2.4.5 · D5Thermodynamics & Statistical Mechanics (Advanced)
Question bank — Chemical potential μ = (∂G - ∂N)_{T,P}
Before we start, one word we will lean on constantly: a quantity is extensive if it scales with system size (double the box, double the quantity — like , , ), and intensive if it does not (like , , and itself). Keep that split in your head; half the traps below are just people confusing the two.
Two small schematics do the visual work for the whole page — glance at them as the cards reference "the peaks picture" and "the coexistence line."
True or false — justify
The chemical potential μ is always negative
False. For an ideal gas can be positive, negative, or zero depending on pressure and the reference; its sign is a convention-plus-conditions matter, not a law.
μ is an intensive quantity
True. It is Gibbs energy per particle ( for a pure substance) — doubling the system leaves it unchanged, exactly like and .
At equilibrium between two connected reservoirs, the particle numbers and are equal
False. What equalizes is , not . A big box and a tiny box can hold vastly different numbers yet sit at identical chemical potential.
If system A has more particles than B, then
False. tracks concentration/pressure and interactions, not raw count. A large dilute box can have a lower than a small dense one.
holds for any thermodynamic system
False. It holds only for a single pure substance, because it relies on being homogeneous degree 1 in one particle number. For mixtures each species has its own partial molar .
Particles can spontaneously move from a region of lower particle energy to one of higher particle energy
True. Flow follows free energy , which contains a entropy term; if entropy gain dominates, matter climbs uphill in bare energy.
also equals μ
False. Only equals μ. The subscripts pick the potential and the held variables; mixing with -held gives a different quantity entirely.
Two phases at coexistence (liquid + its vapor) have equal chemical potential
True. Coexistence is diffusive equilibrium across the phase boundary, so even though densities differ enormously — this is the coexistence line drawn in the second figure, and it connects to Phase Equilibrium and Clausius-Clapeyron.
Compressing an ideal gas at fixed temperature lowers its chemical potential
False. rises like , so higher means higher — the gas becomes "more eager" to escape to lower-pressure regions.
Spot the error
"Since , adding particles doesn't change ."
The error is using the fixed-N first law. Once can vary the correct law is ; the term is precisely the energy carried in by new particles.
", so μ is just a derivative of — any derivative will do."
The subscript is not decoration. Different held variables select different physical quantities; only is μ.
"μ is called a potential and has energy units, so it is the potential energy of one particle."
μ is a free energy per particle, bundling entropy via . Equal energy but different concentration gives different μ, so it is not bare potential energy.
"Particles flow from high concentration to low concentration — that's the fundamental law."
Concentration flow is a special case. The fundamental driver is : high → low . Concentration only tracks μ when nothing else (interactions, external fields) competes.
"Since , if I add particles grows but stays fixed forever."
μ generally depends on (through pressure/concentration). is an instantaneous relation, not a promise that μ is constant as you keep adding particles.
"At the system is in equilibrium, so means automatically for any process."
Only when the constraint makes the sole free variable does force equality. The logic needs that particle-conservation link, not alone.
" was invented for chemical potential."
Gibbs Free Energy G exists independently as the potential natural in ; μ simply emerges as the coefficient of in .
Why questions
Why is μ conjugate to rather than to or ?
In each intensive quantity multiplies the change of its paired extensive variable; μ sits beside , so it is the "force" driving particle number changes — see First Law of Thermodynamics.
Why do we usually write μ from and not from ?
's natural variables are — the quantities we actually control in a lab — whereas needs held fixed, which is experimentally awkward.
Why does Euler's theorem give only for one pure substance?
Euler's homogeneous-function theorem needs to scale linearly in all its extensive arguments; with one species that's just , giving , but multiple species each contribute a term — see Euler Relation and Gibbs-Duhem.
Why does nature push particles from high μ to low μ?
Because that motion lowers total (), and spontaneous processes at fixed decrease — a consequence of the Second Law. This is the "downhill in μ" picture of the first figure.
Why does the ideal-gas μ carry a term at all?
At fixed , ; the ideal-gas law rearranges to , so , and integrating produces the logarithm — it encodes the entropy of spreading particles over more volume.
Why is μ the natural variable in the grand canonical ensemble?
There the system exchanges particles with a reservoir, so we fix the reservoir's μ instead of ; μ becomes the control knob for average particle number — see Grand Canonical Ensemble.
Why does μ appear in Fermi-Dirac and Bose-Einstein occupation numbers?
The occupation depends on the energy cost relative to μ, , because μ sets the reference level at which adding a particle is "free" — see Fermi-Dirac and Bose-Einstein Statistics.
Edge cases
What is μ for a system that cannot exchange particles at all (sealed, no reactions)?
μ is still well-defined as , but it never acts — with the term vanishes and μ has no dynamical role.
As for an ideal gas, what happens to μ?
, because infinite dilution makes adding a particle almost "free" — the entropy gain diverges.
What is μ at absolute zero for a Fermi gas?
It approaches a finite positive value called the Fermi energy; even at adding a fermion costs energy because the low levels are already full (Pauli exclusion) — see Fermi-Dirac and Bose-Einstein Statistics.
If two ideal gases at equal and but different species meet, are their chemical potentials equal?
Not necessarily — each species has its own set by its partial pressure. Equal total says nothing about the individual partial-molar potentials.
Can μ be exactly zero, and does that mean "no particles present"?
Yes it can be zero at a specific (photon gases have ), and it means adding a particle costs no free energy — not that particles are absent.
How do external potentials like gravity or an electric field change μ?
They add a per-particle potential energy, giving a total (electrochemical) potential ; equilibrium then equalizes , not the bare . This is why gas pressure falls with altitude and why a battery can hold charge apart despite unequal internal .
At a first-order phase transition, is μ continuous or discontinuous across the boundary?
Continuous — coexisting phases share the same μ; it is the derivatives of μ (like entropy and volume per particle) that jump — this is what Clausius-Clapeyron quantifies, and it is the kink at the coexistence line in the second figure.
Recall One-line summary of every trap
μ is an intensive free energy per particle, defined by which variables you hold fixed, driving flow from high to low μ (or high to low electrochemical once fields are present) regardless of raw particle count or bare energy.