Problem 8 - Chemistry and Chemical Reactivity

Question

The most prominent line in the emission spectrum of magnesium is \(285.2 \mathrm{nm} .\) Other lines are found at 383.8 and \(518.4 \mathrm{nm}\). In what region of the electromagnetic spectrum are these lines found? Which is the most energetic line? What is the energy of \(1.00 \mathrm{mol}\) of photons with the wavelength of the most energetic line?

Step-by-Step Solution

Answer

285.2 nm and 383.8 nm are UV; 518.4 nm is visible. Most energetic is 285.2 nm; energy is \(4.20 \times 10^{5} \text{ J/mol}\).

Step 1: Identify Spectral Regions

The electromagnetic spectrum includes several regions such as gamma rays, X-rays, ultraviolet (UV), visible light, infrared (IR), microwaves, and radio waves. Wavelengths from 280 nm to 400 nm are in the ultraviolet (UV) range, wavelengths between 400 nm to 700 nm fall into the visible spectrum, and above 700 nm, the spectrum is in the infrared region.

Step 2: Determine Regions for Given Wavelengths

Evaluate the given wavelengths: 285.2 nm is in the UV region, 383.8 nm is also in the UV region, and 518.4 nm is in the visible light region (specifically, green visible light).

Step 3: Identify the Most Energetic Line

The energy of a photon is inversely proportional to its wavelength given by the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant and \( \lambda \) is the wavelength. Therefore, the shortest wavelength corresponds to the most energetic line. Here, 285.2 nm is the shortest wavelength.

Step 4: Calculate Energy of One Photon

Using the equation \( E = \frac{hc}{\lambda} \), where \( h = 6.626 \times 10^{-34} \text{ Js} \) and \( c = 3.00 \times 10^8 \text{ m/s} \), and the wavelength \( \lambda = 285.2 \times 10^{-9} \text{ m} \): \[ E = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{285.2 \times 10^{-9}} \text{ J} \]. Calculate \( E \approx 6.97 \times 10^{-19} \text{ J/photon} \).

Step 5: Calculate Energy for 1.00 Mol of Photons

To calculate the energy for 1 mole of photons, use Avogadro's number \( N_A = 6.022 \times 10^{23} \text{ photons/mole} \): \[ E_{\text{mol}} = 6.97 \times 10^{-19} \text{ J/photon} \times 6.022 \times 10^{23} \text{ photons/mole} \]. This results in \( E_{\text{mol}} = 4.20 \times 10^{5} \text{ J/mol} \).

Key Concepts

Photon Energy

Photon energy is a fundamental concept in understanding the electromagnetic spectrum. The energy of a photon can be calculated using the formula \( E = \frac{hc}{\lambda} \), where \( E \) is the energy, \( h \) is Planck's constant which equals \( 6.626 \times 10^{-34} \text{ Js} \), \( c \) is the speed of light \( 3.00 \times 10^8 \text{ m/s} \), and \( \lambda \) is the wavelength of the photon. This formula shows that photon energy is inversely proportional to its wavelength. - **Short Wavelength = High Energy**: As the wavelength decreases, the energy of the photon increases. Hence, photons with shorter wavelengths in the electromagnetic spectrum are more energetic. - **Energy Variation Across Spectrum**: Energy varies across the electromagnetic spectrum, with gamma rays having the highest energy and radio waves having the lowest.This relationship helps us identify which lines in an emission spectrum are the most energetic based on their wavelengths.

UV Region

The ultraviolet (UV) region of the electromagnetic spectrum is an area where wavelengths range from about 10 nm to 400 nm. UV light is divided into several categories including UVA, UVB, and UVC. - **UVC (100-280 nm)**: Highly energetic and mostly filtered by Earth's atmosphere. - **UVB (280-315 nm)**: Causes sunburn and has various health effects. - **UVA (315-400 nm)**: Less energetic but can contribute to skin aging. The UV region immediately precedes the visible light spectrum and is more energetic than visible light. Many elements emit particular lines in the UV region, making it significant for spectroscopic studies. For instance, elemental magnesium has prominent spectral lines in the UV region like the 285.2 nm line identified in the exercise.

Visible Light Spectrum

The visible light spectrum is the portion of the electromagnetic spectrum that can be detected by the human eye. Wavelengths in this region range from approximately 400 nm to 700 nm. - **Visible Colors**: It includes all the colors you can see, from violet (shorter wavelength) to red (longer wavelength). - **Photons in Light**: Each color has photons with different energies, dictated by their wavelengths. Purple and violet light have the highest energy photons, while red light has the lowest. Understanding the visible light spectrum is crucial in various scientific applications, such as identifying substances through their emission or absorption spectra. In the exercise, the visible spectrum included a line at 518.4 nm, marking it specifically as green light.

Wavelength Calculation

Wavelength calculation is key in determining the position and energy of lines in the electromagnetic spectrum. Using the formula \( E = \frac{hc}{\lambda} \), you can accurately determine photon energy based on wavelength. - **Convert to Correct Units**: Ensure your wavelength is in meters when using this formula. For instance, to convert 285.2 nm to meters, use \( 285.2 \times 10^{-9} \text{ m} \).- **Calculate Energy Per Photon**: Substituting in the constants, you can find the energy for a single photon. This can further be expanded to larger quantities, like moles, by using Avogadro's number.This step helps in both theoretical calculations and practical applications, like calculating how much energy is emitted by a mole of photons for certain wavelengths, as outlined in the given exercise.