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

The relationship between wavelength and frequency of radiant energy is given by the formula \(c = \lambda \nu\), where \(c\) is the speed of light, \(\lambda\) is the wavelength, and \(\nu\) is the frequency. Wavelength and frequency are inversely proportional. The given wavelength range of 210-230 nm falls within the Ultraviolet region of the electromagnetic spectrum.

Step-by-step solution

Part (a): Relationship between wavelength and frequency

The relationship between wavelength and frequency of radiant energy (electromagnetic waves) is given by the formula: \[c = \lambda \nu\] where: \(c\) is the speed of light in vacuum (\(3.0 \times 10^8 m/s\)) \(\lambda\) is the wavelength of the radiant energy \(\nu\) (pronounced "nu") is the frequency of the radiant energy The formula shows that wavelength and frequency are inversely proportional. If the wavelength increases, the frequency decreases, and vice versa.

Part (b): Identifying the electromagnetic spectrum region

We are given the wavelength range of 210-230 nm, and we need to identify in which region of the electromagnetic spectrum this range falls. The regions in the electromagnetic spectrum are categorized as follows: 1. Radio waves: \(10^3 - 10^9 m\) 2. Microwaves: \(10^{-2} - 10^3 m\) 3. Infrared: \(7 \times 10^{-7} - 10^{-2} m\) 4. Visible light: \(4 \times 10^{-7} - 7 \times 10^{-7} m\) 5. Ultraviolet: \(10^{-8} - 4 \times 10^{-7} m\) 6. X-rays: \(10^{-11} - 10^{-8} m\) 7. Gamma rays: less than \(10^{-11} m\) First, let's convert the given range from nm to meters by multiplying by \(10^{-9}\): 210 nm = \(210 \times 10^{-9} m\) 230 nm = \(230 \times 10^{-9} m\) Now, we have the wavelength range of \(210 \times 10^{-9} m\) to \(230 \times 10^{-9} m\). Comparing this range with the regions mentioned above, we can see that it falls within the Ultraviolet region of the electromagnetic spectrum.