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 the wavelength (\(\lambda\)) and frequency (\(\nu\)) of radiant energy is given by the equation \(c = \lambda\nu\), where \(c\) is the speed of light (\(3.0 \cdot 10^8\,\mathrm{m/s}\)). Wavelength and frequency are inversely proportional to each other. (b) The given wavelength range of \(210-230\,\mathrm{nm}\) falls within the Ultraviolet (UV) region of the electromagnetic spectrum.

Step-by-step solution

Understand the relationship between wavelength and frequency of radiant energy

Wavelength and frequency of radiant energy are related through the speed of light, which is a constant value (\(c = 3.0 \cdot 10^8\,\mathrm{m/s}\)). The equation that describes this relationship is: \[c = \lambda\nu\] Where: - \(c\) is the speed of light (\(3.0 \cdot 10^8\,\mathrm{m/s}\)), - \(\lambda\) is the wavelength of the radiant energy, and - \(\nu\) is the frequency of the radiant energy. The equation tells us that as the wavelength increases, the frequency decreases, and vice versa. In other words, wavelength and frequency are inversely proportional to each other.

Determine the region of the electromagnetic spectrum of the given wavelength range

We are given the wavelength range of \(210-230\,\mathrm{nm}\), and we need to determine the region of the electromagnetic spectrum in which this radiation occurs. The electromagnetic spectrum is divided into different regions based on wavelength, as follows: - Radio waves: \(\lambda > 1\,\mathrm{m}\) - Microwaves: \(1\,\mathrm{mm} < \lambda < 1\,\mathrm{m}\) - Infrared (IR): \(700\,\mathrm{nm} < \lambda < 1\,\mathrm{mm}\) - Visible light: \(400\,\mathrm{nm} < \lambda < 700\,\mathrm{nm}\) - Ultraviolet (UV): \(10\,\mathrm{nm} < \lambda < 400\,\mathrm{nm}\) - X-rays: \(0.01\,\mathrm{nm} < \lambda < 10\,\mathrm{nm}\) - Gamma rays: \(\lambda < 0.01\,\mathrm{nm}\) Given that our wavelength range is \(210-230\,\mathrm{nm}\), we can see that it lies within the Ultraviolet region of the electromagnetic spectrum.