Problem 7 - Chemistry and Chemical Reactivity

Question

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

Step-by-Step Solution

Answer

Frequency: \(7.57 \times 10^{14}\, \text{Hz}\); One photon energy: \(5.02 \times 10^{-19}\, \text{J}\); Mole energy: \(3.02 \times 10^{5}\, \text{J/mol}\).

Step 1: Understand the Problem

To solve this problem, we need to find two things: the frequency of the light with a wavelength of \(396.15\, \text{nm}\), and the energy of both one photon and one mole of photons with this wavelength. We'll apply the formulas for wave speed and photon energy.

Step 2: Convert Wavelength to Meters

The wavelength is given in nanometers (\(\text{nm}\)). We need to convert this to meters to use it in our calculations. 1 nanometer equals \(10^{-9}\) meters. So, \(396.15\, \text{nm} = 396.15 \times 10^{-9}\, \text{m}\).

Step 3: Calculate Frequency

The frequency \(f\) can be calculated using the speed of light \(c\) and the wavelength \(\lambda\). Use the formula:\[ f = \frac{c}{\lambda} \]where \(c = 3.00 \times 10^8\, \text{m/s}\) and \(\lambda = 396.15 \times 10^{-9}\, \text{m}\).Plug in the values:\[ f = \frac{3.00 \times 10^8}{396.15 \times 10^{-9}} \approx 7.57 \times 10^{14}\, \text{Hz} \]

Step 4: Calculate Energy of One Photon

The energy \(E\) of a photon is given by the formula:\[ E = h \cdot f \]where \(h\) is Planck's constant, \(6.63 \times 10^{-34}\, \text{J s}\).Using the frequency calculated:\[ E = 6.63 \times 10^{-34} \times 7.57 \times 10^{14} \approx 5.02 \times 10^{-19}\, \text{J} \]

Step 5: Calculate Energy of One Mole of Photons

To find the energy of 1.00 mole of photons, multiply the energy of one photon by Avogadro's number, \(6.022 \times 10^{23}\, \text{mol}^{-1}\).\[ E_{\text{mol}} = 5.02 \times 10^{-19} \times 6.022 \times 10^{23} \approx 3.02 \times 10^{5}\, \text{J/mol} \]

Step 6: Conclusion

The frequency of the line is approximately \(7.57 \times 10^{14}\, \text{Hz}\), the energy of one photon is \(5.02 \times 10^{-19}\, \text{J}\), and the energy of 1.00 mole of photons is \(3.02 \times 10^{5}\, \text{J/mol}\).

Key Concepts

Wavelength to Frequency Conversion

Light travels in waves, and these waves have different wavelengths and frequencies. Wavelength is the distance between two peaks of a wave, usually measured in meters. In this exercise, the wavelength is given as 396.15 nanometers (nm).To convert this to frequency, we need to use the speed of light, which is a constant. The speed of light is approximately \(3.00 \times 10^8 \text{ m/s}\). With the speed of light known, we use the relationship between speed, wavelength, and frequency.The formula to find frequency \(f\) is:

• \(f = \frac{c}{\lambda}\)

where \(c\) is the speed of light and \(\lambda\) is the wavelength.First, convert the wavelength from nanometers to meters because the speed of light is in meters per second. Since 1 nm is equal to \(10^{-9}\) meters, we have:

• \(396.15 \times 10^{-9} \text{ m}\)

Now, plug the values into the formula:

• \(f = \frac{3.00 \times 10^8}{396.15 \times 10^{-9}} \approx 7.57 \times 10^{14} \text{ Hz}\)

This result means the light from the spectrum of aluminum with this wavelength oscillates about \(7.57 \times 10^{14}\) times per second.

Energy of a Photon

Once we have the frequency of light, determining the energy of a photon becomes straightforward. A photon is a particle of light, and its energy is proportional to its frequency.To calculate this energy, we use Planck's equation:

• \(E = h \cdot f\)

where \(E\) is the energy of the photon, \(h\) is Planck's constant \(6.63 \times 10^{-34} \text{ J s}\), and \(f\) is the frequency we previously calculated.Plug in the value of frequency:

• \(E = 6.63 \times 10^{-34} \cdot 7.57 \times 10^{14} \approx 5.02 \times 10^{-19} \text{ J}\)

This result indicates the energy carried by each photon at this specific frequency. Photons can have very low energies because they are extremely small; however, their collective energy can have impactful results, which leads us to the next calculation.

Avogadro's Number in Photon Calculations

Avogadro's number plays a fundamental role when calculating the total energy of a mole of photons. Avogadro's number is approximately \(6.022 \times 10^{23} \, \text{mol}^{-1}\), which represents the number of units (such as atoms or photons) in one mole of a substance.To find the energy of a mole of photons, multiply the energy of a single photon by Avogadro's number:

• \(E_{\text{mol}} = E \times N_A\)

where \(E\) is the energy of one photon and \(N_A\) is Avogadro's number.Using the energy of a single photon we calculated (\(5.02 \times 10^{-19} \, \text{J}\)):

• \(E_{\text{mol}} = 5.02 \times 10^{-19} \times 6.022 \times 10^{23} \approx 3.02 \times 10^{5} \, \text{J/mol}\)

This result tells us how much energy would be contained in one mole of photons of that particular wavelength. It's a significant quantity, illustrating how large collections of photons act as powerful carriers of energy.