Problem 8 - Chemistry and Chemical Reactivity

Question

The most prominent line in the 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 mol of photons with the wavelength of the most energetic line?

Step-by-Step Solution

Answer

The lines are in UV (285.2 nm) and visible regions. The most energetic line is the 285.2 nm UV line. Calculated energy for 1 mol: approximately 422 J/mol.

Step 1: Examine the Wavelengths

First, we identify the given wavelengths: 285.2 nm, 383.8 nm, and 518.4 nm. These measurements are in nanometers (nm).

Step 2: Classify the Regions of the Spectrum

Next, determine the region of the electromagnetic spectrum each wavelength falls into. Wavelengths around 400-700 nm are visible light. 285.2 nm falls in the ultraviolet (UV) range, as it's less than 400 nm. Whereas, 383.8 nm is at the edge between UV and visible, and 518.4 nm is in the visible range (green light).

Step 3: Calculate the Energy of Each Line

The energy of a photon can be calculated using the formula: \[ E = \frac{hc}{\lambda} \]Where \( h \) is Planck’s constant \( (6.626 \times 10^{-34} \, \text{J}\cdot\text{s}) \), \( c \) is the speed of light \( (3.00 \times 10^8 \, \text{m/s}) \), and \( \lambda \) is the wavelength in meters. First, convert the wavelength from nm to meters by dividing by \(10^9\).

Step 4: Identify the Most Energetic Line

Using the formula from Step 3, the energy is inversely proportional to wavelength; thus, the smallest wavelength (285.2 nm) is the most energetic. Perform the calculation to confirm this: \[ E = \frac{(6.626 \times 10^{-34})(3.00 \times 10^8)}{285.2 \times 10^{-9}} \] Calculate \( E \).

Step 5: Calculate Energy Per Mole for the Most Energetic Line

To find the energy per mole, multiply the energy of one photon by Avogadro's number \( (6.022 \times 10^{23} \text{mol}^{-1}) \). \[ E_{\text{mol}} = E \times 6.022 \times 10^{23} \] Compute this value to find the energy for one mole of photons with the 285.2 nm wavelength.

Key Concepts

Photon Energy

Photon energy is an essential concept in understanding how light behaves. Each photon, or particle of light, carries a certain amount of energy. This energy can be calculated using the formula: \[ E = \frac{hc}{\lambda} \]Where:

• \( E \) is the energy of the photon
• \( h \) is Planck’s constant \((6.626 \times 10^{-34} \text{J}\cdot\text{s})\)
• \( c \) is the speed of light \((3.00 \times 10^8 \text{m/s})\)
• \( \lambda \) is the wavelength in meters

The energy of a photon is inversely related to its wavelength. This means that shorter wavelengths correspond to higher energies. For example, ultraviolet light, with its shorter wavelengths, has more energy per photon than visible light. Understanding this relationship helps in identifying which photons are more energetic.

Wavelength Classification

In the electromagnetic spectrum, light is classified based on its wavelength. Different wavelengths correspond to different types of electromagnetic radiation, each with unique properties. Here's a simple classification:

• Radio waves: long wavelengths
• Microwaves: shorter than radio waves
• Infrared: before visible light
• Visible light: around 400-700 nm
• Ultraviolet (UV): shorter than visible light, less than 400 nm
• X-rays: even shorter
• Gamma rays: shortest wavelengths

For magnesium, wavelengths at 285.2 nm fall in the ultraviolet spectrum, making them more energetic. Meanwhile, 383.8 nm is at the edge of ultraviolet and visible, and 518.4 nm is in the visible range (green light). Knowing where a wavelength fits in this classification helps predict its behavior and energy.

Ultraviolet Spectrum

The ultraviolet (UV) spectrum is a part of the electromagnetic spectrum with wavelengths shorter than visible light but longer than X-rays. UV light is invisible to the human eye but plays an essential role in various applications. Here are some important points about UV light:

• Divided into three regions: UVA, UVB, and UVC
• UVA: longest UV wavelengths (closer to visible light)
• UVB: medium wavelengths, can cause sunburn
• UVC: shortest, most energetic, mostly absorbed by the Earth's atmosphere
• Used in sterilization and fluorescent lighting

For magnesium, the 285.2 nm line is in the UV range, indicating it carries more energy than visible light, which can have implications for chemical reactions and visibility under UV illumination.

Planck's Constant

Planck's constant \((h)\) is a fundamental value in physics and appears in the formula for calculating photon energy. It is named after Max Planck, a German physicist who is considered the father of quantum theory. The constant has a value of:\[ h = 6.626 \times 10^{-34} \text{J}\cdot\text{s} \]This constant is crucial for understanding quantum mechanics as it relates energy to the frequency of radiation. Using Planck’s constant in photon energy calculations helps in:

• Determining the energy levels of photons
• Explaining phenomena like photoelectric effect
• Understanding energy distributions at the microscopic level

In the context of magnesium's spectral lines, Planck’s constant allows us to quantify the photon energy for different wavelengths, ensuring precise scientific analysis.