Question

Problem numbers in red indicate that the solution to the problem is given in the Student's Solutions Manual. In exploring the variation of internal energy with temperature for a two-level system with the ground and excited state separated by an energy of \(h v,\) the total energy was found to plateau at a value of \(0.5 N h v .\) Repeat this analysis for an ensemble consisting of \(N\) particles having three energy levels corresponding to \(0, h v,\) and \(2 h v .\) What is the limiting value of the total energy for this system?

Short answer

The limiting value of the total energy for the system of N particles with three energy levels corresponding to \(0, h\nu\), and \(2h\nu\) is \(Nh\nu\).

Step-by-step solution

Boltzmann Distribution

To find the probability distribution of particles in different energy levels, we will use the Boltzmann distribution. The probability for a system to be in a microstate with energy \(E_i\) is given by: \[P(E_i) = \frac{e^{-\beta E_i}}{Z}\] where \(\beta = \frac{1}{kT}\) is the inverse temperature, \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(Z\) is the partition function. The partition function is the sum of the probabilities of all the microstates: \[Z = \sum_i e^{-\beta E_i}\] For our system, there are 3 energy levels, with energies \(E_0 = 0\), \(E_1 = h\nu\), and \(E_2 = 2h\nu\). Therefore, we have: \[Z = e^{-\beta(0)} + e^{-\beta h\nu} + e^{-\beta(2h\nu)} = 1 + e^{-\beta h\nu} + e^{-2\beta h\nu}\]

Average Energy per Particle

To find the average energy per particle, we can compute the expectation value of the energy: \[\langle E \rangle = \sum_i P(E_i) E_i = \frac{1}{Z} \sum_i E_i e^{-\beta E_i}\] With our energy levels, this becomes: \[\langle E \rangle = \frac{1}{Z} (0 + h\nu e^{-\beta h\nu} + 2h\nu e^{-2\beta h\nu}) = \frac{h\nu(e^{-\beta h\nu} + 2e^{-2\beta h\nu})}{1 + e^{-\beta h\nu} + e^{-2\beta h\nu}}\]

Total Energy

The total energy of the system is the average energy per particle multiplied by the number of particles N: \[E_{total} = N \langle E \rangle = N \frac{h\nu(e^{-\beta h\nu} + 2e^{-2\beta h\nu})}{1 + e^{-\beta h\nu} + e^{-2\beta h\nu}}\]

Find the Limiting Value of Total Energy

To find the limiting value of the total energy as the temperature approaches infinity, we will study the behavior of the total energy as \(\beta\) approaches zero: \[\lim_{\beta\to 0} E_{total} = N \lim_{\beta\to 0} \frac{h\nu(e^{-\beta h\nu} + 2e^{-2\beta h\nu})}{1 + e^{-\beta h\nu} + e^{-2\beta h\nu}}\] As \(\beta\) approaches zero, the exponentials go to 1. We then have: \[\lim_{\beta\to 0} E_{total} = N \frac{h\nu(1 + 2)}{1 + 1 + 1} = Nh\nu\] So, the limiting value of the total energy for this system is \(Nh\nu\).

Key concepts

Internal Energy

In statistical mechanics, internal energy represents the total energy stored within a system. It encompasses all forms of energy possessed by particles due to not only motion but also interaction. In the context of the Boltzmann distribution, internal energy is particularly significant because it helps us understand how energy distributes across particles at different energy levels when thermal equilibrium is established. This equilibrium is crucial to determine with accuracy the state of a system, as particles continuously exchange energy via collisions or interactions.

• It includes kinetic energy, potential energy, and any other forms of energy within a system.
• Changes in internal energy often lead to changes in measurable quantities like temperature or pressure.

Understanding how energy shifts between levels allows us to predict behavior of a system when exposed to different thermal conditions. This knowledge is foundational for evaluating macroscopic properties such as the specific heat or magnetization of materials.

Three Energy Levels

In many physical systems, energy is quantized, meaning it can only take on certain discrete values. In this analysis, particles can occupy one of three distinct energy levels: 0, \(hv\), and \(2hv\). The existence of these discrete levels greatly dictates how energy is shared among particles.

• Energy Level 0: Represents the ground state where no energy is absorbed, so \(E_0 = 0\).
• Energy Level \(hv\): Represents the first excited state, with an energy level of \(E_1 = hv\).
• Energy Level \(2hv\): The second excited state, with two units of energy absorbed, so \(E_2 = 2hv\).

These quantized levels influence how particles are distributed among these states, which, in turn, affects the calculation of other thermodynamic properties like entropy and heat capacity. Knowing how particles settle among these energy levels helps understand material properties on a macroscopic scale.

Partition Function

The partition function, denoted as \(Z\), is a critical mathematical concept in statistical mechanics that provides key insights into the thermodynamic properties of a system. It's essentially a sum over all possible states, weighted by their respective Boltzmann factors. In our context:
\[ Z = e^{-\beta(0)} + e^{-\beta hu} + e^{-2\beta hu} = 1 + e^{-\beta hu} + e^{-2\beta hu} \]

• It acts as a normalization constant for probabilities, ensuring they add up to one.
• A higher partition function value indicates more accessible states for the system.
• It is vital for computing macroscopic properties such as pressure, free energy, and entropy.

By serving as a cornerstone for calculating average energies and other observables, the partition function allows us to derive detailed insights about the system’s behavior under various conditions of temperature and pressure.

Limiting Value of Total Energy

As we examine the behavior of the total energy for our system of particles at high temperatures, we look to find its limiting value. This is where the Boltzmann distribution becomes very informative, particularly as the inverse temperature (\(\beta\)) approaches zero. At this limit, energy distribution becomes even across all available states.

• When \(\beta \to 0\), temperature approaches infinity, leading exponentials \(e^{-\beta hv}\) and \(e^{-2\beta hv}\) to tend towards 1.
• This results in a simplified calculation showing that potential energy configurations are equally probable.
• For the system: \[ \lim_{\beta\to 0} E_{total} = N hu \], the limiting total energy becomes directly proportional to the number of particles and the energy quanta \(hu\).

This result is fundamental as it showcases how, regardless of how the system was initially distributed, the high-temperature behavior converges to a predictable total energy, underlining universal principles of thermodynamics and statistical mechanics.