Problem 42 - Chemistry An Atoms First Approach

Question

A photon of ultraviolet (UV) light possesses enough energy to mutate a strand of human DNA. What is the energy of a single UV photon and 1 mole of UV photons having a wavelength of \(25 \mathrm{nm} ? 1\) mol UV photons \(=6.022 \times 10^{23} \mathrm{UV}\) photons.

Step-by-Step Solution

Answer

The energy of a single UV photon with a wavelength of \(25 \ nm\) is approximately \(7.98 \times 10^{-19} \ J\), and the energy of 1 mole of UV photons is approximately \(4.80 \times 10^5 \ J/mol\).

Step 1: Identify the constants and variables given in the problem.

In this problem, we are given the wavelength of UV light λ = 25 nm. We will need to convert this to meters for our calculations. We also know the speed of light c = 3.00 x 10^8 m/s, Planck's constant h = 6.63 x 10^(-34) Js, and Avogadro's number N_A = 6.022 x 10^23 particles/mol.

Step 2: Convert the wavelength from nm to meters.

To convert the wavelength from nm to meters, we use the conversion factor 1 nm = 10^(-9) m. Therefore: \(λ = 25 \ nm \times \frac{10^{-9} \ m}{1 \ nm} = 25 \times 10^{-9} \ m\)

Step 3: Use the energy-wavelength formula to calculate the energy of a single UV photon.

The energy of a light wave can be calculated using the formula: \(E = \frac{hc}{λ}\) where E is the energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. We already have all the values, so we can substitute them into the formula: \(E = \frac{(6.63 \times 10^{-34} \ Js)(3.00 \times 10^8 \ m/s)}{25 \times 10^{-9} \ m}\) Now, calculate the energy of a single UV photon: \(E ≈ 7.98 \times 10^{-19} \ J \)

Step 4: Calculate the energy of 1 mole of UV photons.

Now that we have the energy of a single UV photon, we can find the energy of 1 mole of UV photons by multiplying the energy of a single photon with Avogadro's number: \(E_{mole} = E \times N_A\) \(E_{mole} = (7.98 \times 10^{-19} \ J)(6.022 \times 10^{23} \ photons/mol)\) Now, calculate the energy of 1 mole of UV photons: \(E_{mole} ≈ 4.80 \times 10^5 \ J/mol \) The energy of a single UV photon is approximately \(7.98 \times 10^{-19} \ \text{J}\) and the energy of 1 mole of UV photons is approximately \(4.80 \times 10^5 \ \text{J/mol}\).

Key Concepts

Energy-Wavelength Relationship

Understanding the energy-wavelength relationship is crucial when studying the properties of light, especially in the context of the photoelectric effect and various spectroscopic techniques. The energy (\(E\)) of a photon is inversely proportional to its wavelength (\( \text{lambda} \(\text{λ}\)\)), which means that photons with shorter wavelengths carry more energy than those with longer wavelengths. This relationship is expressed mathematically by the equation \[E = \frac{hc}{λ}\] where \(h\) is Planck's constant, \(c\) is the speed of light in a vacuum, and \(λ\) is the wavelength of the photon. For example, ultraviolet (UV) photons, which have relatively short wavelengths, contain enough energy to cause chemical changes, such as mutating DNA. To calculate the energy of a UV photon with a wavelength of 25 nm, we can apply this formula after converting the wavelength to meters, since the units of \(h\) and \(c\) are in joule seconds (Js) and meters per second (m/s) respectively.

Planck's Constant

Planck's constant (\(h\)) is a fundamental quantity in quantum mechanics with a value of approximately \(6.63 \times 10^{-34}\) joule seconds (Js). Named after the physicist Max Planck, who first introduced it, this constant plays a pivotal role in the quantization of energy. It represents the proportionality factor between the energy of a photon and the frequency of its associated electromagnetic wave. In the formula \[E = \frac{hc}{λ}\] mentioned earlier, Planck's constant is essential for calculating the energy of photons.
Planck's constant helps bridge the gap between classical and quantum physics, introducing the concept that energy transfer can only happen in discrete amounts, or 'quanta,' rather than in a continuous flow. This quantization of energy leads to the conclusion that light exhibits both wave and particle characteristics — a duality key to understanding phenomena such as the photoelectric effect.

Avogadro's Number

Avogadro's number, denoted as \(N_A\), is a constant that represents the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. The approximate value of Avogadro's number is \(6.022 \times 10^{23}\) particles per mole, a fundamental quantity in chemistry for converting between numbers of particles and amount of substance. When calculating the energy of a mole of UV photons (as in our example problem), we multiply the energy of a single photon by Avogadro's number to switch our perspective from the microscopic to the macroscopic scale.
This constant not only simplifies calculations in chemistry and physics but also provides insight into the molecular and atomic scales by relating the macroscopic quantities we can measure directly to the number of particles that interact on the much smaller, imperceptible scale.

Photoelectric Effect

The photoelectric effect is a phenomenon in which electrons are emitted from a material, typically a metal, when it absorbs electromagnetic radiation, such as light. One of the pivotal experiments in demonstrating quantum mechanics, this effect provided evidence that light can behave as a particle and not just a wave. In the context of UV light and its interactions with matter, the energy possessed by UV photons is often sufficient to overcome the binding energy that holds electrons in place within atoms or molecules.
This process underscores the application of the energy-wavelength relationship, Planck's constant, and the quantization of light; energy must be absorbed in specific quantized amounts, or 'photon packets', for the photoelectric effect to occur. Notably, it was Albert Einstein who offered a definitive explanation of the photoelectric effect, for which he was awarded the Nobel Prize in Physics, solidifying the concept of light quanta or photons.