Question

The thermodynamic probability for corrected Maxwell-Boltzmann statistics is given by $$ W_{C M B}=\prod_{j} \frac{g_{j}^{N_{j}}}{N_{j} !} $$ where \(N_{j}\) is the number of particles and \(g_{j}\) is the degeneracy of the \(j\) th energy level. a. Using the methods of statistical thermodynamics, show that the equilibrium particle distribution is $$ N_{j}=g_{j} e^{-\alpha} e^{-\beta \varepsilon_{j}} . $$ b. Defining the molecular partition function \(Z=\sum_{j} g_{j} e^{-\beta \varepsilon_{j}}\), show that $$ S=k \beta E+k N\left[\ln \left(\frac{Z}{N}\right)+1\right] . $$ c. Employing the Helmholtz free energy and presuming that \(\beta=1 / k T\), verify that $$ P=N k T\left(\frac{\partial \ln Z}{\partial V}\right)_{T} . $$

Short answer

a. Apply the method of Lagrange multipliers to the logarithm of \( W_{CMB} \) considering constraints and solve to find \( N_{j}=g_{j} e^{-\alpha} e^{-\beta \varepsilon_{j}} \). b. Define entropy using statistical mechanics and express it in terms of molecular partition function \( Z \) to get \( S=k \beta E+k N[\ln (\frac{Z}{N})+1] \). c. Use the relation of Helmholtz free energy to pressure and differentiate with respect to volume to show \( P=N k T(\frac{\partial \ln Z}{\partial V})_{T} \).

Step-by-step solution

Use the Lagrange multipliers to maximize thermodynamic probability

To find the equilibrium particle distribution, consider the maximization of thermodynamic probability, subject to constraints on the total number of particles, and the total energy. The constraints are given as \[ \sum_{j} N_{j} = N \] (total number of particles) and \[ \sum_{j} N_{j} \varepsilon_{j} = E \] (total energy), where \( \varepsilon_{j} \) is the energy of the jth level. Introduce the Lagrange multipliers \( \alpha \) and \( \beta \) for these constraints and maximize the log of \( W_{CMB} \) with respect to \( N_j \), which will be \( \ln W_{CMB} - \alpha (\sum_{j} N_{j} - N) - \beta (\sum_{j} N_{j} \varepsilon_{j} - E) \).

Differentiate and set to zero for maximization

Differentiating the quantity with respect to \( N_j \) gives, \[ \frac{d}{dN_{j}} (\ln W_{CMB} - \alpha (\sum_{j} N_{j} - N) - \beta (\sum_{j} N_{j} \varepsilon_{j} - E)) = 0 \]. Solve for each \( N_{j} \) to find its equilibrium value.

Solve for the equilibrium particle distribution

The differentiation yields the expression for the equilibrium particle distribution \( N_{j} = g_{j} e^{-\alpha} e^{-\beta \varepsilon_{j}} \), as the derivative of \( \ln W_{CMB} \) with respect to \( N_j \) involves terms with \( g_{j}^{N_{j}}/(N_{j}!) \), and includes the exponential terms from the constraints involving \( \alpha \) and \( \beta \).

Define the molecular partition function and express the entropy

First, define the molecular partition function \( Z \) as given by \( Z=\sum_{j} g_{j} e^{-\beta \varepsilon_{j}} \). Then use the definition of entropy in the context of statistical mechanics, \( S = -k \sum_{j} P_j \ln P_j \), where \( P_j \) is the probability of occupation of the jth level. We can express \( P_j \) in terms of \( N_j \) and \( Z \).

Derive the expression for entropy

After expressing \( P_j \) in terms of the molecular partition function and the number of particles, substitute these into the formula for entropy, and simplify to obtain \( S=k \beta E+k N[\ln (\frac{Z}{N})+1] \).

