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Chapter 8: Problem 3

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

Expert verified
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

01

Frequency Comparison

The frequency (\(f\)) of a wave is inversely proportional to its wavelength (\(λ\)). 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.
02

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.
03

Light Speed in Vacuum

The speed of light in a vacuum is a constant (\(3.0 \times 10^8 \mathrm{m/s}\)) 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.
04

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.

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