Question

A thermodynamic system consists of \(N\) independent, distinguishable particles. Each particle has three energy levels at \(0, \varepsilon\), and \(2 s\), with degeneracies of 1,3 , and 5 , respectively. The system is in thermal equilibrium with a heat reservoir of absolute temperature \(T=\varepsilon / k\), where \(k\) is Boltzmann's constant. a. Calculate the partition function for this system. b. What fraction of the particles resides in each energy level? c. Determine the average particle energy and the associated mean particle entropy- d. At what temperature would the population of the energy level having \(2 \varepsilon\) be equal to that at \(\varepsilon\) ?

Short answer

Partition function: \(q = 1 + 3e^{-1} + 5e^{-2}\); Fraction of particles at each level: calculate using step 4; Average energy: calculate using step 5; Mean particle entropy: calculate using step 6; Temperature for equal population: solve using step 7.

Step-by-step solution

Define the partition function for one particle

The partition function for a single particle, denoted as q, is the sum over all the energy states, weighted by their degeneracy and the Boltzmann factor. This can be defined as: \(q = g_0e^{-E_0 / (kT)} + g_1e^{-E_1 / (kT)} + g_2e^{-E_2 / (kT)}\), where \(g_i\) is the degeneracy and \(E_i\) is the energy of the i-th level. For the given system, it becomes: \(q = 1e^{-(0) / (kT)} + 3e^{- (\varepsilon) / (kT)} + 5e^{- (2 \times \varepsilon) / (kT)}\). Since \(T = \frac{\varepsilon}{k}\), the Boltzmann factor for each energy level can be simplified.

Simplify the partition function for one particle

Using the temperature provided we get: \(q = 1e^0 + 3e^{-1} + 5e^{-2}\). Calculating the exponentials, the partition function for a single particle simplifies to: \(q = 1 + 3e^{-1} + 5e^{-2}\).

Calculate the partition function for the entire system

The partition function for the system of N particles, \(Z\), assuming that they are independent and distinguishable, is given by the partition function of a single particle raised to the power of the total number of particles, N. So, we have \(Z = q^N\), where \(q\) is the partition function for one particle.

Calculate fraction of particles in each energy level

The fraction of particles in an energy level is given by the ratio of the degeneracy of that level times its Boltzmann factor, over the partition function of one particle. The fraction for each energy level \(i\) is \(f_i = \frac{g_i e^{-E_i / (kT)}}{q}\). We calculate the fractions using the energy levels given.

Determine the average particle energy

The average energy, \(\langle E \rangle\), of a particle in the system can be found by considering the energy of each level weighted by its associated fraction. \(\langle E \rangle = f_0 \times 0 + f_1 \times \varepsilon + f_2 \times (2\varepsilon)\). Use the fractions from Step 4 to calculate \(\langle E \rangle\).

Determine the mean particle entropy

The mean particle entropy for the system can be calculated using the formula \(S = k \ln{Z} + \frac{\text{average energy}}{T}\). Where \(Z\) is the partition function of the system, and average energy has been calculated in Step 5.

Calculate the temperature for equal population levels

To find the temperature at which the populations of \(\varepsilon\) and \(2\varepsilon\) levels are equal, we set the fractions equal to each other: \(f_1 = f_2\). Substituting the expressions for \(f_1\) and \(f_2\), we get an equation involving the temperature: \(3e^{-\varepsilon/(kT')} = 5e^{-2\varepsilon/(kT')}\). Taking the natural logarithm of both sides and solving for \(T'\), will give the desired temperature.

Key concepts

Partition Function

In the realm of statistical thermodynamics, the partition function is a fundamental concept as it is instrumental in deducing the statistical properties of a system at thermal equilibrium.

Consider a thermodynamic system with multiple energy levels; the partition function, often represented as q for a single particle or Z for the entire system, is a sum of the exponential factors associated with these energy levels. Mathematically, for a single particle, it's given by \[ q = \textstyle\sum g_i e^{-E_i / (kT)} \],where g_i represents the degeneracy of the i-th energy level, E_i is the energy of that level, k is the Boltzmann constant, and T is the temperature in kelvins.

For our system of N particles with distinct energy levels, we extrapolate to find the partition function of the entire system as \[ Z = q^N \].
This expression encapsulates all possible states of the system and serves as a cornerstone for further thermodynamic calculations, such as determining probabilities of particle distributions across various energy levels and calculating macroscopic properties like internal energy and entropy.

Boltzmann Constant

The Boltzmann constant, symbolized as k or k_B, serves as a bridge between the microscopic and macroscopic worlds in thermodynamics. It appears in various statistical and thermodynamic equations, reflecting its core role in these fields.

This constant essentially converts temperature into energy and vice versa, and its value is approximately\[ k = 1.380649 \times 10^{-23} \frac{\text{J}}{\text{K}} \],where J stands for joules and K for kelvins. In our exercise, the Boltzmann constant allows us to calculate the exponentials in the partition function, translating the effect of temperature on the energy distribution among the particles.

Average Energy in Thermodynamics

The average energy of particles in a thermodynamic system at equilibrium is crucial to understanding and predicting how a system behaves and interacts with its surroundings.

The average energy, denoted \( \langle E \rangle \), can be found by summing the product of each energy level's probability (or the fraction of particles at that level) with its corresponding energy. Formally, it's expressed as\[ \langle E \rangle = \frac{1}{Z} \sum_{i} g_i E_i e^{-E_i / (kT)} \],where the sum is over all energy states. This relationship allows us to calculate the average energy based on the known energies and degeneracies of each state and directly relates to observable quantities like temperature and heat capacity.

Entropy in Thermodynamics

In statistical thermodynamics, entropy is a measure of disorder or randomness within a thermodynamic system. It is a central concept and is associated with the number of microscopic configurations that correspond to a system's macroscopic state.

Entropy, represented by S, can be calculated using the partition function through the Boltzmann's entropy formula, \[ S = k \ln(Z) + \frac{\text{average energy}}{T} \],for a system of independent particles. This equation underscores the dual nature of entropy — as an inherent property of matter arising from its microscopic states, represented by k ln(Z), and as an energy-related property, adjusted by the average energy over temperature.

Such an understanding of entropy allows scientists and engineers to predict how energy will be distributed within a system, and how that system will respond to changes in its environment.