Question

The most prominent line in the emission spectrum of aluminum is at 396.15 nm. What is the frequency of this line? What is the energy of one photon with this wavelength? Of 1.00 mol of these photons?

Short answer

The frequency is \(7.57 \times 10^{14}\) Hz. Energy of one photon is \(5.01 \times 10^{-19}\) J. Energy of 1.00 mol of photons is \(3.02 \times 10^{5}\) J/mol.

Step-by-step solution

Convert Wavelength to Frequency

To find the frequency \( f \) of the electromagnetic radiation, use the equation \( c = \lambda f \), where \( c \) is the speed of light \((3.00 \times 10^8 \text{ m/s})\) and \( \lambda \) is the wavelength. Given \( \lambda = 396.15 \text{ nm} = 396.15 \times 10^{-9} \text{ m} \), we rearrange the formula to \( f = \frac{c}{\lambda} \). Substituting the values gives:\[ f = \frac{3.00 \times 10^8 \text{ m/s}}{396.15 \times 10^{-9} \text{ m}} \approx 7.57 \times 10^{14} \text{ Hz} \].

Calculate Energy of One Photon

To find the energy \( E \) of one photon, use the equation \( E = hf \), where \( h \) is Planck's constant \((6.626 \times 10^{-34} \text{ J}\cdot\text{s})\) and \( f \) is the frequency. From Step 1, \( f = 7.57 \times 10^{14} \text{ Hz} \). Substituting the values gives:\[ E = 6.626 \times 10^{-34} \text{ J}\cdot\text{s} \times 7.57 \times 10^{14} \text{ Hz} \approx 5.01 \times 10^{-19} \text{ J} \].

Calculate Energy of 1 Mol of Photons

To find the energy of 1.00 mol of these photons, multiply the energy of one photon by Avogadro's number \((6.022 \times 10^{23} \text{ mol}^{-1})\). From Step 2, the energy of one photon is \( 5.01 \times 10^{-19} \text{ J} \). Substituting the values gives:\[ E_{\text{mol}} = 5.01 \times 10^{-19} \text{ J} \times 6.022 \times 10^{23} \text{ mol}^{-1} \approx 3.02 \times 10^{5} \text{ J/mol} \].

Key concepts

Emission Spectrum

The emission spectrum is like a fingerprint of light emitted by an element. When an element, like aluminum in our context, is heated or excited in some way, it releases energy in the form of light. This light can be separated into its component wavelengths to form what we call the emission spectrum. Each element's spectrum is unique, consisting of specific 'lines,' each representing a particular wavelength of light emitted. These lines occur because electrons in atoms jump to higher energy levels when excited and then fall back down, releasing energy at specific wavelengths.

• Each line corresponds to light of a single wavelength.
• The most prominent line for an element is typically the brightest and the easiest to detect in the spectrum.

Understanding emission spectra is crucial in fields like astrophysics and chemistry, as it helps identify the elements present in stars and distant galaxies, or even in unknown materials here on Earth.

Wavelength to Frequency Conversion

Converting a wavelength into frequency is a key step in understanding electromagnetic waves. The relationship between a wave's wavelength (\(\lambda\)), frequency (\(f\)), and the speed of light (\(c\)) is given by the formula:\[f = \frac{c}{\lambda}\] Here, \(c\) stands for the speed of light, which is approximately \(3.00 \times 10^8 \, \text{m/s}\). This equation demonstrates that the frequency is inversely proportional to the wavelength—meaning as one increases, the other decreases.

• Use it to find the frequency if you know the wavelength.
• It's essential for characterizing electromagnetic radiation, including visible light.
• This calculation is vital for determining photon energy.

So, if the wavelength of the light emitted from aluminum is 396.15 nm, first convert it to meters (\(396.15 \times 10^{-9} \, \text{m}\)) then apply the formula to find its frequency.

Planck's Constant

Planck's constant (\(h\)) is a fundamental feature in quantum mechanics and pivotal in calculating photon energy. It is a small number meaning it governs the physics of particles at the smallest scales, such as photons. Planck's constant has a value of approximately \(6.626 \times 10^{-34} \, \text{J} \cdot \text{s}\). It appears in the formula to calculate the energy (\(E\)) of a photon given its frequency (\(f\)):\[E = h \cdot f\]

• This equation was a key discovery in understanding atomic and subatomic processes.
• It relates energy to frequency, key for technologies like lasers and LEDs.
• Planck's work led to the development of quantum theory—a cornerstone in modern physics.

By using the frequency calculated from the wavelength of aluminum's emission line, you can determine the energy of each photon it emits using this constant.

Avogadro's Number

Avogadro's number bridges the microscopic and the macroscopic world by allowing us to calculate how many of the tiny particles are in a more conceivable amount. This constant, approximately \(6.022 \times 10^{23} \, \text{mol}^{-1}\), represents the number of atoms, ions, or molecules in one mole of a substance.

• It allows chemists to count entities by weighing samples.
• Avogadro's number is fundamental in converting measurements from the atomic scale to the kilogram scale.
• It provides a link between the macroscopic scales of chemistry and biology and the microscopic scales of atoms and molecules.

In this exercise, multiplying the energy of one photon by Avogadro's number gives the energy of a mole of photons. This calculation is crucial for understanding how much energy is associated with larger amounts of particles, which helps quantify reactions and processes at practical scales in labs and industry.