Consider a solid surface to be a two-dimensional lattice with \(N_{\mathrm{s}}\) sites. Assume that \(N_{\mathrm{a}}\) atoms \(\left(N_{a} \ll N_{s}\right)\) are adsorbed on the surface, so that each lattice site has either zero or one adsorbed atom. An adsorbed atom has energy \(E=-\varepsilon\), where \(\varepsilon>0\). Assume the atoms on the surface do not interact with one another. If the surface is at temperature \(T\), compute the chemical potential of the adsorbed atoms as a function of \(T, \varepsilon\), and \(N_{\mathrm{a}} / N_{\mathrm{s}}\) (use the canonical ensemble).

In statistical thermodynamics, the canonical ensemble is a fundamental concept. It is a collection of systems in thermal equilibrium with a heat bath at a fixed temperature. Each system within this ensemble can exchange energy with the surroundings, but the total energy of the ensemble remains constant.

Key attributes of the canonical ensemble include:

• A fixed number of particles.
• A fixed volume and temperature.

The primary goal of using the canonical ensemble is to calculate the statistical properties of a system, like energy, pressure, or chemical potential, based on a probability distribution.

To work with the canonical ensemble, we often use functions like the partition function, which summarizes all possible states of the system and their probabilities.

The partition function is a cornerstone in statistical mechanics and thermodynamics. It is a mathematical function that encapsulates the statistical properties of a system in thermodynamic equilibrium. For a system with multiple possible energy states, the partition function, denoted as \(Z\), is given by:

Z = \sum_i e^{-\beta E_i}\, where:

• \(E_i\) is the energy of the \(i\)th state.
• \(\beta = 1/(k_B T)\), with \(k_B\) being the Boltzmann constant and \(T\) the temperature.

In the problem above, the partition function for a single site is:
\(Z_1 = e^{\beta \varepsilon} + 1 \)

For a system with non-interacting particles, the total partition function is the product of the individual partition functions of all sites. This multiplication simplifies the calculation of various thermodynamic quantities. The partition function is crucial for deriving properties such as energy, entropy, and chemical potential.

It helps us understand how the probability of a system occupying a certain energy state changes with temperature and other factors.

Chemical potential is a vital concept in thermodynamics and statistical mechanics. It represents the change in the free energy of a system when the number of particles changes, at constant temperature and volume.

For the system in the problem, the chemical potential \(\mu\) of the adsorbed atoms can be obtained using the formula:
\(\frac{N_a}{N_s} = \frac{e^{\beta (\mu + \varepsilon)}}{1 + e^{\beta (\mu + \varepsilon)}} \). Solving for \(\mu\), we get:

\(\mu = k_B T \, \ln \, \left(\frac{N_a/N_s}{1 - N_a/N_s}\right) - \varepsilon\).

This equation demonstrates how the chemical potential depends on temperature \(T\), energy \(\varepsilon\), and the ratio of the number of adsorbed atoms \(N_a\) to the total lattice sites \(N_s\).

The chemical potential plays a key role in determining the distribution of particles in different energy states, and it is crucial for understanding many phenomena in both classical and quantum systems. It helps in predicting how a system will evolve when particles are added, removed, or when other external conditions change.