Problem 132 - Introductory Chemistry Atoms First

Question

Electromagnetic radiation emitted by magnesium has a wavelength of \(285.2 \mathrm{~nm}\). (a) Is this radiation visible to the eye? (b) What is the energy of this radiation?

Step-by-Step Solution

Answer

The radiation with a wavelength of 285.2 nm emitted by magnesium is not visible to the eye, since the visible light range is approximately from 380 nm to 750 nm. The energy of this radiation is approximately \(6.97 × 10^{-19} J\).

Step 1: Calculate the frequency of the radiation

We are given the wavelength, λ, of the electromagnetic radiation as 285.2 nm. We will need to find the frequency, ν, of the radiation. We can do this using the speed of light, c, and the equation: \(c = λν\) The speed of light is approximately \(3.00 × 10^8 m/s\). First, convert the wavelength to meters: \(285.2\,\mathrm{nm} × \frac{1\,\mathrm{m}}{10^9 \,\mathrm{nm}} = 2.852 × 10^{-7}\, \mathrm{m}\) Now, we can rearrange the equation to solve for the frequency: \(ν = \frac{c}{λ}\)

Step 2: Find the frequency of the radiation

Plug in the values of the speed of light (c) and the wavelength (λ) into the equation: \(ν = \frac{3.00 × 10^8 m/s}{2.852 × 10^{-7} m} \approx 1.052 × 10^{15} Hz\) So, the frequency of the electromagnetic radiation is approximately \(1.052 × 10^{15} Hz\).

Step 3: Determine if the radiation is visible to the eye

The range of wavelengths for visible light is approximately from 380 nm to 750 nm. Since the given wavelength of magnesium's radiation is 285.2 nm, which is outside the visible range, the radiation is not visible to the eye.

Step 4: Calculate the energy of the radiation

To calculate the energy of the electromagnetic radiation, we can use the Planck's constant, \(h = 6.626 × 10^{-34} Js\), and the frequency, ν, in the following equation: \(E = hν\) Plug in the values of Planck's constant (h) and the frequency (ν) into the equation: \(E = (6.626 × 10^{-34} Js)(1.052 × 10^{15} Hz) \approx 6.97 × 10^{-19} J\) The energy of the electromagnetic radiation emitted by magnesium is approximately \(6.97 × 10^{-19} J\).

Key Concepts

Wavelength

The wavelength of electromagnetic radiation is the distance between successive peaks (or troughs) in a wave, often measured in nanometers (nm). In the case of the electromagnetic radiation emitted by magnesium, the wavelength is given as 285.2 nm. This value needs to be converted to meters when performing calculations with the speed of light. To convert from nanometers to meters, remember that 1 nm equals 10⁻⁹ meters. Therefore, \(285.2 \, nm = 285.2 \, \times \, 10^{-9} \, m = 2.852 \, \times \, 10^{-7} \, m\).
Understading wavelength is crucial because it is directly related to the energy and frequency of the wave. Smaller wavelengths correspond to higher energy radiation.

Frequency

Frequency refers to the number of wave cycles that pass a point per second, measured in hertz (Hz). It's a crucial part of understanding electromagnetic radiation.
To find the frequency of magnesium's radiation, we use the equation: \(c = \lambda u\), where \(c\) is the speed of light (\(3.00 \, \times \, 10^8 \, m/s\), \(\lambda\) is the wavelength, and \(u\) is the frequency.
Rearranging for frequency gives us: \(u = \frac{c}{\lambda}\). Substituting the values, \(u = \frac{3.00 \, \times \, 10^8 \, m/s}{2.852 \, \times \, 10^{-7} \, m} \approx 1.052 \, \times \, 10^{15} \, Hz\).
Frequency is directly related to energy, as higher frequencies imply more energetic waves, which is evident in different types of electromagnetic radiation.

Visible Light Spectrum

The visible light spectrum is the range of electromagnetic wavelengths that can be detected by the human eye. It spans from approximately 380 nm to 750 nm. This range includes all the colors seen in a rainbow, with violet having the shortest wavelength and red the longest. It is a small part of the entire electromagnetic spectrum, which ranges from gamma rays to radio waves.
In the exercise, the wavelength of magnesium's radiation, 285.2 nm, is outside the visible range, meaning it falls within the ultraviolet part of the electromagnetic spectrum, which is invisible to the naked eye.

Energy Calculation

Energy calculation in the context of electromagnetic radiation involves finding the energy of a photon. The energy, \(E\), of a photon is directly proportional to its frequency, \(u\), through Planck's formula: \(E = hu\), where \(h\) is Planck’s constant, \(6.626 \, \times \, 10^{-34} \, Js\).
For our magnesium radiation, substituting \(u = 1.052 \, \times \, 10^{15} \, Hz\) yields: \(E = (6.626 \, \times \, 10^{-34} \, Js)(1.052 \, \times \, 10^{15} \, Hz) \approx 6.97 \, \times \, 10^{-19} \, J\).
This energy calculation helps in understanding the magnitude of energy carried by electromagnetic waves and is crucial in fields like spectroscopy and quantum mechanics.