Problem 17 - Chemistry The Central Science

Question

The dissociation energy of a carbon-bromine bond is typically about \(210 \mathrm{~kJ} / \mathrm{mol}\). (a) What is the maximum wavelength of photons that can cause \(\mathrm{C}-\mathrm{Br}\) bond dissociation? (b) Which kind of electromagnetic radiation-ultraviolet, visible, or infrared-does the wavelength you calculated in part (a) correspond to?

Step-by-Step Solution

Answer

The maximum wavelength of photons that can cause C-Br bond dissociation is \(5.69 \times 10^{-7}\) m or 569 nm, which corresponds to visible electromagnetic radiation.

Step 1: Use the energy-wavelength relationship for photons

Recall the energy-wavelength relationship for photons given by the Planck's equation: \[E = h \frac{c}{\lambda}\] where E is the energy of the photon, h is the Planck's constant (\(6.63 \times 10^{-34} \mathrm{~Js}\)), c is the speed of light (\(3.00 \times 10^8 \mathrm{~m/s}\)), and λ is the wavelength of the photon. We are given the dissociation energy of the C-Br bond, which is \(210 \mathrm{~kJ/mol}\). We will first convert this energy into Joules and then into energy per photon using Avogadro's constant (number of entities per mol: \(6.022 \times 10^{23} \mathrm{~entities/mol}\)).

Step 2: Convert the dissociation energy to Joules per photon

To convert the dissociation energy to Joules per photon, we can follow the steps below: 1. Multiply by 1000 to convert kJ to J: \(210 \mathrm{~kJ/mol} \times 1000 = 210,000 \mathrm{~J/mol}\) 2. Divide by Avogadro's constant to get J per photon: \(\frac{210,000 \mathrm{~J/mol}}{6.022 \times 10^{23} \mathrm{~entities/mol}} = 3.49 \times 10^{-19} \mathrm{~J/photon}\) Therefore, the dissociation energy per photon is \(3.49 \times 10^{-19} \mathrm{J}\).

Step 3: Calculate the maximum wavelength

Now that we have the dissociation energy per photon (E), we can use the Planck's equation to find the maximum wavelength (λ): E = h(c/λ) → λ = hc/E Plugging in the values: λ = \(\frac{6.63 \times 10^{-34} \mathrm{~Js} \times 3.00 \times 10^8 \mathrm{~m/s}}{3.49 \times 10^{-19} \mathrm{~J}} = 5.69 \times 10^{-7}\) m The maximum wavelength of photons that can cause C-Br bond dissociation is \(5.69 \times 10^{-7}\) m or 569 nm.

Step 4: Identify the type of electromagnetic radiation

Now we need to determine the type of electromagnetic radiation. The range of wavelengths for different types of electromagnetic radiation are: - Ultraviolet radiation: \(10 \; \mathrm{nm} \leq \lambda < 400 \; \mathrm{nm}\) - Visible radiation: \(400 \; \mathrm{nm} \leq \lambda < 750 \; \mathrm{nm}\) - Infrared radiation: \(\lambda \geq 750 \; \mathrm{nm}\) Since the calculated maximum wavelength (569 nm) falls within the range of visible radiation (\(400 \leq \lambda < 750\) nm), the type of electromagnetic radiation is visible light. So, the maximum wavelength of photons that can cause the C-Br bond dissociation is 569 nm, which corresponds to visible electromagnetic radiation.

Key Concepts

Bond Dissociation

When a bond such as carbon-bromine (C-Br) breaks, it requires a specific amount of energy known as bond dissociation energy. This energy is necessary to overcome the forces that hold the atoms together in a molecule. Typically, we measure this energy in kilojoules per mole (kJ/mol). In this context, the C-Br bond requires 210 kJ/mol for dissociation.

But what does this mean in practical terms? It tells us how much energy is needed to split one mole of these bonds. If you're wondering why this matters, think of chemical reactions like cooking. You need to know how much of each ingredient, or in this case, energy, is required to achieve the desired change!

Planck's Equation

Planck's equation is crucial for understanding how energy relates to light. The equation is given by \[E = h \frac{c}{\lambda}\] where:

• \(E\) is the photon energy
• \(h\) is Planck's constant, \(6.63 \times 10^{-34}\,\mathrm{Js}\)
• \(c\) is the speed of light, \(3.00 \times 10^8\, \mathrm{m/s}\)
• \(\lambda\) is the wavelength of light

The energy of a photon is inversely proportional to its wavelength. That means, shorter wavelengths have higher energy. This relationship helps us calculate the energy needed to dissociate bonds, just like those in the C-Br bond example. All we need is the wavelength of the light that can provide the energy for bond dissociation.

Electromagnetic Radiation

Electromagnetic radiation spans a vast range of wavelengths and energies, from gamma rays to radio waves. It's not just mysterious fields; it's the light and heat we experience every day.

The electromagnetic spectrum includes several types, such as:

• Ultraviolet (UV) radiation: Short wavelengths from 10 to 400 nm
• Visible light: Wavelengths from 400 to 750 nm
• Infrared radiation: Wavelengths beyond 750 nm

Each type of electromagnetic radiation has different applications and effects. For example, UV light can cause sunburns, while infrared is often used in remote controls. The type of radiation depends on the energy a photon carries, which ties back to its wavelength.

Visible Light

Visible light is the portion of the electromagnetic spectrum that human eyes can see, ranging from 400 to 750 nm. It's the reason why we perceive colors.

In the context of bond dissociation, the C-Br bond breaks when exposed to a certain wavelength of light. In the earlier calculation, we found the necessary wavelength to be 569 nm, which falls squarely within the visible light range.

This information helps us understand how substances interact with light. By knowing the type of light involved, scientists can design experiments and technologies that use visible light for various reactions or processes.