SCIENCE · INVESTIGATE AND EXPLAIN

Distinguish analytical concentration from thermodynamic activity

Analytical concentration describes how much solute is assigned per solution volume.

Learning outcome: Compare gamma .5,1 and1.5 at c=.08 M. Separate RT ln(c/c°) and RT ln gamma; construct two distinct concentration/gamma pairs giving equal activity.

Starting recap

Use chemical amount, dimensional ratios and positive interval endpoints. Distinguish original formula units, independently moving species and a chosen thermodynamic standard.

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Try

At c=.080 M, gamma=.75 and c°=1 M, activity=.060. Relative to the same standard state, μ−μ°=8.314×298×ln(.060)≈−6970 J/mol.

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Predict

Can two samples have equal concentration but different activity without changing the number of analytical moles?

Your prediction and reason

Experiment

Compare gamma .5,1 and1.5 at c=.08 M. Separate RT ln(c/c°) and RT ln gamma; construct two distinct concentration/gamma pairs giving equal activity.

Distinguish dimensionless activity from analytical concentration

Worked teaching model · synthetic values · One neutral dissolved species with c°=1 mol/L standard concentration, a=γc/c°. Supplied γ is an empirical hypothesis, not predicted from c. At298 K, μ−μ°=RT ln a with R8.314 J/(mol K). No single-ion absolute activity measurement or change of standard state is implied.

Analytical neutral-solute concentration: 0.08 mol/L Supplied concentration-scale activity coefficient: 0.75 ratio

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Axes rescale. Connected samples represent declared equilibrium or activity relationships, not elapsed time. Dashed lines show the concentration-scale gamma=1 reference or ideal Raoult total pressure as named in the model.

Numerical diagram coordinates

analytical concentration (mol/L)	dimensionless activity
0.01	0.0075
0.015	0.01125
0.02	0.015
0.025	0.01875
0.03	0.0225
0.035	0.02625
0.04	0.03
0.045	0.03375
0.05	0.0375
0.055	0.04125
0.06	0.045
0.065	0.04875
0.07	0.0525
0.075	0.05625
0.08	0.06
0.085	0.06375
0.09	0.0675
0.095	0.07125
0.1	0.075

Dashed reference
analytical concentration (mol/L)	dimensionless activity
0.01	0.01
0.015	0.015
0.02	0.02
0.025	0.025
0.03	0.03
0.035	0.035
0.04	0.04
0.045	0.045
0.05	0.05
0.055	0.055
0.06	0.06
0.065	0.065
0.07	0.07
0.075	0.075
0.08	0.08
0.085	0.085
0.09	0.09
0.095	0.095
0.1	0.1

Calculated evidence under the stated assumptions
Quantity	Value	Unit
Dimensionless activity	0.06	ratio
Ideal concentration-scale activity	0.08	ratio
Chemical potential minus selected standard	-6970.428	J/mol
Nonideality contribution RT ln gamma	-712.753	J/mol

Equal analytical concentration does not imply equal chemical potential when the activity coefficient differs.

The standard concentration makes the logarithm argument dimensionless. Neither activity nor μ−μ° is a free-standing count of particles; their numerical convention depends on the chosen standard state. Holding supplied gamma fixed over this plot is an illustrative comparison, not a fitted concentration-dependent activity law.

Retain the species, standard state and ideal/nonideal assumptions with every number. This mathematical packet is not a solution-preparation procedure or evidence of observed laboratory technique.

Equal analytical concentration does not imply equal chemical potential when the activity coefficient differs.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Separate an amount account from a thermodynamic descriptor

Analytical concentration describes how much solute is assigned per solution volume. Activity packages the thermodynamic effect of composition and interactions relative to a chosen standard state. It is not a second conserved species count and should not be substituted into an atom balance as though it were moles.

Make the standard explicit

For this neutral-solute illustration define a=γc/c°, with c°=1 mol/L. The ratio is dimensionless. A supplied γ=.75 and c=.080 mol/L gives a=.060. The same concentration with γ=1 gives.080; a different concentration.060 M with γ=1 again gives activity.060. Equal activity does not require equal analytical concentration.

Interpret the supplied chemical-potential relation

At298 K use μ−μ°=RT ln a. Chemical potential has energy-per-mole units; a is dimensionless. Splitting the logarithm gives RT ln(c/c°)+RT lnγ. The latter is the nonideality contribution relative to the selected convention. Here RT≈2477.57 J/mol and RT ln(.75)≈−712.8 J/mol. No absolute single-ion activity is measured by this neutral-species exercise.

Do not mistake a control for an empirical law

The graph holds the selected γ constant while concentration varies. Real coefficients can depend on composition, temperature and the chosen standard state; this illustrative line does not fit those dependencies. The symmetric vapor model in a separate lesson uses a different, pure-component mole-fraction convention, so its coefficients must not silently be copied into this concentration-scale equation.

Transfer and check

At c=.040 M and γ=1.5, predict activity before calculating μ: a=.060, the same as.080 M withγ=.75. Under the same standard and temperature these have equal μ−μ°, although their analytical concentrations differ. Changing the standard changes the numerical decomposition, not the underlying physical chemical potential.

Sources and evidence boundary

Conceptual references checked 30 October 2026: IUPAC activity definition (indexed primary entry). All cases, prose and diagrams are original. Supplied and calculated data are not observed laboratory evidence. No chemical handling, heating, pressurized gas, membrane experiment or clinical application is requested. Retain predictions, contrasting recipes and a changed-case explanation; numerical agreement does not certify independent mastery.

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Why does taking the logarithm of a dimensional concentration without a reference standard obscure the convention?

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Previous: Reconcile solvent, solution and entity denominators Next: Test ideal and nonideal vapor-composition predictions

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