Problem 30 - Chemistry The Central Science In Si Units

Question

The energy from radiation can be used to rupture chemical bonds. A minimum energy of \(192 \mathrm{~kJ} / \mathrm{mol}\) is required to break the bromine- bromine bond in \(\mathrm{Br}_{2}\). 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 is 624 nm, which falls under visible red light.

Step 1: Understand the Problem

We are asked to find the longest wavelength of radiation that can provide enough energy to break a bromine-bromine bond. The energy required for bond rupture is given as 192 kJ/mol.

Step 2: Convert Energy to Joules per Molecule

Given energy is in kJ/mol. First, we need to convert it to joules. 1 kJ = 1000 J. Thus, energy per mole = 192 kJ/mol = 192,000 J/mol. Next, divide by Avogadro's number (A eq 6.022 imes 10^{23}) to find energy per molecule: \( E_{molecule} = \frac{192,000 \text{ J/mol}}{6.022 \times 10^{23} \text{ molecules/mol}} \approx 3.19 \times 10^{-19} \text{ J/molecule} \).

Step 3: Use the Energy-Wavelength Relationship

The energy of a photon is related to its wavelength by the equation \( E = \frac{hc}{\lambda} \), where \( h = 6.626 \times 10^{-34} \text{ J·s} \) is Planck's constant and \( c = 3.00 \times 10^{8} \text{ m/s} \) is the speed of light.Rearranging for wavelength: \( \lambda = \frac{hc}{E} \).

Step 4: Calculate the Longest Wavelength

Substitute in the values to find the wavelength:\( \lambda = \frac{(6.626 \times 10^{-34} \text{ J·s})(3.00 \times 10^{8} \text{ m/s})}{3.19 \times 10^{-19} \text{ J}} \approx 6.24 \times 10^{-7} \text{ m} \).Converted to nanometers (1 m = 10^9 nm), the wavelength is approximately 624 nm.

Step 5: Determine the Type of Radiation

Radiation with a wavelength of 624 nm falls within the visible light spectrum. Specifically, it corresponds to red light, which is the longest wavelength in the visible spectrum.

Key Concepts

Wavelength Calculation

When we talk about the wavelength of electromagnetic radiation, we refer to the distance between successive peaks of this wave. This is crucial because the wavelength is inversely related to the energy of the photons that make up the light. To calculate it, knowing the energy involved is essential, as shown in the relationship \( E = \frac{hc}{\lambda} \). Here, \( E \) is the energy of a photon, \( h \) is Planck's constant, and \( c \) is the speed of light. Rearranging this relationship for \( \lambda \) gives us the formula for the wavelength: \( \lambda = \frac{hc}{E} \). This tells us that the longer the wavelength, the lower the energy of the photons. To find the longest wavelength capable of breaking a chemical bond, it's necessary to determine the least amount of energy capable of doing so, which is what this equation allows us to find.

Electromagnetic Spectrum

The electromagnetic spectrum is a continuum of all electromagnetic waves arranged according to frequency or wavelength. This range is vast and includes various types of radiation, each with different energy levels. Some of the prominent sections of the electromagnetic spectrum include:

• Radio Waves: Longest wavelengths and the lowest energy
• Microwaves: Slightly shorter wavelengths than radio waves
• Infrared: Often related to heat
• Visible Light: The small part of the spectrum that is visible to human eyes
• Ultraviolet: Shorter wavelengths than visible light and more energetic
• X-Rays and Gamma Rays: Extremely short wavelengths with high energy

Visible light, which includes the calculated wavelength of 624 nm, sits just between the infrared and ultraviolet ranges. Red light constitutes the longest wavelengths in visible light, showcasing the correlation between wavelength, frequency, and energy.

Photon Energy

Photon energy is the energy carried by a single photon, which can be calculated using the equation \( E = \frac{hc}{\lambda} \). This equation, derived from quantum mechanics, demonstrates that photon's energy is inversely proportional to its wavelength. Thus, the shorter the wavelength, the higher the energy a photon carries.

The energy required to break a bond between atoms in a molecule is described in terms of bond energy, often measured in kilojoules per mole. By converting this amount of energy to the energy per molecule and applying the photon energy formula, one can find the wavelength of light that would be needed to supply enough energy to rupture a chemical bond. Photon energy plays a critical role in areas like photoelectric effects, photovoltaics, and even photosynthesis.

Planck's Constant

Planck's constant is a fundamental constant in quantum mechanics that relates the energy of a photon to the frequency of its electromagnetic wave, with a value of \( 6.626 \times 10^{-34} \text{ J·s} \). It is pivotal in bridging the gap between the energy of atomic and subatomic particles and the classical physics that governs macroscopic phenomena.

Discovered by Max Planck in the early 20th century, it revolutionized the understanding of atomic and quantum phenomena by introducing the notion that energy is quantized and transferred in discrete amounts called quanta. These insights laid the groundwork for quantum mechanics, transforming how scientists perceive the interactions of particles at the smallest scale. For the problem at hand, Planck's constant helps us express the energy-wavelength relationship clearly, giving insight into how different wavelengths of light possess varying energy levels.