Question

The magnesium spectrum has a line at $266.8\mathrm{nm}$. Which of these statements about this radiation is (are) correct? Explain. (a) It has a higher frequency than radiation with wavelength $402\mathrm{nm}$ (b) It is visible to the eye. (c) It has a greater speed in a vacuum than does red light of wavelength $652\mathrm{nm}$ (d) Its wavelength is longer than that of X-rays.

Short answer

Statement (a) and statement (d) are correct. Statement (b) is not correct because $266.8\mathrm{nm}$ is not within the visible spectrum for the human eye. Statement (c) is not correct because the speed of light is the same in vacuum regardless of its wavelength or frequency.

Step-by-step solution

Frequency Comparison

The frequency ($f$) of a wave is inversely proportional to its wavelength ($\lambda$). Therefore, as the wavelength increases, the frequency decreases and vice versa. Since $266.8\mathrm{nm}$ is less than $402\mathrm{nm}$, it means the frequency of the radiation with wavelength $266.8\mathrm{nm}$ is higher than the radiation with wavelength $402\mathrm{nm}$. So, the statement (a) is correct.

Visibility To The Eye

The human eye can perceive radiation of wavelengths ranging from approximately $400\mathrm{nm}$ (violet) to $700\mathrm{nm}$ (red). Since $266.8\mathrm{nm}$ is less than $400\mathrm{nm}$, radiation with wavelength $266.8\mathrm{nm}$ is not visible to the human eye. Therefore, the statement (b) is not correct.

Light Speed in Vacuum

The speed of light in a vacuum is a constant ($3.0\times {10}^8\mathrm{m}/\mathrm{s}$) and is independent of its wavelength or frequency. Hence, statement (c) is incorrect. The speed of both the radiation with wavelength $266.8\mathrm{nm}$ and the red light with wavelength $652\mathrm{nm}$ in vacuum will be the same.

Wavelength Comparison with X-rays

On the electromagnetic spectrum, X-rays have wavelengths ranging from $0.01\mathrm{nm}$ to $10\mathrm{nm}$. Therefore, the radiation with wavelength $266.8\mathrm{nm}$ is longer than that of X-rays, making statement (d) correct.

Key concepts

Wavelength and Frequency Relationship

Understanding the relationship between wavelength and frequency is crucial in the study of electromagnetic waves. Wavelength ( $\lambda$ ) is the distance between consecutive peaks of a wave, while frequency ( $f$ ) is the number of waves that pass a given point per second. These two properties are inversely related, meaning when one increases, the other decreases. This is expressed through the formula: $f=\frac{c}{\lambda }$ , where $c$ is the speed of light in a vacuum, approximately $3\times {10}^8\mathrm{m}/\mathrm{s}$ .
If you have a shorter wavelength, like $266.8\mathrm{nm}$ , then the frequency must be higher compared to a longer wavelength such as $402\mathrm{nm}$ . This relationship helps scientists understand and compare different types of electromagnetic radiation based on either their wavelength or frequency.

Visible Light Spectrum

The visible light spectrum is a narrow band within the electromagnetic spectrum that can be perceived by the human eye. It ranges approximately from $400\mathrm{nm}$ (violet) to $700\mathrm{nm}$ (red). This means that any wavelength within this range will be visible as a particular color.
Wavelengths shorter than about $400\mathrm{nm}$ , such as $266.8\mathrm{nm}$ , fall into the ultraviolet range and are not visible to the naked eye. Similarly, wavelengths longer than $700\mathrm{nm}$ are in the infrared range, which is also invisible. Understanding where visible light fits into the electromagnetic spectrum helps us discern why certain wavelengths are seen as colors, and why others remain unseen.

Speed of Light in Vacuum

The speed of light in a vacuum is one of the fundamental constants of nature, denoted by $c$ . It is equal to approximately $3\times {10}^8\mathrm{m}/\mathrm{s}$ . This means that all electromagnetic radiation, regardless of wavelength or frequency, travels at this same speed when in a vacuum.
Therefore, whether we're talking about $266.8\mathrm{nm}$ ultraviolet radiation or $652\mathrm{nm}$ red light, they both move through a vacuum at the same speed. This constant speed factor is important for the calculation of other properties and effects, such as the time it takes for light to travel from one point to another.

X-rays Wavelengths

X-rays are a form of electromagnetic radiation that have much shorter wavelengths compared to visible light. They typically range from $0.01\mathrm{nm}$ to $10\mathrm{nm}$ , which makes them highly penetrating and useful in medical imaging, among other applications. Because of their very short wavelengths, X-rays have higher frequencies than visible light or ultraviolet light.
When compared to something with a longer wavelength, like $266.8\mathrm{nm}$ from ultraviolet radiation, it is clear that such radiation is much longer in wavelength. These attributes of X-rays explain why they can pass through objects like skin and soft tissues, allowing the capture of internal structures, unlike much longer wavelength types of electromagnetic waves.