Problem 6 - Chemistry and Chemical Reactivity

Question

Violet light has a wavelength of about \(410 \mathrm{nm}\). What is its frequency? Calculate the energy of one photon of violet light. What is the energy of 1.0 mol of violet photons? Compare the energy of photons of violet light with those of red light. Which is more energetic?

Step-by-Step Solution

Answer

The energy of violet photons is higher than red photons.

Step 1: Convert Wavelength to Meters

First, convert the given wavelength from nanometers to meters since standard SI units are necessary for calculations. The wavelength of violet light is \(410 \text{ nm}\), which is \(410 \times 10^{-9} \text{ m}\).

Step 2: Calculate the Frequency

Use the speed of light equation to find the frequency of violet light. The formula is \( c = \lambda u \), where \( c \) is the speed of light \(3 \times 10^8 \text{ m/s}\), \( \lambda \) is the wavelength, and \( u \) is the frequency. Solve for \( u \):\[ u = \frac{c}{\lambda} = \frac{3 \times 10^8}{410 \times 10^{-9}} \approx 7.32 \times 10^{14} \text{ Hz} \]

Step 3: Calculate the Energy of One Photon

Use the formula\(E = h u \) where \( E \) is the energy of one photon, \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ J} \cdot \text{s} \), and \( u \) is the frequency calculated above.\[ E = 6.626 \times 10^{-34} \times 7.32 \times 10^{14} \approx 4.85 \times 10^{-19} \text{ J} \]

Step 4: Calculate the Energy of 1.0 Mol of Photons

Use Avogadro's number \(6.022 \times 10^{23}\) to find the total energy for one mole of photons:\[ E_{\text{mol}} = E \times 6.022 \times 10^{23} \]\[ E_{\text{mol}} = 4.85 \times 10^{-19} \times 6.022 \times 10^{23} \approx 2.92 \times 10^{5} \text{ J/mol} \]

Step 5: Compare Energies of Violet and Red Light Photons

Red light has a longer wavelength than violet light, typically around \(650 \text{ nm}\). Longer wavelengths correspond to lower frequencies and hence lower energies. Since violet light has a shorter wavelength than red light, it also has a higher frequency and is therefore more energetic.

Key Concepts

Wavelength and Frequency Relationship

The wavelength and frequency of light are intricately connected. They describe the nature of light as a wave. The wavelength (\(\lambda\)) is the distance between consecutive peaks of the wave, measured in meters. Frequency (\(u\)) is how many wave peaks pass a point in one second, measured in Hertz (Hz). The relationship between them is captured by the equation:

• \( c = \lambda u \)

where \(c\) represents the speed of light in a vacuum, approximately \(3 \times 10^8 \text{ m/s}\). This formula tells us that wavelength and frequency are inversely related:

• If wavelength decreases, the frequency increases.
• If wavelength increases, the frequency decreases.

This means violet light, with a smaller wavelength of \(410 \text{ nm}\), will have a higher frequency compared to red light, which has a longer wavelength.

Planck's Constant

Planck's constant is a fundamental constant used in quantum mechanics. It appears in many equations related to energy and frequency. Denoted as \(h\), it has a value of \(6.626 \times 10^{-34} \text{ J}\cdot\text{s}\). This constant is used to calculate the energy of a photon, which is a particle of light.The energy of a photon can be calculated using the equation:

• \( E = hu \)

where \(E\) is the energy, \(h\) is Planck's constant, and \(u\) is the frequency. This formula shows that the energy of a photon is directly proportional to its frequency:

• Higher frequency means higher energy.
• Lower frequency means lower energy.

In our calculation, the energy of one photon of violet light (\(7.32 \times 10^{14} \text{ Hz}\)) is around \(4.85 \times 10^{-19} \text{ J}\), illustrating how a relatively high frequency results in higher photon energy.

Avogadro's Number

Avogadro's number is a key concept in chemistry, providing a link between the micro-world of atoms and molecules and the macro-world of grams and liters. It is defined as the number of units, such as atoms or molecules, in one mole of a substance, expressed as \(6.022 \times 10^{23}\) units/mol.When dealing with photons, Avogadro's number helps us calculate the energy of a mole of photons. By multiplying the energy of a single photon by Avogadro's number, we obtain the energy for one mole:

• \( E_{\text{mol}} = E \times 6.022 \times 10^{23} \)

For violet photons, this results in an energy of \(2.92 \times 10^5 \text{ J/mol}\). This calculation allows chemists and physicists to work with more comprehensible amounts of energy when dealing with large quantities of photons.