A gas is studied at its Boyle temperature. A chemist concludes that 'the gas behaves ideally at this temperature because intermolecular forces cancel out.' What is wrong with this conclusion?
AThe Boyle temperature is where B₂ is at its maximum, not zero
BIntermolecular forces do not actually cancel — B₂ = 0 is a coincidental balance of repulsive and attractive contributions to the integral, not an absence of forces
CThe gas behaves ideally at all temperatures, not just the Boyle temperature
DAt the Boyle temperature, only repulsive forces remain, so the gas is not truly ideal
B₂(T_B) = 0 means that the positive contribution from short-range repulsion and the negative contribution from attractive interactions in the integral ∫[exp(−u(r)/kT) − 1] 4πr² dr happen to cancel exactly. Intermolecular forces are still very much present — the Mayer f-function is nonzero at many separations. The gas obeys PV = NkT to first order, but this is a cancellation, not absence of interactions. Higher-order virial terms (B₃, etc.) are still nonzero.
Question 2 Multiple Choice
At low temperatures, B₂(T) is negative. What physical process causes this?
AAt low temperatures, molecular velocities are too low for molecules to collide, so there is no repulsive contribution
BThe hard-core repulsion disappears at low temperatures because kT is small
CThe attractive potential well becomes significant relative to kT, causing molecules to linger near each other and reducing pressure below the ideal value
DLow temperatures reduce molecular volume, making the ideal gas approximation more accurate and B₂ smaller
B₂(T) = −½ ∫ f(r) 4πr² dr, where f(r) = exp(−u(r)/kT) − 1. At low temperatures, kT is small relative to the depth of the attractive well |ε|, so exp(−u(r)/kT) >> 1 in the attractive region, making f(r) large and positive. This positive integrand dominates the negative contribution from the repulsive core, making B₂ < 0. Physically: molecules slow down, spend more time in each other's attractive well, and the average pressure is reduced below the ideal value.
Question 3 True / False
A gas with B₂ = 0 at its Boyle temperature has no intermolecular interactions at that temperature.
TTrue
FFalse
Answer: False
B₂ = 0 does not mean the intermolecular potential u(r) is zero. The integral ∫[exp(−u(r)/kT) − 1] 4πr² dr = 0 because the positive contribution from repulsive interactions (short distances) and the negative contribution from attractive interactions (intermediate distances) exactly cancel. The forces are fully operative — this is a mathematical cancellation in the virial coefficient, not an absence of molecular interactions.
Question 4 True / False
At high temperatures, the second virial coefficient B₂(T) is typically positive because repulsive interactions dominate.
TTrue
FFalse
Answer: True
At high temperatures, kT >> |u(r)| for the attractive well, so the Mayer f-function in the attractive region is approximately zero (the exponential collapses to near 1). Only the hard-core repulsive region contributes significantly: there, u(r) >> kT makes exp(−u(r)/kT) ≈ 0, so f(r) ≈ −1, contributing a positive term to B₂ (since B₂ = −½ ∫ f(r) 4πr² dr). The result is B₂ > 0, meaning pressure exceeds the ideal prediction — molecules effectively exclude each other's volume.
Question 5 Short Answer
Why is measuring B₂(T) at many temperatures more useful than a single measurement, and what can be learned from its temperature dependence?
Think about your answer, then reveal below.
Model answer: B₂(T) as a function of temperature encodes the shape of the intermolecular pair potential u(r). At different temperatures, different regions of u(r) dominate: high-T data constrain the repulsive core, low-T data constrain the attractive well, and T_B constrains where the two contributions balance. By fitting a model potential (such as the Lennard-Jones potential) to B₂(T) data across temperatures, one can extract molecular parameters ε (well depth) and σ (effective diameter) that cannot easily be measured directly.
This is the key bridge B₂(T) provides between macroscopic thermodynamic measurements and microscopic molecular physics. A single B₂ value at one temperature is just a correction factor; a full B₂(T) curve is a fingerprint of the intermolecular potential. This is why Lennard-Jones potential parameters were historically refined by fitting to B₂(T) datasets rather than from direct force measurements.