Question

Demonstrate that the molar enthalpy is equal to the molar energy for a collection of one-dimensional harmonic oscillators.

Short answer

In order to demonstrate that the molar enthalpy is equal to the molar energy for a collection of one-dimensional harmonic oscillators, we first calculate the energy levels using quantum mechanics. Next, we find the partition function as a sum of Boltzmann factors. We then use the partition function to determine the molar internal energy and calculate the PV term, which equals 0 for one-dimensional harmonic oscillators. Finally, we calculate the molar enthalpy as the sum of molar internal energy and PV term and find that it is equal to the molar internal energy, thus proving the statement. Specifically, both molar enthalpy and molar internal energy are given by the expression: \[H = U = \frac{N_A \hbar \omega}{2} \frac{1}{\tanh(\frac{\hbar\omega}{2k_BT})}\]

Step-by-step solution

Quantum energy levels of a one-dimensional harmonic oscillator

Using quantum mechanics, we can find that the energy levels, E, for a one-dimensional harmonic oscillator are given by: \[ E_n = (n + \frac{1}{2})\hbar\omega \] where \(n\) is a non-negative integer (n=0, 1, 2, ...), \(\hbar\) is the reduced Planck constant, and \(\omega\) is the angular frequency.

Calculate partition function for one-dimensional harmonic oscillators

The partition function, Z, for one-dimensional harmonic oscillators can be obtained considering each available energy level along with the Boltzmann factor, given by: \[Z = \sum_{n=0}^{\infty} e^{-\beta E_n}\] where \(\beta\) is the inverse temperature, defined as \(\beta = \frac{1}{k_BT}\), with \(k_B\) being the Boltzmann constant, and T is the temperature. Plugging the energy levels from Step 1, we get: \[Z = \sum_{n=0}^{\infty} e^{-(n+\frac{1}{2})\hbar\omega\beta}\] This is a geometric series, and its sum is given by: \[Z = \frac{e^{-\frac{1}{2}\hbar\omega\beta}}{1-e^{-\hbar\omega\beta}}\]

Calculate the molar internal energy, U

The molar internal energy, U, can be calculated using the derivative of the partition function with respect to the inverse temperature times the Boltzmann constant and the Avogadro number, N_A: \[U = N_A k_B T^2 \frac{\partial \ln Z}{\partial T}\] Using the partition function from Step 2, we can calculate: \[U = N_A k_B T^2 \frac{\partial \ln{(\frac{e^{-\frac{1}{2}\hbar\omega\beta}}{1-e^{-\hbar\omega\beta}})}}{\partial T}\] After performing the partial derivative and simplification, we find that: \[U = \frac{N_A \hbar \omega}{2} \frac{1}{\tanh(\frac{\hbar\omega}{2k_BT})}\]

Calculate the PV term

For a one-dimensional harmonic oscillator, we don't have a continuous volume, so no pressure. Therefore, the PV term (P times the molar volume, V) is equal to 0.

Calculate the molar enthalpy, H

The molar enthalpy, H, can be calculated as the sum of molar internal energy, U, and the product of pressure and molar volume, PV, using the results from Steps 3 and 4: \[H = U + PV\] Since the PV term is 0, we have: \[H = U = \frac{N_A \hbar \omega}{2} \frac{1}{\tanh(\frac{\hbar\omega}{2k_BT})}\] By comparing the expressions of molar enthalpy and molar internal energy, it is demonstrated that they are equal for a collection of one-dimensional harmonic oscillators.

Key concepts

Quantum Energy Levels

In quantum mechanics, energy levels are discrete rather than continuous. This means that a system like a one-dimensional harmonic oscillator can only exist in specific energy states. These states are defined by the quantum number \( n \), which is a non-negative integer (\( n = 0, 1, 2, \ldots \)). The energy levels \( E_n \) for a harmonic oscillator are given by:\[E_n = (n + \frac{1}{2})\hbar\omega\]

• \( \hbar \) is the reduced Planck constant, a fundamental constant in quantum mechanics.
• \( \omega \) is the angular frequency of the oscillator.

These quantized energy levels indicate that the oscillator can never have zero energy; even at its lowest energy state, it has the zero-point energy \( \frac{1}{2}\hbar\omega \). This concept is crucial in understanding the behaviors of microscopic systems in quantum mechanics.

Partition Function

The partition function \( Z \) is a central concept in statistical mechanics. It encapsulates all possible energy states of a system and significantly impacts thermodynamic properties. For one-dimensional harmonic oscillators, the partition function is given by:\[Z = \sum_{n=0}^{\infty} e^{-\beta E_n}\]

• \( \beta = \frac{1}{k_B T} \), where \( k_B \) is the Boltzmann constant and \( T \) is the temperature.

Substituting energy levels from quantum mechanics, we recognize the series as a geometric one:\[Z = \frac{e^{-\frac{1}{2}\hbar\omega\beta}}{1-e^{-\hbar\omega\beta}}\]This formula allows us to calculate other thermodynamic properties, like internal energy and free energy, by linking the microscopic energies to macroscopic observables.

Molar Internal Energy

Molar internal energy \( U \) is a macroscopic energy measure over Avogadro's number of particles. For one-dimensional harmonic oscillators, it can be determined using the partition function:\[U = N_A k_B T^2 \frac{\partial \ln Z}{\partial T}\]

• \( N_A \) is Avogadro's number, representing the number of entities in a mole.

After performing the required differentiation and simplifications, we derive:\[U = \frac{N_A \hbar \omega}{2} \frac{1}{\tanh(\frac{\hbar\omega}{2k_BT})}\]Molar internal energy helps determine the energy contained in a mole of quantum harmonic oscillators and is equal to the molar enthalpy as shown in the exercise.

One-Dimensional Harmonic Oscillators

One-dimensional harmonic oscillators are foundational models in physics to study simple harmonic motion. This model is represented by a particle moving back and forth along a single line under the influence of a restoring force proportional to its displacement from a point of equilibrium:

• The potential energy is quadratic, leading to a constant frequency \( \omega \).
• Its quantized energy states are directly influenced by classical mechanics principles combined with quantum transitions.

Harmonic oscillators provide insights into molecular vibrations and phonons in solid-state physics, serving as building blocks for more complex systems.

Quantum Mechanics

Quantum mechanics is a fundamental theory describing physical phenomena at microscopic scales, where classical mechanics fails to accurately predict behaviors. It incorporates the quantization of physical properties such as energy:

• Introduces concepts like wave-particle duality, uncertainty principle, and superposition.
• Utilizes mathematical frameworks involving operators, wave functions, and observable probabilities.

For one-dimensional harmonic oscillators, quantum mechanics explains the discrete energy levels they can occupy and provides tools for calculating important thermodynamic properties. Understanding how quantum mechanics impacts such systems is key to advancing technology in various fields, including electronics and nanotechnology.