# Thermodynamic activity
**Thermodynamic activity** is a dimensionless measure of the "effective concentration" of a chemical species, defined by the IUPAC equation
\[ a = e^{\frac{\mu - \mu^{o}}{R\,T}} \]
where \( \mu \) is the chemical potential of the species, \( \mu^{o} \) its standard chemical potential, \( R \) the gas constant, and \( T \) the thermodynamic temperature[1](https://goldbook.iupac.org/terms/view/A00115). Because the definition depends on the choice of standard state, activity is always a relative quantity, sometimes called the relative activity[1](https://goldbook.iupac.org/terms/view/A00115) • [2](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.07%3A_Activity_of_an_Uncharged_Species). It lets thermodynamics treat real gases, liquids, and solutions with the same formalism that works for ideal ones.
| Key fact | Detail |
|---|---|
| Definition | \( a = \exp[(\mu - \mu^{o})/(RT)] \), dimensionless, relative to a chosen standard state[1](https://goldbook.iupac.org/terms/view/A00115) |
| Relation to composition | \( a = a_{\mathrm{ideal}} \cdot \gamma \), where \( \gamma \) is the activity coefficient describing deviation from ideal behavior[3](https://books.rsc.org/books/monograph/2559/chapter/9011748/Introduction) |
| Pure phases | \( a = 1 \) for pure liquids and solids; for gases \( a_{\mathrm{gas}} \cong p/p_{0} \) with \( p_{0} = 1 \) bar[3](https://books.rsc.org/books/monograph/2559/chapter/9011748/Introduction) |
| Gas standard state | Ideal gas at exactly 100 kPa (0.1 MPa); with this choice the activity of a gas is its fugacity expressed in bars[4](https://www.sciencedirect.com/topics/mathematics/state-condition) |
| Equilibrium | The thermodynamic equilibrium constant is \( K = \prod_i (a_i)^{\nu_i} \), and \( \Delta_{r}G^{\circ} = -RT \ln K \)[5](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.08%3A_The_Thermodynamic_Equilibrium_Constant) |
| Water activity | In practice measured as equilibrium relative humidity, \( a_{w} = \mathrm{ERH}/100 \)[6](https://pmp.errc.ars.usda.gov/WaterActivity.aspx) |
| Single ions | Individual ion activities are not independently measurable; the IUPAC pH definition is therefore notional[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC4501315/) |
## Chemical potential and the standard state
The chemical potential \( \mu \), with unit J mol⁻¹, measures the tendency of a substance to undergo physical or chemical change, and activity is the dimensionless variable through which composition enters it[3](https://books.rsc.org/books/monograph/2559/chapter/9011748/Introduction). For each kind of mixture component the chemical potential is written as a standard part plus an activity term, for example \( \mu_i = \mu_i^{\circ} + RT \ln a_i \) for a substance in a gas, liquid, or solid mixture, and \( \mu_{\mathrm{B}} = \mu_{\mathrm{B}}^{\circ} + RT \ln a_{\mathrm{B}} \) for a solute on a mole fraction, concentration, or molality basis[2](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.07%3A_Activity_of_an_Uncharged_Species).
The standard chemical potential \( \mu^{o} \) is defined in terms of a specified state of aggregation, a specified temperature, and a specified standard pressure[8](https://publications.iupac.org/pac/1994/pdf/6603x0533.pdf). IUPAC defines standard states for gases, pure and mixed condensed phases, solvents, and solutes[8](https://publications.iupac.org/pac/1994/pdf/6603x0533.pdf). The usual choices are summarized in the standard-state table of the Royal Society of Chemistry text: a pure compound for a solid or liquid, an ideal gas at \( p_{0} \) for a gas, the pure solvent for a solvent, and for a solute either the hypothetical pure solute ([Henry's law](https://www.edgepedia.org/henrys-law) basis), an ideal 1 M solution, or an ideal 1 molal solution[3](https://books.rsc.org/books/monograph/2559/chapter/9011748/Introduction). IUPAC introduced 1 bar as the standard pressure only in 1982; older literature and tables use 1 atm[3](https://books.rsc.org/books/monograph/2559/chapter/9011748/Introduction).
