Question

In \(\mathrm{CF}_{3} \mathrm{Cl}\) the \(\mathrm{C}-\mathrm{Cl}\) bond- dissociation energy is \(339 \mathrm{~kJ} / \mathrm{mol}\), In \(\mathrm{CCl}_{4}\) the \(\mathrm{C}-\mathrm{Cl}\) bond-dissociation energy is \(293 \mathrm{~kJ} / \mathrm{mol}\). What is the range of wavelengths of photons that can cause \(\mathrm{C}\) - Cl bond rupture in one molecule but not in the other?

Short answer

To find the range of wavelengths of photons that can selectively break the C-Cl bond in one molecule but not in the other, we first convert the bond-dissociation energies to energy per photon and then find the corresponding wavelengths using Planck's constant and the speed of light. After calculating the wavelengths for each compound, we subtract the wavelengths to find the range of wavelengths that can selectively break the C-Cl bond. The range of wavelengths is given by \(\Delta\lambda = \lambda_2 - \lambda_1\).

Step-by-step solution

Convert bond-dissociation energies to energy per photon

To do this, we will use the relationship between energy and frequency which is given by the formula: \[E = h\nu\] where \(E\) is the energy per photon, \(h\) is Planck's constant (\(6.626 \times 10^{-34}\) Js), and \(\nu\) is the frequency of the photon. For CF3Cl: \[E_1 = 339 \text{ kJ/mol} = 339000 \text{ J/mol}\] For CCl4: \[E_2 = 293 \text{ kJ/mol} = 293000 \text{ J/mol}\] Since these energies are given per mole, we will first divide them by Avogadro's number (\(6.022 \times 10^{23}\) particles/mol) to get the energy per photon: \[E_1' = \frac{E1}{6.022\times10^{23}\, \text{particles/mol}} \] \[E_2' = \frac{E2}{6.022\times10^{23}\,\text{particles/mol}} \]

Find the corresponding wavelengths of photons using the speed of light

Using the relationship between speed of light (\(c\)), wavelength (\(\lambda\)), and frequency (\(\nu\)), we have: \[c = \lambda\nu\] Solving for \(\nu\) and substituting into the energy-frequency relationship, we get: \[E = h \frac{c}{\lambda}\] Now, we can solve for the wavelength of each compound: \[\lambda_1 = \frac{hc}{E_1'}\] \[\lambda_2 = \frac{hc}{E_2'}\]

Calculate the range of wavelengths that can selectively break the C-Cl bond

Since the C-Cl bond rupture in CF3Cl requires higher energy than that in CCl4, we expect the photon wavelength to be in the range of \(\lambda_2\) to \(\lambda_1\). To find the range of wavelengths, subtract the wavelengths of photons required for each compound: \[\Delta\lambda = \lambda_2 - \lambda_1\] After calculating these values, we will find the range of wavelengths of photons that can selectively break the C-Cl bond in one molecule but not in the other.

Key concepts

Photochemical Bond Rupture

Photochemical bond rupture refers to the breaking of chemical bonds in molecules due to the absorption of light. When a molecule absorbs a photon whose energy is at least equal to the bond-dissociation energy of a particular bond, that bond can break, leading to a chemical reaction. This process is the cornerstone of photochemistry and has wide applications, including the synthesis of complex molecules and photolithography.

For the bond rupture to be selective, the photon's energy must be carefully matched to the bond-dissociation energy. If the energy is too low, the bond will not break; if it's too high, other bonds might break as well. This selectivity is particularly important in processes like photodynamic therapy, where light is used to target specific cells without damaging surrounding healthy tissue.

Planck's Constant

Planck's Constant is a fundamental constant denoted as 'h', with a value of approximately \(6.626 \times 10^{-34}\) joule-seconds (\text{Js}). It plays a pivotal role in quantum mechanics, linking the energy of a photon to its frequency through the equation \(E = hu\). This constant is not only a cornerstone in the photoelectric effect, which explains how light can eject electrons from a material, but also in determining the energy of photons required to break specific chemical bonds in photochemical processes.

Understanding the relationship between the energy of photons and the frequency of the light they represent helps to predict and explain the behavior of molecules when they interact with light. In our exercise, Planck's constant is key to finding the energy per photon that is necessary to break a molecular bond.

Avogadro's Number

Avogadro's Number, designated as \(6.022 \times 10^{23}\), represents the number of particles (such as atoms or molecules) in one mole of a substance. It's a fundamental constant in chemistry that allows scientists to bridge the macroscopic and microscopic worlds, meaning it connects the quantity of material we can see and measure with the number of atoms or molecules it contains.

In the context of the problem, Avogadro's number is used to convert the energy required to break bonds from a per-mole basis to a per-particle basis. This conversion is necessary to determine the energy of individual photons needed to cause bond rupture, as photons interact with molecules on a one-to-one basis.

Speed of Light

The speed of light, commonly symbolized as \(c\), is a physical constant describing how fast light travels in a vacuum. Its value is approximately \(3.00 \times 10^{8}\) meters per second. The speed of light is paramount in the equations that relate the energy and momentum of photons with their frequency and wavelength.

This means that in calculations involving the photochemical rupture of bonds, the speed of light helps us establish the wavelength of light needed to supply the energy for breaking a specific bond. Since the speed of light is a maximum and unvarying speed in the universe, it provides a stable reference point for these kinds of calculations.

Photon Wavelength

Photon wavelength, represented with \(\lambda\), is the distance over which a light wave's shape repeats. It is inversely related to the frequency of the light, and consequently, to the energy of the photons that compose the light. The longer the wavelength, the lower the frequency, and the less energy each photon has.

In our problem, the wavelength is critical to determining which photons can induce the rupture of the \text{C}-\text{Cl} bond. Since different bonds require different energies to break, photons with wavelengths within a specific range are needed to selectively break one bond while leaving others intact. Calculating this precise range of wavelengths is essential in applications where specific photochemical reactions are desired.