Problem 30 - Chemistry: The Central Science

Question

The energy from radiation can be used to cause the rupture of chemical bonds. A minimum energy of \(242 \mathrm{~kJ} / \mathrm{mol}\) is required to break the chlorine-chlorine bond in \(\mathrm{Cl}_{2}\). What is the longest wavelength of radiation that possesses the necessary energy to break the bond? What type of electromagnetic radiation is this?

Step-by-Step Solution

Answer

The longest wavelength of radiation that possesses the necessary energy to break the chlorine-chlorine bond is \(4.96 \times 10^{-7} ~\text{m}\) (496 nm). This type of electromagnetic radiation falls within the visible light range.

Step 1: Note the given energy

The given minimum energy required to break the Cl-Cl bond is \(242~\text{kJ/mol}\).

Step 2: Convert energy per mole to energy per photon

To find the energy per photon, we need to divide the given energy by Avogadro's number: \(E = \dfrac{242~\text{kJ/mol}}{6.022 \times 10^{23} ~\text{photons/mol}} = \dfrac{242 \times 10^3 ~\text{J/mol}}{6.022 \times 10^{23} ~\text{photons/mol}}\)

Step 3: Calculate energy per photon in Joules

Now, compute the energy per photon in Joules: \(E = \dfrac{242 \times 10^3 ~\text{J/mol}}{6.022 \times 10^{23} ~\text{photons/mol}} = 4.02 \times 10^{-19} ~\text{J}\)

Step 4: Use the energy-wavelength equation

Now, we'll use the energy-wavelength equation to find the wavelength, \(\lambda\). The equation is: \(E = \dfrac{hc}{\lambda}\) where \(E\) is the energy of the photon, \(h\) is Planck's constant (\(6.626 \times 10^{-34} ~\text{Js}\)), \(c\) is the speed of light (\(3.00 \times 10^8 ~\text{m/s}\)), and \(\lambda\) is the wavelength of the electromagnetic radiation.

Step 5: Rearrange the equation to find the wavelength

Rearrange the energy-wavelength equation to find the wavelength: \(\lambda = \dfrac{hc}{E}\)

Step 6: Calculate the wavelength

Substitute the known values into the equation and calculate the wavelength: \(\lambda = \dfrac{(6.626 \times 10^{-34} ~\text{Js})(3.00 \times 10^8 ~\text{m/s})}{4.02 \times 10^{-19} ~\text{J}} = 4.96 \times 10^{-7} ~\text{m}\)

Step 7: Identify the type of electromagnetic radiation

The wavelength we found, \(4.96 \times 10^{-7} ~\text{m}\), falls within the visible light range (around 400 to 700 nm), which means that the electromagnetic radiation required to break the Cl-Cl bond is visible light.

Key Concepts

Bond Dissociation Energy

Bond dissociation energy refers to the amount of energy required to break a particular chemical bond and separate atoms in a gas-phase molecule. It's a specific case of bond energy because it applies to the breaking of a single bond in a diatomic molecule.

For instance, the exercise mentions the chlorine-chlorine bond in \(\mathrm{Cl}_{2}\) which requires 242 kJ/mol to break. This value signifies the strength of the bond — the higher the bond dissociation energy, the stronger the bond, and the more energy is necessary to overcome the attraction between the two chlorine atoms. Understanding the concept of bond dissociation energy is crucial because it gives insight into the reactivity and stability of a molecule.

Electromagnetic Radiation

Electromagnetic radiation consists of waves of electric and magnetic fields that propagate through space carrying energy. It covers a broad spectrum that includes, in order of increasing frequency and decreasing wavelength: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

In the context of the exercise, we're dealing with the energy carried by these electromagnetic waves. The radiation must have enough energy, quantified per photon, to break chemical bonds, as in the case of the Cl-Cl bond where visible light provides the necessary energy. This concept is inherently linked to the photoelectric effect and the quantum nature of light.

Photochemistry

Photochemistry is the branch of chemistry concerned with the chemical effects of light. It studies the interaction of electromagnetic radiation with substances causing them to undergo a chemical change. A fundamental principle in photochemistry is that, for a chemical reaction to occur, the molecule must first absorb energy. When a molecule absorbs a photon whose energy matches the bond dissociation energy, a bond breakage can occur.

The exercise demonstrates photochemical principles by calculating the exact type of radiation needed to break a bond. Different chemical processes require different wavelengths of light because the energy of photons varies inversely with their wavelength.

Avogadro's Number

Avogadro's number, usually denoted as \(6.022 \times 10^{23}\), is the number of constituent particles, typically atoms or molecules, that are contained in one mole of a substance. It's a dimensionless quantity that provides a bridge between the macroscopic and microscopic worlds of chemistry.

In our exercise, Avogadro's number is used to convert the bond dissociation energy from per mole to per photon, which is crucial when relating macroscopic quantities we can measure to microscopic properties that explain chemical processes at the atomic or molecular level.