\( \newcommand{\xrightleftharpoons}[2]{\overset{#1}{\underset{#2}{\rightleftharpoons}}} \) \( \newcommand{\conc}[1]{\left[\mathrm{#1}\right]} \) \( \newcommand{\chem}[1]{\mathrm{#1}} \) \( \newcommand{\expect}[1]{\left< #1 \right>} \)

Statistical Mechanics and Molecular Interactions

Shaun Williams, PhD

Molecular Interactions

Gas Phase

Intermolecular Potential Energy

Repulsion

Attractions

Dipole-Dipole Interaction

The Dipole-Dipole Interaction

Interactions utilize the axis that connects their centers of mass, set at the z-axis of the system. Theta are the angle to that axis while phi is the rotation angle of the molecule about that axis.

Dipole-Dipole Interactions Continued

Beginning the Derivation

Normalizing the Function

Simplifying the Equation

Induced Dipole

Dispersion Force

Model #1

The Lennard-Jones Potential

The Lennard-Jones potential curve has very high energy at small distances. The energy rapidly drops as the distance expands. Eventually the energy reaches a minimum values before increasing again albeit at a slower rate. The energy the becomes asymptotic to zero energy at long distances. The energy at the minimum is the bond energy. The distance at the minimum is the equilibrium bond distance. The distance where the initial steep descent cross y=0 is the Lennard-Jones distance.

Model #2

Square Well Potential

At very small distances the energy is infinite. Then it immediately drops to zero between the inner square distance and the outer square distance. Beyond that, the energy is 0.

Model #3

Hard Sphere Model

The energy is inifite inside the hard sphere distance. Beyond that the energy is zero.

Solids

Liquids

Pair Correlation Function

The number of particles directly neighboring a particle is small but they are very correlated. As you move outward, there are more and more neighbors at these levels but they are less will correlated.

Pressure of a Non-Ideal Gas

Pressure of a Non-Ideal Gas

Entropy in a Non-Ideal Gas

Now Some Highlights of the Derivation

The Pressure of a Non-Ideal Gas

The Virial Expansion

Virial Expansion Continued

Virial Expansion Examples

System \(B_2\, @\, 273\,\mathrm{K} \\ (\mathrm{L\,mol^{-1}})\) \(V_m\, @\, 1\,\mathrm{bar} \\ (\mathrm{L})\) \(V_m\, @\, 10\,\mathrm{bar} \\ (\mathrm{L})\)
Helium 0.0222 22.733 2.293
Argon -0.0279 22.683 2.243
Ideal Gas 22.711 2.271

The van der Waals Coefficients

Lennard-Jones Parameters, van der Waals Coefficients, and Second Virial Coefficients

Gas \( \bfrac{\varepsilon}{k_\mathrm{B}} \\ (\mathrm{K}) \) \( R_\chem{LJ} \\ (\mathrm{\AA}) \) \( a \\ (\mathrm{L^2\,bar\,mol^{-1}}) \) \( b \\ (\mathrm{L\,mol^{-1}}) \) \( B_2(298\,\mathrm{K}) \\ (\mathrm{L\,mol^{-1}}) \) \( B_2(298\,\mathrm{K})\text{(calc)} \\ (\mathrm{L\,mol^{-1}}) \)
\(\chem{He}\) 10 2.58 0.0346 0.0238 0.012 0.020
\(\chem{Ne}\) 36 2.95 0.208 0.01672 0.011 0.022
\(\chem{Ar}\) 120 3.44 1.355 0.03201 -0.016 -0.004
\(\chem{Kr}\) 190 3.61 2.325 0.0396 -0.051 -0.042
\(\chem{H_2}\) 33 2.97 0.2453 0.02651 0.015 0.023
\(\chem{N_2}\) 92 3.68 1.370 0.0387 -0.004 0.011
\(\chem{O_2}\) 113 3.43 1.382 0.03186 -0.016 -0.001
\(\chem{CO}\) 110 3.59 1.472 0.03948 -0.008 0.001
\(\chem{CO_2}\) 190 4.00 3.658 0.04286 -0.126 -0.057
\(\chem{CH_4}\) 137 3.82 2.300 0.04301 -0.043 -0.016
\(\chem{C_2H_2}\) 185 4.22 4.516 0.05220 -0.214 -0.062
\(\chem{C_2H_4}\) 205 4.23 4.612 0.05821 -0.139 -0.080
\(\chem{C_2H_6}\) 230 4.42 5.570 0.06499 -0.181 -0.115
\(\chem{C_6H_6}\) 440 5.27 18.82 0.1193 -1.454 -0.542

Example 4.1

Estimate the van der Waals coefficient for helium and argon using their Lennard-Jones parameters.

Conversion to a Liquid

The Pair Correlation Function

Pair Correlation Function

Position Probability Distribution Function

Manipulating This Expression

$$ \begin{align} \mathcal{G}(R) &= \frac{N-1}{4\pi \rho R^2}\mathcal{P}_R(R) = \left( \frac{V(N-1)}{4N\pi R^2} \right) \mathcal{P}_R(R) \\ &= \left(\frac{V}{4\pi R^2}\right) \mathcal{P}_R(R) \\ &= \left(\frac{V}{4\pi R^2}\right) \frac{4\pi R^2V \int_0^a \cdots \int_0^c e^{-\frac{U}{k_\mathrm{B}T}}\,dx_3\dots dz_N}{Q'_U(T,V)} \\ &= \frac{V^2 \int_0^a \cdots \int_0^c e^{-\frac{U}{k_\mathrm{B}T}}\,dx_3\dots dz_N}{Q'_U(T,V)} \end{align} $$

More Manipulations

$$ \begin{align} \mathcal{G}(R) &= V^2 \frac{\int_0^a \cdots \int_0^c e^{-\frac{U}{k_\mathrm{B}T}}\,dx_3\dots dz_N}{\int_0^a \cdots \int_0^c e^{-\frac{U}{k_\mathrm{B}T}}\,dx_1\dots dz_N} \\ &= V^2 \frac{V^{N-2}e^{-\frac{u}{k_\mathrm{B}T}}\left[ 1+\frac{\mathcal{I}(T)}{V} \right]^{\frac{N(N+1)}{2}-1}}{V^N\left[ 1+\frac{\mathcal{I}(T)}{V} \right]^{\frac{N(N+1)}{2}}} \\ &= e^{-\frac{u(R)}{k_\mathrm{B}T}}\left[ 1+\frac{\mathcal{I}(T)}{V} \right]^{-1} \approx e^{-\frac{u(R)}{k_\mathrm{B}T}} \end{align} $$

Approximate Pair Correlation Function

The approximate pair correlation function decreases in intensity as the temperature increases. Also, the approximate pair correlation function has its maximum near the minimum in the Lennard-Jones potential energy.

Bose-Einstein and Fermi-Dirac Statistics

Indistinguishable Particles

Helium-4

Graphical Representation of Two Boson Wavefunctions

When the wavefunctions of two boson wavefunctions hit each other, the constructively interfere cause a single larger wavefunction until the two wavefunctions pass each other.

Counting States

Consider Two Particles Limited to Five Values

\(I_b\)
\(I_a\) 1 2 3 4 5
1 (11) (12) (13) (14) (15)
2 (21) (22) (23) (24) (25)
3 (31) (32) (33) (34) (35)
4 (41) (42) (43) (44) (45)
5 (51) (52) (53) (54) (55)

Some States Are Available for Bosons But Not Fermions

Bose-Einstein Condensates

Superfluid \(\chem{{}^4He}\)

Superfluidic Helium Video

/