Problem 2 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

The Cl-Cl bond in \(\mathrm{Cl}_{2}\) has a bond energy of \(242 \mathrm{kJmol}^{-1}\) Assuming that absorption of photons of this energy will break the bond, what is the minimum frequency of electromagnetic radiation that is required? What part of the electromagnetic spectrum does this correspond to? (Section 3.2).

Step-by-Step Solution

Answer

The minimum frequency is \(6.07 \times 10^{14}\) Hz, corresponding to violet light in the visible spectrum.

Step 1: Understand Bond Energy

The bond energy given is 242 kJ/mol. This energy is required to break one mole of Cl-Cl bonds. We need to use this energy to find the minimum frequency of electromagnetic radiation that can break these bonds.

Step 2: Convert Bond Energy to Single Bond Energy

Since the bond energy is given in kJ/mol, we first convert it to energy per single bond in joules. Use Avogadro's number (\(6.022 imes 10^{23} \) mol\(^{-1}\)) to calculate the energy per bond:\[\text{Energy per bond} = \frac{242,000 \text{ J/mol}}{6.022 \times 10^{23} \text{ mol}^{-1}} = 4.02 \times 10^{-19} \text{ J/bond}\]

Step 3: Use Energy-Frequency Relation

Planck's equation relates energy per photon to frequency: \[E = hu\] where \(E\) is energy per photon, \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ Js}\), and \(u\) is frequency. Solve for \(u\):\[u = \frac{E}{h} = \frac{4.02 \times 10^{-19} \text{ J}}{6.626 \times 10^{-34} \text{ Js}} = 6.07 \times 10^{14} \text{ Hz}\]

Step 4: Electromagnetic Spectrum Correspondence

The calculated frequency \(6.07 \times 10^{14} \text{ Hz}\) falls within the visible light range of the electromagnetic spectrum. Specifically, it is in the violet region, which ranges from about \(7.5 \times 10^{14}\) Hz to \(4.3 \times 10^{14}\) Hz.

Key Concepts

Electromagnetic Spectrum

The electromagnetic spectrum is a range that includes all types of electromagnetic radiation, classified by wavelength or frequency.
Electromagnetic radiation consists of waves of electric and magnetic fields that travel through space. It spans a broad spectrum, from very low-frequency radio waves to high-frequency gamma rays. Here are some key parts of the spectrum:

• Radio Waves: The longest wavelengths and lowest energies. Used in communication technologies like radios and televisions.
• Microwaves: Shorter wavelengths than radio waves, used in microwave ovens and radar technology.
• Infrared: Just below visible light in frequency, associated with heat. Used in thermal imaging and remote controls.
• Visible Light: The only part of the spectrum observable to the human eye, ranging from red (longest wavelength) to violet (shortest wavelength).
• Ultraviolet Light (UV): Beyond visible light, associated with sunburn. Used in sterilization and fluorescence.
• X-rays: Used in medical imaging, with wavelengths shorter than UV light.
• Gamma Rays: The highest energy, often emitted from nuclear reactions or radioactive decay.

Knowing where a specific frequency falls within the electromagnetic spectrum helps us understand its applications and effects. For example, the energy to break a bond in the Cl-Cl molecule is within the visible range, which helps us identify the approximate color and application of such energy.

Photon Energy

Photon energy describes the energy carried by a single photon, the elementary particle of light and other electromagnetic radiation. Photons are massless and travel at the speed of light. The energy of a photon is directly related to its frequency or inversely related to its wavelength, according to the formula:\[E = h \cdot u\]Where:

• E stands for energy (in joules).
• h is Planck's constant (6.626 \times 10^{-34} \, \text{Js}).
• u represents the frequency of the photon (in Hertz).

When a molecule absorbs a photon, its energy can promote reactions such as breaking chemical bonds. In the case of the Cl-Cl bond, the energy for one bond to break is calculated at 4.02 \times 10^{-19} \text{ J/bond}. By isolating the relevant frequency, we can determine what type of electromagnetic radiation it corresponds to and its potential to cause changes at the atomic or molecular level.

Planck's Constant

Planck's constant is a fundamental physical constant that describes the quantized nature of energy exchange. It serves as a bridge between the macroscopic and quantum worlds. This constant is fundamental in Planck's equation, which relates the energy of photons to their frequency:\[E = h \cdot u\]Where:

• h is Planck's constant, approximately 6.626 \times 10^{-34} \, \text{Js}.
• E is energy (in joules).
• u is frequency (in Hertz).

Though initially discovered in the context of blackbody radiation, Planck's constant has broad implications across various scientific fields, especially in quantum mechanics. It is crucial in calculations involving the atomic and molecular scales because it highlights how energy levels are quantized, existing at discrete rather than continuous values. This concept is essential when calculating the properties of photons interacting with matter, such as determining the minimum frequency of radiation needed to break a chemical bond, as seen in the Cl-Cl bond exercise.