Microstate Probability#

• The probability that the system has energy $E$ and particle number $N$ is proportional to the number of microstates of the reservoir with $E_R=E_{\text{tot}}-E$, $N_R=N_{\text{tot}}-N$:

• Define entropy of the reservoir $S_R=k_B\ln {\mathrm{\Omega }}_R$. Then:

• Expand to first order:

• Use thermodynamic definitions:

Thermodynamics and Legendre transform#

• In thermodynamics grand potential $\mathrm{\Psi }$ is derived from the internal energy $E(S,V,N)$ via a Legendre transform that replaces the natural variables $S\to T$ and $N\to \mu$:

  $$E(S,V,N)\to E-\left(\frac{\partial E}{\partial S}\right)S-\left(\frac{\partial E}{\partial N}\right)N=E-TS-\mu N$$

Grand Potential and Pressure#

• Using Euler’s relation for extensive thermodynamic variables:

  $$E=TS+PV+\mu N$$

• Substituting into the definition of $\mathrm{\Psi }$, we find:

  $$\mathrm{\Psi }=E-TS-\mu N=-PV$$

• Hence, the grand potential is directly related to the pressure and volume:

  $$\mathrm{\Psi }(T,V,\mu )=-PV$$

Grand Canonical Partition Function#

• To account for fluctuations in particle number, we compute the grand canonical partition function:

• Fugacity: $z=e^{\beta \mu }$, a dimensionless measure of how favorable it is to add a particle to the system.

• The average particle number is then obtained by taking first derivative withr espect to N

Chemical Potential in Terms of Pressure and Density#

• We can invert the relation for $⟨N⟩$ to express the chemical potential:

• Noting that for an ideal gas $p=\frac{N{k}_{B}T}{V}$, this becomes:

Number Fluctuations#

• In the grand canonical ensemble, number fluctuations are given by:

• So the standard deviation ${\sigma }_N\sim \sqrt{N}$, and relative fluctuations vanish as $N^{-1/2}$ in the thermodynamic limit.

Isothermal Compressibility#

• We relate number fluctuations to the isothermal compressibility ${\kappa }_T$:

• This is the well-known result for the ideal gas compressibility:

• ${\kappa }_T=\frac{1}{p}\ge 0$, a positive quantity ensuring mechanical stability.

One Site–One Molecule Model#

We consider a simple model where each adsorption site can hold at most one molecule. The surface is in thermal and chemical equilibrium with a gas reservoir at temperature $T$ and chemical potential $\mu$.

• Each site has two possible states:

  • Empty: energy $E=0$

  • Occupied: energy $E=ϵ$

• Then the grand partition function for a single site is:

$N$ Independent Sites#

• If there are $N$ independent and identical adsorption sites, then the total grand partition function is:

• Since the sites do not interact, their contributions multiply.

Average Number of Adsorbed Molecules#

The average occupation number per site is:

The total number of adsorbed molecules is:

Average Energy#

The average energy per site is:

The total energy of the system is:

Connecting to Pressure#

• For an ideal gas in the reservoir, the chemical potential is related to pressure via:

• Substituting this into the expression for $⟨n⟩$:

• and thus the total number of adsorbed molecules becomes:

• This is the Langmuir adsorption isotherm, describing how the fraction of occupied adsorption sites depends on gas pressure. It captures saturation behavior as $p\to \mathrm{\infty }$, where $⟨n⟩\to 1$.

Show code cell source Hide code cell source

import matplotlib.pyplot as plt
import numpy as np

# Parameters
T = 298  # Temperature in K
epsilon = 10 * 1.38e-23  # Adsorption energy in J (10 k_B T)
k_B = 1.38e-23  # Boltzmann constant
beta = 1 / (k_B * T)
p0 = 1  # Reference pressure (arbitrary units)

# Pressure range
p = np.linspace(0, 5, 500)

# Langmuir isotherm
n_avg = p / (p0 * np.exp(beta * epsilon) + p)

# Plotting
plt.figure(figsize=(8, 5))
plt.plot(p, n_avg, lw=2)
plt.xlabel('Pressure (p / p₀)', fontsize=12)
plt.ylabel('Average Occupancy ⟨n⟩', fontsize=12)
plt.title('Langmuir Adsorption Isotherm', fontsize=14, weight='bold')
plt.grid(True)
plt.ylim(0, 1.05)
plt.xlim(0, 5)
plt.tight_layout()
plt.show()

Equilibrium via minimization of free energy#

• Now, we consider a chemical reaction:

• We introduce an extent of reaction $\xi$ which measures how far reaction has progressed from reactants to products:

• We minimize $F(\xi )$ at constant $T,V$ by setting:

• Computing the derivative:

• with ${\nu }_i$ the stoichiometric coefficient ($<0$ for reactants, $>0$ for products). Or we can write it in terms of positive stochimoetric coefficients:

• So the equilibrium condition is:

• Exponentiating both sides we relate ration of equilibrium numbers or concentrations to partition functions

$NVT-NVE-\mu VT$#

1. Consider a three level single particle system with five microstates with energies 0, ε, ε, ε, and 2ε. What is $\mathrm{\Omega }(ϵn)$ n=0,1,2 for this system? What is the mean energy of the system if is in equilibrium with a heat bath at temperature T ?

2. Derive an expression for the chemical potential of an ideal gas using clssical mechanics model for energy $E=\frac{p^2}{2m}$ in the $\mu VT$ ensemble evaluate the fluctuations in particle number.

3. Consider a system in equilibrium with a heat bath at temperature $T$ and a particle reservoir at chemical potential $\mu$. The system can have a minimum of one and a maximum of four distinguishable particles. The particles in the system do not interact and can be in one of two states with energies zero or $\mathrm{\Delta }$. Determine the (grand) partition function of the system.

4. Combine the Gibbs formula of Entropy $S=-k_B\sum _i{p}_{i}log{p}_i$ with the Grand canonical prbability distribution $P(E_i,N)=\frac{e^{-\beta E_i+\beta \mu N}}{Z_G}$ to show that $\beta PV=log{Z}_G$

5. Derive partition function for a pressure ensemble $(T,p,N)$ and show its connection with microcanonical ensemble $NVE$

6. At a given temperature T a surface with $N_0$ adsorption centers has on average $N\ne N_0$ number of adsorbed molecules. Suppose hat there are no interactions between molecules.

   • Show that the chemical potential of adsorbed gas is given by: $\mu =k_{B}Tlog\frac{N}{N_0-Na(T)}$

   • What is the meaning of $a(T)$