Problem 105 - General Chemistry: Principles and Modern Applications

Question

A molecule of chlorine can be dissociated into atoms by absorbing a photon of sufficiently high energy. Any excess energy is translated into kinetic energy as the atoms recoil from one another. If a molecule of chlorine at rest absorbs a photon of 300 nm wavelength, what will be the velocity of the two recoiling atoms? Assume that the excess energy is equally divided between the two atoms. The bond energy of \(\mathrm{Cl}_{2}\) is \(242.6 \mathrm{kJ} \mathrm{mol}^{-1}\)

Step-by-Step Solution

Answer

The velocity of each chlorine atom after the bond breaks is given by the above calculations. If the photon's energy is insufficient to break the bond, the velocity is zero.

Step 1: Calculate photon energy

Calculate the energy (E) of the photon using the relation between energy, wavelength, and Planck's constant, given by the formula \(E = \frac{hc}{λ}\), where h is Planck's constant (6.626 x 10^-34 J.s), c is the speed of light (3.0 x 10^8 m/s), and λ is the wavelength. Remember to convert wavelength to meters.

Step 2: Calculate bond energy

The bond energy given is in kJ/mol, which needs to be converted to Joules (J) per molecule. For this, multiply the provided bond energy (242.6 kJ/mol) by 10^3 to convert kJ to J and divide it by Avogadro's number (6.022 x 10^23 molecules/mol), resulting in the bond energy per molecule.

Step 3: Calculate excess energy

Excess energy is calculated by subtracting the bond energy of the molecule from the energy of the photon. If the result is positive, the photon has enough energy to break the bond, and the excess energy is converted into kinetic energy.

Step 4: Calculate velocity

Now calculate the velocity (v) of each atom. Since the molecule was initially at rest, and mass and velocity are inversely proportional (according to the law of conservation of momentum), both atoms move with the same speed but in opposite directions. The velocity is given by \(v= \sqrt{\frac{2KE}{m}}\), where KE is the kinetic energy (which is half of the excess energy, as it is divided equally between the two atoms), and m is the mass of a chlorine atom.

Key Concepts

Bond Dissociation

Bond dissociation is the process where a chemical bond breaks, resulting in separate atoms or smaller molecules. In this exercise, we are looking at the dissociation of a chlorine molecule \(\mathrm{Cl}_2\). Bond energy is required to break these chemical bonds. The given bond energy for chlorine is 242.6 kJ/mol, which marks the energy needed to disconnect the two chlorine atoms in each molecule.

Let's break it down into easy steps:

• The bond energy is given in multi-molecule terms, measured as energy per mole.
• Since a mole contains a vast number of molecules, Avogadro's number (6.022 x 10^23 molecules/mol) helps us convert this energy to a single molecule basis.
• By multiplying by 1000, we switch from kJ to J, then dividing by Avogadro's number, we obtain joules per molecule.

This energy calculation ensures we know exactly how much energy each individual molecule requires in the bond-breaking process. This is a vital step in understanding how energy exchange affects reactions and atomic behavior.

Kinetic Energy Calculation

Once a bond dissociates, there may be excess energy leftover beyond what was needed to break the bond. This excess turns into kinetic energy, giving the fragments of the molecule their motion post dissociation. In our scenario, imagine a strong photon hitting the chlorine molecule. When photons are of sufficient energy to exceed the bond energy, the energy not used to break the bond contributes to kinetic energy.

To calculate kinetic energy, consider these points:

• The photon energy is calculated initially using the formula: \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant and \(c\) is the speed of light.
• Subtract the bond energy per molecule from photon energy—this computes the excess energy.
• The excess energy is then split evenly between the two atoms since both receive equivalent energy amounts.

In calculations, the energy leftover that transitions to kinetic form ensures the atoms speed away from each other after the bond breaks. This straightforward connection between energy absorption and kinetic movement demonstrates key principles of the conservation of energy.

Chlorine Molecules

Chlorine molecules consist of two chlorine atoms covalently bonded, expressed as \(\mathrm{Cl}_2\). These are diatomic molecules, meaning two atoms bond together. \(\mathrm{Cl}_2\) is known for being a green gas with significant chemical usage and precise atomic behavior in physics and chemistry.

Understanding chlorine at the atomic level includes:

• Each chlorine atom has a significant atomic mass, approximately 35.5 unified atomic mass units.
• When considering kinetic energy results, each atom's mass comes into play strongly influencing their velocity upon separation.
• This velocity is viewed by applying kinetic energy principles, ensuring these atoms move away from each other synchronously but in opposite directions when examined through kinetic calculations.

Chlorine's molecular understanding is crucial in many industrial and scientific functions. Demonstrating how and why these molecules dissociate under energy influence deepens comprehension of fundamental reaction dynamics and practical chemistry insights.