A molecule pair has an LJ potential with σ = 3.4 Å. At what separation distance does the potential energy reach its minimum (equilibrium) value?
Ar = σ = 3.4 Å, because σ is defined as the equilibrium separation
Br = 2^(1/6) × σ ≈ 3.82 Å, just beyond the zero-crossing
Cr = 2σ = 6.8 Å, where the long-range attraction is strongest
Dr = σ/2 = 1.7 Å, inside the repulsive core
σ is the collision diameter — the distance at which the LJ potential crosses zero (repulsion and attraction exactly balance). The actual minimum occurs slightly farther out, at r_min = 2^(1/6)σ ≈ 1.12σ. This is a common confusion: σ is a zero-crossing, not an equilibrium. The well depth ε (positive) is the magnitude of the potential at r_min, and the potential energy there is −ε.
Question 2 Multiple Choice
Why is the r⁻¹² repulsion term used in the Lennard-Jones potential rather than an exponential function, even though the exponential is physically more accurate?
AQuantum mechanics rigorously derives a 12th-power repulsion from Pauli exclusion between electron clouds
BExperimental data for noble gases precisely fit a 12th-power law at short range
CThe exponent 12 is the square of 6, making the repulsion term the square of the attractive term and drastically simplifying computation
DAn exponential function cannot produce the steep repulsive wall observed in molecular collisions
The r⁻¹² choice is computational convenience, not physical rigor. Because (σ/r)¹² = [(σ/r)⁶]², the repulsive term can be computed by squaring an already-computed quantity — halving the number of expensive power operations in molecular simulation. Exponential repulsion (the Buckingham potential) is physically more accurate but harder to compute. The key insight is that the r⁻¹² exponent has no deep theoretical justification; it was chosen to make the math tractable.
Question 3 True / False
The second virial coefficient B(T) is negative at low temperatures for gases modeled by the Lennard-Jones potential.
TTrue
FFalse
Answer: True
At low temperatures, kT is small relative to the well depth ε, so thermal energy cannot overcome the attractive part of the potential. Molecules spend more time near each other than a purely repulsive gas would, making the gas more compressible than ideal. This corresponds to negative B(T). At high temperatures, kT >> ε, the attraction becomes negligible, and the hard repulsive core dominates — B(T) becomes positive. This temperature dependence is what makes B(T) measurements across a range of T so useful for simultaneously fitting both ε and σ.
Question 4 True / False
The parameter ε in the Lennard-Jones potential represents the total potential energy of two molecules at their equilibrium separation.
TTrue
FFalse
Answer: False
ε (positive) is the depth of the energy well — the magnitude of the minimum potential energy. The actual potential energy at the minimum is −ε (negative, since it represents attraction). The sign matters: the well depth ε tells you how strongly the molecules attract each other, but the potential energy there is −ε. Confusing ε with the total energy leads to sign errors when calculating thermodynamic quantities from the pair potential.
Question 5 Short Answer
Why does the second virial coefficient B(T) provide an experimental route to determining the Lennard-Jones parameters ε and σ? What is the physical logic connecting a macroscopic PVT measurement to a microscopic pair potential?
Think about your answer, then reveal below.
Model answer: B(T) = −2πN_A∫[exp(−u(r)/kT)−1]r²dr. This integral directly encodes how much two molecules deviate from independent (ideal) behavior due to their pairwise interaction. At each temperature, B(T) is a single number derived from PVT data. Because the LJ potential has two parameters (ε, σ) and B(T) depends on temperature, measuring B across a range of temperatures gives a curve whose shape and magnitude can only be matched by specific values of ε and σ. The bridge is statistical mechanics: the Mayer f-function exp(−u/kT)−1 weights the interaction energy by Boltzmann factors, connecting the microscopic energy landscape to the macroscopic equation of state.
The key insight is that PVT deviations from ideality encode pair interaction information. Ideal gas molecules don't interact; real gas deviations are caused by interactions. B(T) captures the pairwise contribution. Because the LJ potential changes shape with ε and σ, different parameter choices produce differently shaped B(T) curves. Fitting experimental B(T) data constrains both parameters simultaneously — turning PVT measurements taken with a pressure gauge into knowledge about molecular interaction strength and size.