Relate the Helmholtz free energy with pressure and volume

The Helmholtz free energy is related to the molecular partition function by \( F = -kT \ln Z \). To find how pressure \( P \) relates to \( F \), we consider the derivative of \( F \) with respect to volume \( V \), while keeping temperature \( T \) constant. Then knowing \( P = -\left(\frac{\partial F}{\partial V}\right)_{T} \), and substituting for \( F \), we show that \( P=N k T(\frac{\partial \ln Z}{\partial V})_{T} \).

Key concepts

Statistical Thermodynamics

Statistical thermodynamics, also known as statistical mechanics, provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed at the human scale, such as temperature, pressure, and volume.

At the heart of statistical thermodynamics is the notion that bulk properties are the result of the motions and interactions of particles within the material. The Maxwell-Boltzmann statistics offer one way to describe the distribution of particles among various energy states. For instance, in an ideal gas, the energy levels are continuous and widely spaced, and the particles may be distributed among them according to their energy.

The fundamental postulate of statistical thermodynamics is that all accessible microstates of a system are equally probable at equilibrium. This assumption allows for the calculation of thermodynamic properties using probability theory. The ultimate goal is to derive expressions for macroscopic thermodynamic quantities, such as entropy or Helmholtz free energy, from a knowledge of the microscopic motion of the particles in the system.

Thermodynamic Probability

The concept of thermodynamic probability, or the number of ways a particular state can be achieved, plays a crucial role in statistical thermodynamics. The equation

\(W_{CMB} = \prod_{j} \frac{g_{j}^{N_{j}}}{N_{j}!}\)

represents the thermodynamic probability for corrected Maxwell-Boltzmann statistics, accounting for the indistinguishability of particles. In this equation, \(N_{j}\) is the number of particles occupying the \(j\)th energy level, and \(g_{j}\) is the degeneracy of that level—essentially, the number of ways the energy state \(j\) can be populated.

By maximizing this thermodynamic probability, subject to constraints like the conservation of the total number of particles and total energy, one can derive the most probable distribution of particles among different energy levels, which characterizes the system at equilibrium.

Molecular Partition Function

The molecular partition function, represented as

\(Z = \sum_{j} g_{j} e^{-\beta \varepsilon_{j}}\)

is a key quantity in statistical thermodynamics that provides a link between the microscopic and macroscopic properties. Here, \(\beta = 1/kT\), where \(k\) is the Boltzmann constant and \(T\) is the temperature, while \(\varepsilon_{j}\) is the energy of the \(j\)th level.

The partition function sums over all possible states and can be thought of as a measure of the number of states available to a system at a given temperature. Knowing this function allows us to calculate observable quantities, such as the average energy, entropy, and pressure of the system.

Helmholtz Free Energy

The Helmholtz free energy (\(F\)) is a thermodynamic potential that measures the work obtainable from a closed thermodynamic system at a constant temperature. In the realm of statistical thermodynamics, the Helmholtz free energy is given by

\(F = -kT \ln Z\)

where \(Z\) is the molecular partition function. The Helmholtz free energy is significant because it accounts for both the energy and the entropy of a system, offering insight into the balance between energy dispersal and order within the system.

Moreover, it is instrumental in determining equilibrium as a system naturally progresses toward a state of minimum free energy. For an ideal gas, changes in the Helmholtz free energy can be related to changes in volume and pressure, providing a bridge between microscopic and macroscopic descriptions.

Entropy

Entropy, symbolized by \(S\), is a fundamental concept in both thermodynamics and statistical thermodynamics. It quantitatively measures the amount of disorder or randomness in a system.

Using the Maxwell-Boltzmann distribution along with the molecular partition function, the formula for entropy

\(S=k \beta E+k N\left[\ln \left(\frac{Z}{N}\right)+1\right]\)

relates the macroscopic definition of entropy with its microscopic statistical counterpart by considering the distribution of particles among various energy levels and the inherent probability of these configurations.

Entropy can be interpreted as a measure of the number of microstates that correspond to a macrostate, effectively linking the microscopic details of particle arrangements to macroscopic observables. This connection between the microscopic and macroscopic worlds is a key feature of statistical thermodynamics that deepens our understanding of the fundamental laws of physics and chemistry governing the behavior of all matter.