Chemistry · Lesson

Activities and non-ideal solutions

Express nonideal behavior through dimensionless activities and matched reference states. Separate a local supplied activity coefficient from a thermodynamically consistent composition model.

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By the end, you can

Construct a dimensionless concentration or fugacity activity.
Keep activity coefficients paired with their standard-state convention.
Explain why nonideal coefficients and electrolyte quantities require constraints.

The logarithm needs a dimensionless reference ratio

At a stated temperature and pressure-reference convention, write μᵢ = μᵢref + RT ln aᵢ. R is the molar gas constant and aᵢ is dimensionless. A concentration convention can use aᵢ = γc,ᵢ(cᵢ/c°), with c° = 1 mol L⁻¹ in this example. An activity is not simply a concentration with its unit silently removed.

The standard-state choice specifies a reference, not a universal temperature of 298.15 K. Molality, concentration and mole-fraction conventions have different coefficients and matched reference potentials. Converting a convention requires converting both together. Activities need not lie between zero and one.

Normalize before taking a logarithm

Local convention	Supplied inputs	Dimensionless a	μ − μref = RT ln a
Concentration; ideal approximation	c = 0.100 mol L⁻¹; c° = 1 mol L⁻¹; γc = 1	0.100	-5.708 kJ mol⁻¹
Concentration; supplied nonideal value	c/c° = 0.100; γc = 0.800	0.0800	-6.261 kJ mol⁻¹
Gas fugacity reference	p = 5.00 bar; φ = 0.900; p° = 1.00 bar	f/p° = 4.50	3.729 kJ mol⁻¹

Three separate invented local examples at 298.15 K, each with its stated matched reference. The concentration rows compare γc = 1 and a supplied γc = 0.800 at the same concentration. The gas row uses f/p°. These local inputs are not an independently specified full-mixture model.

Gas fugacity and pure-phase references also need a convention

For a gas convention, a = f/p°, where fugacity f has pressure units and p° is the specified reference pressure. With f = φp, a supplied fugacity coefficient φ corrects the ideal-gas pressure relation. The ideal-gas limit has φ approaching 1.

A pure condensed phase can have activity 1 when referenced to that same pure phase at the same T,p. If its standard potential instead uses fixed p°, a pressure correction may remain: μ(T,p) − μ(T,p°) = ∫ V̄ dp at fixed T for the phase. Setting that correction to zero is an approximation, not a general identity at arbitrary pressure.

Use a consistent model across composition

For an ideal binary mixture under a same-pressure pure-component convention, the molar mixing Gibbs energy is Δg_mix = RT[x ln x + (1 − x) ln(1 − x)], where x is the mole fraction of B. A regular-solution teaching model adds RTχx(1 − x), with dimensionless interaction parameter χ at the stated temperature.

That single Gibbs function supplies both component potentials consistently. In this model ln γA = χx² and ln γB = χ(1 − x)² under the matched mole-fraction convention. These composition functions satisfy Gibbs–Duhem. A supplied coefficient at one composition is useful for a local calculation; it is not enough to invent an independently constant coefficient for every component over an entire mixture range.

Electrolytes add an individual-ion limitation

Bulk thermodynamic measurements access electrically neutral combinations such as mean ionic activities. An individual-ion activity cannot be determined independently by ordinary bulk thermodynamic measurements without an additional convention. A known mean ionic coefficient does not prove the cation and anion coefficients are individually equal.

For a completely dissociated 1:1 salt model, the neutral salt chemical potential contains the sum of cation and anion potentials and their two-ion stoichiometry. Ion pairing, specific interactions and concentration limits can require a richer model. Activity coefficients summarize a chosen thermodynamic description rather than every microscopic interaction directly.

Worked examples

Use a supplied local activity coefficient

At 298.15 K, take c = 0.100 mol L⁻¹, c° = 1 mol L⁻¹ and γc = 0.800 in the matched concentration convention. Calculate a and μ − μref.

Normalize concentration: c/c° = 0.100.
Multiply by the coefficient: a = 0.800 × 0.100 = 0.0800.
Use μ − μref = RT ln a with R ≈ 8.314462618 J mol⁻¹ K⁻¹.
The logarithm is ln 0.0800 ≈ −2.52573, so the shift is approximately −6261 J mol⁻¹.

a = 0.0800 and μ − μref ≈ −6.26 kJ mol⁻¹ under this supplied convention and local coefficient.

Compare two ideal-gas pressures on one reference

For a fixed-temperature ideal-gas model at 298.15 K, double the pressure from p₁ to p₂ = 2p₁. What is the chemical-potential change?

Subtract the same-reference expressions μ₂ and μ₁.
The reference potential cancels, giving RT ln(p₂/p₁).
The pressure ratio is dimensionless and equals 2.
RT ln 2 ≈ 1718 J mol⁻¹.

μ₂ − μ₁ ≈ +1.72 kJ mol⁻¹. A real-gas comparison instead uses its matched fugacity ratio.

Try it yourself

For γc = 0.500 and c/c° = 0.200, what is the activity in this concentration convention?
Show the answer and reasoning
0.100.
Multiply the dimensionless concentration ratio by its matched coefficient.
For f = 4.50 bar and p° = 1.00 bar, may the dimensionless gas activity exceed 1?
Show the answer and reasoning
Yes; here it is 4.50.
Activity is a reference ratio rather than a probability.
Is pure-phase activity exactly 1 at arbitrary p if the chosen standard potential remains at fixed p° and the pressure correction is retained?
Show the answer and reasoning
Not in general.
An exact same-pressure reference and a fixed-p° standard must be distinguished.
Does measuring a mean ionic coefficient determine both individual ion coefficients separately?
Show the answer and reasoning
No.
Individual-ion assignments require conventions beyond the neutral bulk measurement.

What to carry forward

State the reference before using RT ln a.
Pair each coefficient with its concentration, molality or mole-fraction convention.
A Gibbs model constrains component composition functions together.
Treat individual-ion and pure-phase pressure assumptions explicitly.

References

Teaching resources and reference material supporting this lesson’s explanations and examples.