## Activity coefficients
The activity coefficient \( \gamma \) measures the deviation of a real mixture from ideal behavior: \( a = a_{\mathrm{ideal}} \cdot \gamma \), with \( \gamma = 1 \) under ideal conditions[3](https://books.rsc.org/books/monograph/2559/chapter/9011748/Introduction). For a gas the corresponding quantity is the fugacity coefficient \( \phi \), and using the ideal-gas standard state the activity is \( a = f/p_{0} \), where fugacity \( f \) is the corrected pressure that reduces to the partial pressure as \( p \to 0 \)[3](https://books.rsc.org/books/monograph/2559/chapter/9011748/Introduction) • [9](https://goldbook.iupac.org/terms/view/F02543) • [18](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/15:_Chemical_Potential_Fugacity_Activity_and_Equilibrium/15.02:_The_Chemical_Potential_and_Activity_of_a_Gas).
**Limits of the coefficient.** The coefficient approaches unity in the limit that defines the reference state: \( \gamma \to 1 \) as \( x \to 1 \) for a solvent or pure compound, as \( p \to 0 \) for a gas, and as \( c \to 0 \) or \( m \to 0 \) for a solute[3](https://books.rsc.org/books/monograph/2559/chapter/9011748/Introduction). The 1 M and 1 molal solute standard states are hypothetical, because at those concentrations most solutions behave far from ideally[3](https://books.rsc.org/books/monograph/2559/chapter/9011748/Introduction).
For ions, electrostatic interactions dominate at low concentration. The Debye-Hückel limiting law gives \( \ln \gamma_{\pm} = -\alpha\, c^{1/2} \), where \( \gamma_{\pm} \) is the mean ionic activity coefficient and \( \alpha = 0.507\ (\mathrm{l/mol})^{1/2} \) for water at 25 °C[10](https://www.sciencedirect.com/science/article/abs/pii/S0009261498002723). The law depends on the ionic strength of the solution, and the logarithm of the coefficient varies with the square of ionic charge[11](https://recherche-expertise.asnr.fr/sites/default/files/documents/larecherche/formation_recherche/theses/theses-soutenues/dei/2004-these-Denison.pdf). Modeling this range is a substantial enterprise: the widely used Pitzer model requires about 22,000 parameters to be extracted from experimental data at a single temperature for the 28 most important cations and 16 anions in salt chemistry, and newer electrolyte equations of state aim to reduce this to a few adjustable parameters[12](https://pubs.aip.org/aip/adv/article/9/1/015214/1069482/A-generalized-Debye-Huckel-theory-of-electrolyte).
## Activities in equilibrium constants
The central application of activity is the definition of equilibrium constants[2](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.07%3A_Activity_of_an_Uncharged_Species). For a reaction, the reaction quotient is the proper quotient of activities,
\[ Q_{\mathrm{rxn}} = \prod_i a_i^{\nu_i} \]
with products in the numerator and reactants in the denominator, each raised to its stoichiometric number \( \nu_i \)[5](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.08%3A_The_Thermodynamic_Equilibrium_Constant). At equilibrium \( Q_{\mathrm{rxn}} = K \), the thermodynamic equilibrium constant, and
\[ \Delta_{r}G^{\circ} = -RT \ln K \]
so \( K \) depends only on temperature and the choice of standard states, not on pressure or composition[5](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.08%3A_The_Thermodynamic_Equilibrium_Constant). Because activities are dimensionless, \( K \) is dimensionless too[5](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.08%3A_The_Thermodynamic_Equilibrium_Constant).
This is why pure solids and liquids are omitted from equilibrium-constant expressions: their activities equal 1 by definition, since the standard state of a pure substance is the pure substance itself[3](https://books.rsc.org/books/monograph/2559/chapter/9011748/Introduction) • [13](https://stacks.cdc.gov/view/cdc/206360/cdc_206360_DS1.pdf). For dissolved species, the activity \( a \) relates to concentration \( c \) through the dimensionless coefficient \( \gamma \), and for aqueous species the standard state is a hypothetical ideal solution of 1 mol solute per kilogram of water with \( \gamma = 1 \)[13](https://stacks.cdc.gov/view/cdc/206360/cdc_206360_DS1.pdf). When activity coefficients are known, \( K \) written in activities can be converted into a concentration-based constant, and over narrow concentration ranges the coefficient quotient \( K_{\gamma} \) is often approximately constant, making the measured concentration constant \( K_c \) transferable between similar systems[14](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_%28Ellgen%29/15%3A_Chemical_Potential_Fugacity_Activity_and_Equilibrium/15.07%3A_Chemical_Potential_Activity_and_Equilibrium).
