Question

Consider a monatomic ideal gas having the molecular partition function $$ Z=g .\left(\frac{2 \pi m k T}{h^{2}}\right)^{3 / 2} V, $$ where \(g_{\circ}\) is the degeneracy of the ground electronic state. a. Develop expressions for both \(\ln Q\) and \(\ln \Xi\), where \(Q\) is the canonical partition function and \(\Xi\) is the grand canonical partition function. b. For the canonical ensemble, the primary statistical and classical relations are $$ A=-k T \ln Q \quad d A=-S d T-P d V+\mu d N . $$ Employing the canonical partition function, develop expressions for \(S, P\), and \(\mu\) for the monatomic ideal gas. c. Show that the ideal gas equation of state is obtained if \(Q=f(T) V^{N}\), where \(f(T)\) is any function of temperature. What are the implications of this result? d. For the grand canonical ensemble, the primary statistical and classical relations are $$ P V=k T \ln \Xi \quad d(P V)=S d T+N d \mu+P d V . $$ Using the grand canonical ensemble, develop expressions for \(N, P\), and \(S\) for the monatomic ideal gas. Show that your results are the same as those obtained using (1) the canonical ensemble and (2) the Maxwell-Boltzmann method.

Short answer

The expressions for \( S, P, \) and \( \mu \) using the canonical partition function are \( S = k \left(\frac{5}{2} N + N \ln \left(\frac{V}{N} \left(\frac{mkT}{2\pi \hbar^{2}}\right)^{\frac{3}{2}}\right)\right) \) for entropy, \( P = \frac{NkT}{V} \) for pressure, and \( \mu = -kT \ln \left(\frac{V}{N} \left(\frac{mkT}{2\pi \hbar^{2}}\right)^{\frac{3}{2}}\right) \) for the chemical potential. For part c, using \( Q = f(T) V^{N} \) leads us to the ideal gas law, \( PV=NkT \) which implies that the thermodynamic properties are extensive. Using the grand canonical partition function, we find the expressions for \( N, P, \) and \( S \) are consistent with the results from the canonical ensemble and the Maxwell-Boltzmann method.

Step-by-step solution

Express \( \ln Q \) for the Canonical Partition Function

Starting with the molecular partition function for a monatomic ideal gas, \( Z=g \left(\frac{2 \pi m k T}{h^2}\right)^{\frac{3}{2}} V \), the canonical partition function \( Q \) for \( N \) particles is given by \( Q = Z^N/N! \) due to indistinguishability of particles. Taking the logarithm of \( Q \) and using Stirling's approximation, \( \ln N! \approx N \ln N - N \) when \( N \) is large, we get \( \ln Q = N \ln Z - N \ln N + N \)

Key concepts

Statistical Thermodynamics

Statistical thermodynamics provides a microscopic understanding of thermodynamic phenomena, bridging the gap between the macroscopic thermodynamic quantities and the microscopic behaviors of individual particles.

By using probability and statistical methods, this branch of physics explores how the properties of atoms and molecules in thermodynamic systems can lead to macroscopic laws, such as the laws of thermodynamics. Fundamental to this field is the concept of the ensemble, which is a vast number of virtual copies of a system considered all at once to simplify the mathematics of complex systems. There are various types of ensembles, with the canonical ensemble and grand canonical ensemble being particularly important for understanding systems at constant temperature and varying particle numbers respectively.

A central tool in statistical thermodynamics is the partition function, a quantity that encapsulates all possible states of a system and thus allows the computation of thermodynamic variables. The behavior of an ideal gas, given its simplicity, often serves as an entry point to understanding these more complex concepts.

Grand Canonical Partition Function

The grand canonical partition function, denoted as \( \Xi \), is crucial in situations where both the particle number and energy can fluctuate, such as in an open system in contact with a large reservoir.

Exploring the concept from the step-by-step solution provided, \( \Xi \) is related to the pressure (\( P \) in a thermodynamic context. This relationship can be seen in the equation \( P V = k T \ln \Xi \), where \( V \) is the volume, \( T \) is the temperature, and \( k \) is Boltzmann's constant. When comparing to the canonical partition function \( Q \) for a closed system, \( \Xi \) takes into account the additional degree of freedom for particle number variation. This is particularly relevant for systems where the exchange of particles with the surroundings cannot be ignored, such as in the context of chemical reactions or in systems not isolated from their environment.

Monatomic Ideal Gas

A monatomic ideal gas is a theoretical gas composed entirely of single-atom molecules, which are considered to be point particles. These particles move freely and do not interact with each other except through elastic collisions.

In the context of the exercise, the molecular partition function for this gas leads to expressions for various thermodynamic properties. The assumptions of the ideal gas model allow us to derive elegant formulas, such as the ideal gas law \( PV=nRT \) and expressions for internal energy that depend solely on temperature. The model simplifies the complex behaviors of real gases by ignoring intermolecular forces and volume occupied by gas particles, which can be a useful approximation under many conditions. This simplification is what allows students to understand more complicated thermodynamic and statistical principles, as demonstrated in the solutions of \( S, P, \) and \( \mu \) for monatomic ideal gas using both the canonical and grand canonical partitions.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution is a statistical law that describes the distribution of speeds among particles in a monatomic ideal gas. This distribution is fundamental to the kinetic theory of gases, providing insight into the average speed, the most probable speed, and the distribution of kinetic energy within a gas.

It is a result derived under the assumptions that the gas particles do not interact and are distributed randomly in space. This distribution function helps to connect statistical properties of a gas to observable macroscopic quantities, like temperature and pressure. This connection is particularly important when looking at non-equilibrium states or predicting the rates of molecular collisions and reactions. In the broader context of statistical thermodynamics, the Maxwell-Boltzmann distribution is an example of an equilibrium distribution for a system described well by classical mechanics as opposed to quantum mechanics.