Problem 41 - University Physics with Modern Physics

Question

Chemists use infrared absorption spectra to identify chemicals in a sample. In one sample, a chemist finds that light of wavelength 5.8\(\mu \mathrm{m}\) is absorbed when a molecule makes a transition from its ground harmonic oscillator level to its first excited level. (a) Find the energy of this transition. (b) If the molecule can be treated as a harmonic oscillator with mass \(5.6 \times 10^{-26} \mathrm{kg},\) find the force constant.

Step-by-Step Solution

Answer

(a) The energy of the transition is approximately \( 3.42 \times 10^{-20} \ J \). (b) The force constant is approximately \( 58.7 \ N/m \).

Step 1: Determine the Frequency of Light

First, calculate the frequency \( u \) of the absorbed light using the wavelength \( \lambda = 5.8 \mu m = 5.8 \times 10^{-6} m \). Use the formula \( u = \frac{c}{\lambda} \) where \( c = 3 \times 10^8 \ m/s \) is the speed of light. \[u = \frac{3 \times 10^8 \ m/s}{5.8 \times 10^{-6} \ m} \] The resulting frequency is approximately \( u \approx 5.17 \times 10^{13} \ Hz \).

Step 2: Calculate the Energy of the Transition

Using the frequency calculated, find the energy of the transition. This can be done using Planck's equation: \( E = h u \), where \( h = 6.626 \times 10^{-34} \ J\cdot s \) is Planck's constant. \[E = (6.626 \times 10^{-34} \ J\cdot s) \times (5.17 \times 10^{13} \ Hz) \]The energy of the transition is approximately \( E \approx 3.42 \times 10^{-20} \ J \).

Step 3: Calculate the Angular Frequency

Next, determine the angular frequency \( \omega \) of the harmonic oscillator, using the energy expression \( E = \hbar \omega \) for a quantum harmonic oscillator, where \( \hbar = \frac{h}{2\pi} \). Rearrange to find \( \omega \): \[\omega = \frac{E}{\hbar} = \frac{E}{\frac{h}{2\pi}} = \frac{E \cdot 2\pi}{h}\]Substitute \( E = 3.42 \times 10^{-20} \ J \):\[\omega = \frac{3.42 \times 10^{-20} \ J \cdot 2\pi}{6.626 \times 10^{-34} \ J\cdot s} \]The calculated \( \omega \approx 3.24 \times 10^{14} \ rad/s \).

Step 4: Calculate the Force Constant

Using the angular frequency and the given mass, find the force constant \( k \) of the harmonic oscillator. The relation between angular frequency, mass \( m \), and force constant is \( \omega = \sqrt{\frac{k}{m}} \). Solve for \( k \):\[k = m \omega^2 \]Substitute \( m = 5.6 \times 10^{-26} \ kg \) and \( \omega = 3.24 \times 10^{14} \ rad/s \):\[k = (5.6 \times 10^{-26} \ kg) \times (3.24 \times 10^{14} \ rad/s)^2 \]The force constant \( k \approx 5.87 \times 10^{1} \ N/m \).

Key Concepts

Infrared Absorption Spectra

Infrared absorption spectra are powerful tools used by chemists to analyze the molecular composition of substances. When infrared light passes through a sample, certain wavelengths of light are absorbed by the molecules. This absorption occurs because the energy of the infrared light matches the energy needed for a molecule to transition between vibrational levels.
These transitions typically involve moving from the ground state to an excited state, which can be visualized as a molecule stretching and contracting like a spring—aligning well with the concept of a harmonic oscillator. By observing which wavelengths are absorbed, chemists can determine specific characteristics of the molecules, such as their structure and the kinds of bonds they contain.
This method is especially helpful because different functional groups in organic compounds have distinct absorption patterns, allowing for detailed chemical identification through their unique 'fingerprints' in the infrared region.

Energy Transition

Energy transitions in molecules involve a change in the energy state, often of an electron or within the molecular vibrations. When a molecule absorbs electromagnetic radiation, like infrared light, it undergoes a transition from a lower energy state to a higher one.
For our exercise, the molecule acts like a harmonic oscillator, meaning it has quantized energy levels. The energy difference between these levels can be calculated using the formula: \[ E = h u \] where \( E \) is the energy of the transition, \( h \) is Planck's constant, and \( u \) is the frequency of the absorbed light. This absorption results in a measurable peak in the infrared spectrum, indicating which specific energy level transition has occurred.
The understanding of these transitions is crucial in fields such as spectroscopy, where precise detection and measurement of energy states lead to insights about molecular structure and dynamics.

Force Constant

The force constant is a fundamental parameter in understanding the behavior of a molecule modeled as a harmonic oscillator. In simple terms, it represents the stiffness of the bond connecting two atoms. A higher force constant indicates a stiffer, stronger bond, while a lower force constant implies a weaker bond.
Mathematically, the force constant \( k \) can be determined through the relationship with angular frequency \( \omega \) and mass \( m \), given by: \[ \omega = \sqrt{\frac{k}{m}} \] Rearranging for \( k \): \[ k = m \omega^2 \] In the context of the provided exercise, we analyze the bond strength by calculating \( k \) using the derived angular frequency and given molecular mass. This calculation offers insights into how tightly bound the atoms within a molecule are, impacting its vibrational motion and, consequently, its spectral properties.

Planck's Constant

Planck's constant is a fundamental constant used extensively in quantum mechanics. It establishes the scale at which quantum effects become significant, acting as a bridge between the macroscopic and quantum worlds.
Defined as \( h = 6.626 \times 10^{-34} \ J \cdot s \), Planck's constant is crucial for calculating the energy of photons associated with electromagnetic radiation. This constant appears prominently in the equation for energy transitions: \[ E = h u \] Here, \( E \) is the energy, \( h \) is Planck's constant, and \( u \) is the frequency of the radiation.
Essentially, Planck's constant demonstrates the quantized nature of energy levels, a fundamental principle of quantum mechanics. It explains why molecules can only absorb or emit discrete packets of energy, rather than continuous amounts, underscoring the structure of atomic and molecular spectra.