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 $\mathrm{C}-\mathrm{Cl}$ bond rupture in one molecule but not in the other?

Short answer

The range of wavelengths of photons that can cause C-Cl bond rupture in one molecule but not the other is at or below 260 nm.

Step-by-step solution

Calculate the Energy Difference

First, calculate the energy difference between the two bond dissociation energies: Energy difference = Bond dissociation energy (CF3Cl) - Bond dissociation energy (CCl4) Energy difference = 339 kJ/mol - 293 kJ/mol = 46 kJ/mol Convert the energy difference to Joules: Energy difference = 46 kJ/mol × (1000 J/1 kJ) = 46,000 J/mol

Convert Energy Difference to Wavelength Range

We need to convert the energy difference to a wavelength range using the energy-wavelength relationship: $E=h\ast c/\lambda$ Where E is the energy, h is Planck's constant (6.626 x 10^-34 J s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength. First, we need to determine the energy per photon. We can convert the energy difference per mole to the energy per photon using Avogadro's number (6.022 x 10^23 mol^-1): Energy per photon (in Joules) = Energy difference (in Joules) / Avogadro's number Energy per photon = (46,000 J/mol) / (6.022 x 10^23 mol^-1) = 7.64 x 10^-20 J Now, using the energy-wavelength relationship, we can calculate the wavelength: $\lambda =h\ast c/E$ $\lambda =(6.626\ast {10}^{-34}Js)\ast (3.00\ast {10}^{8}m/s)/(7.64\ast {10}^{-20}J)$ $\lambda =2.60\ast {10}^{-7}m$ Since the wavelength is in meters, we can convert this to nanometers (1 nm = 1 x 10^-9 m): $\lambda =2.60\ast {10}^{-7}m\ast (1\ast {10}^{9}nm/1m)=260nm$ This means that photons with a wavelength of 260 nm or less can break the C-Cl bond in CF3Cl, but not in CCl4. Hence, the range of wavelengths of photons that can cause C-Cl bond rupture in one molecule but not in the other is at or below 260 nm.

Key concepts

Chemical Bond Energy

Chemical bond energy is a fundamental concept that refers to the amount of energy required to break a bond between two atoms in a molecule. This energy is indicative of the bond's strength—the higher the bond dissociation energy, the stronger the bond, and conversely, lower energy signifies a weaker bond.

In the given exercise, we're provided with the bond dissociation energies for the C-Cl bond in two different compounds, CF3Cl and CCl4. These energies, measured in kilojoules per mole (kJ/mol), represent the energy needed to break one mole of C-Cl bonds in each compound. For CF3Cl, it is 339 kJ/mol, and for CCl4, it is 293 kJ/mol. Understanding that the energy required to break these bonds varies with the molecular environment is crucial to anticipating the reactivity of different compounds.

Wavelength and Energy Relationship

The relationship between wavelength and energy is described by an elegant equation in quantum physics: $E=h\ast \frac{c}{\lambda }$, where E is the energy of a photon, h is Planck's constant, c is the speed of light, and $\lambda$ is the wavelength.

As per this equation, there is an inverse relationship between the energy of light and its wavelength: photons with shorter wavelengths have higher energies, and those with longer wavelengths have lower energies. In practice, this means that to break a bond, we need a photon with enough energy—corresponding to a particular wavelength or shorter. The exercise asked to find the wavelength of photons that could cause C-Cl bond rupture in one molecule but not the other. This question is closely linked to the energy of those bonds and demonstrates the practical application of the wavelength-energy relationship in predicting chemical reactions triggered by different frequencies of light.

Planck's Constant

Planck's constant, denoted by h, is a fundamental constant in the realm of quantum mechanics and has a value of approximately 6.626 x 10^-34 Joule seconds (Js). Its importance can hardly be overstated as it relates the energy carried by a photon to its frequency and, as a consequence, to its wavelength.

The constant is utilized in the aforementioned exercise to convert the energy difference calculated (46 kJ/mol) into a wavelength. Planck's constant is the pivotal element in the equation $E=h\ast \frac{c}{\lambda }$ which was used to deduce the specific wavelengths capable of initiating a chemical bond rupture. Through this constant, we can quantify the infinitesimally small energies associated with individual photons and match them with the macroscopic energies involved in breaking chemical bonds in a molecule.