01 \mathrm{nm}\) 7. Gamma rays: \(\lambda < 0.01 \mathrm{nm}\) The given wavelength range of \(210-230 \mathrm{nm}\) falls under the ultraviolet region of the electromagnetic spectrum, as it is between \(400 \mathrm{nm}\) and \(10 \mathrm{nm}\).