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 C-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 \(5.68 \times 10^{-7} \, \mathrm{m}\) or 568 nm. This wavelength corresponds to the visible light region of the electromagnetic spectrum.

Step 1: Understanding the relation between energy and wavelength

We need to calculate the wavelength of a photon that has enough energy to break the C-Br bond. The energy of a photon (E) can be related to its wavelength (λ) using the equation: \[E = \frac{h \times c}{\lambda}\] where h is Planck's constant (\(6.626 \times 10^{-34} \, \mathrm{J} \cdot \mathrm{s}\)), c is the speed of light (\(3.0 \times 10^8 \, \mathrm{m/s}\)), and λ is the wavelength.

Step 2: Convert dissociation energy to energy per photon

We can calculate the dissociation energy per photon by converting the given 210 kJ/mol to Joules and dividing it by Avogadro's number (6.022 x \(10^{23}\) particles/mol). \[E_{\mathrm{dissociation}} = \frac{210 \times 10^3 \ \mathrm{J/mol}}{6.022 \times 10^{23} \ \mathrm{particles/mol}} = 3.486 \times 10^{-19} \ \mathrm{J}\]

Step 3: Calculate the maximum wavelength

We can now calculate the maximum wavelength (λ) that can cause C-Br bond dissociation. Rearrange the energy equation from step 1 to isolate λ and substitute the calculated energy: \[\lambda = \frac{h \times c}{E_{\mathrm{dissociation}}} = \frac{6.626 \times 10^{-34} \, \mathrm{J} \cdot \mathrm{s} \times 3.0 \times 10^8 \, \mathrm{m/s}}{3.486 \times 10^{-19} \ \mathrm{J}} = 5.68 \times 10^{-7} \ \mathrm{m}\] So the maximum wavelength of photons that can cause C-Br bond dissociation is \(5.68 \times 10^{-7} \, \mathrm{m}\) or 568 nm.

Step 4: Determine the type of electromagnetic radiation

Now we need to identify which type of electromagnetic radiation the calculated wavelength belongs to. The electromagnetic spectrum is divided into different regions based on wavelength: - Ultraviolet: 10 nm - 400 nm - Visible light: 400 nm - 700 nm - Infrared: 700 nm - 1mm Since the calculated wavelength of 568 nm falls within the range of visible light, the type of radiation that corresponds to the maximum wavelength required to dissociate a C-Br bond is visible light.

Key Concepts

carbon-bromine bond

The carbon-bromine (C-Br) bond is a type of covalent bond that involves a carbon atom and a bromine atom. These bonds are commonly found in organic chemistry, notably in alkyl halides, which are of great industrial and chemical significance. Understanding the strength of a bond, such as the C-Br bond, is crucial because it indicates how much energy is required to break it. This energy is known as the bond dissociation energy.
To break the C-Br bond, a certain amount of energy must be provided, typically in the form of electromagnetic radiation such as light. The bond dissociation energy of the C-Br bond is often around 210 kJ/mol. This value indicates that to break all C-Br bonds in a mole of compound, 210 kJ of energy is needed. Knowing this energy helps us understand what type of photon, in terms of wavelength and energy, is necessary to cleave the bond when using light.

photon energy

Photon energy is the energy carried by a single photon, a particle of light. It is directly linked to the frequency and inversely related to the wavelength of light. This concept is grounded in the formula: \[E = \frac{h \times c}{\lambda}\] where \(E\) is the energy of the photon, \(h\) is Planck’s constant \( (6.626 \times 10^{-34} \, \mathrm{J} \cdot \mathrm{s}) \), \(c\) is the speed of light \((3.0 \times 10^8 \, \mathrm{m/s})\), and \(\lambda\) is the wavelength.
In the context of bond dissociation, the energy of the photon must be equivalent to or greater than the bond dissociation energy for effective breaking. For a carbon-bromine bond, which needs 210 kJ/mol, this means calculating the energy per photon and ensuring the photon wavelength corresponds to this energy. As shorter wavelengths have higher photon energies, they are more capable of breaking strong bonds.

electromagnetic spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from very short gamma rays to very long radio waves. Each region of the spectrum is associated with a different wavelength range. For instance, visible light, which is what our eyes can see, falls within a narrow range of wavelengths, from about 400 nm to 700 nm.
When we consider bond dissociation, knowing where the required wavelength falls on the spectrum helps us identify which type of electromagnetic radiation is necessary. In the C-Br bond example, the maximum wavelength needed is calculated to be 568 nm.

• This wavelength fits within the range of visible light.
• The visible spectrum is broader than that of ultraviolet (10-400 nm) but narrower than infrared (700 nm - 1 mm).

Thus, photons from the visible light region of the spectrum are appropriate for dissociating a carbon-bromine bond, as they offer just the right amount of energy for the purpose without overwhelming or insufficient energy.