## Water activity
A prominent practical case is the activity of water in food. The water activity \( a_{w} \) is the ratio of the water vapor pressure of the food to the vapor pressure of pure water under the same conditions, expressed as a fraction, and in practice it is usually measured as equilibrium relative humidity[6](https://pmp.errc.ars.usda.gov/WaterActivity.aspx). The FDA states the relation as \( a_{w} = \mathrm{ERH}/100 \)[15](https://chem.academy/notes/water-activity.html). [Water activity](https://www.edgepedia.org/water-activity) governs microbial growth and spoilage, and the span from fresh food to unspoiled food covers a range of about 0.4 in \( a_{w} \), an osmotic effect corresponding to nearly 700 atm of pressure[15](https://chem.academy/notes/water-activity.html).
## Single-ion activities and pH
Activities of individual ions occupy a special position. Because pH involves the single-ion hydrogen activity, a quantity that cannot be measured independently, the IUPAC pH definition is notional and rests on a convention (the Bates-Guggenheim convention) for single-ion activity coefficients[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC4501315/) • [16](https://publications.iupac.org/ci/2003/2504/pac3_baucke.html). Only mean ionic activity coefficients, combining the cation and anion of an electrolyte, are directly accessible to thermodynamic measurement[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC4501315/). Some thermodynamic activity-coefficient models used in atmospheric science and geochemistry do compute single-ion coefficients within their mathematical framework, but their single-ion values remain model-dependent[17](https://acp.copernicus.org/articles/20/4809/2020/index.html).
## References
1. [IUPAC Gold Book, "activity" (A00115)](https://goldbook.iupac.org/terms/view/A00115)
2. [DeVoe, Thermodynamics and Chemistry, Sec. 9.7: Activity of an Uncharged Species, Chemistry LibreTexts](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.07%3A_Activity_of_an_Uncharged_Species)
3. [Electrochemistry: Fundamentals, Cells and Applications, Introduction, Royal Society of Chemistry](https://books.rsc.org/books/monograph/2559/chapter/9011748/Introduction)
4. [State Condition, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/state-condition)
5. [DeVoe, Thermodynamics and Chemistry, Sec. 11.8: The Thermodynamic Equilibrium Constant, Chemistry LibreTexts](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.08%3A_The_Thermodynamic_Equilibrium_Constant)
6. [Water Activity in Food, Pathogen Modeling Program, USDA Agricultural Research Service](https://pmp.errc.ars.usda.gov/WaterActivity.aspx)
7. [Meaning and Measurability of Single-Ion Activities, the Thermodynamic Foundations of pH, and the Gibbs Free Energy for the Transfer of Ions between Dissimilar Materials, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC4501315/)
8. [Standard quantities in chemical thermodynamics, Pure and Applied Chemistry 66, 533 (1994), IUPAC](https://publications.iupac.org/pac/1994/pdf/6603x0533.pdf)
9. [IUPAC Gold Book, "fugacity" (F02543)](https://goldbook.iupac.org/terms/view/F02543)
10. [A new derivation of approximate limiting laws for ionic activities, Chemical Physics Letters](https://www.sciencedirect.com/science/article/abs/pii/S0009261498002723)
11. [Denison thesis on the Debye-Hückel limiting law, Université de Provence](https://recherche-expertise.asnr.fr/sites/default/files/documents/larecherche/formation_recherche/theses/theses-soutenues/dei/2004-these-Denison.pdf)
12. [A generalized Debye-Hückel theory of electrolyte solutions, AIP Advances 9, 015214 (2019)](https://pubs.aip.org/aip/adv/article/9/1/015214/1069482/A-generalized-Debye-Huckel-theory-of-electrolyte)
13. [The activity of a pure solid or liquid species, CDC Stacks](https://stacks.cdc.gov/view/cdc/206360/cdc_206360_DS1.pdf)
14. [Ellgen, Thermodynamics and Chemical Equilibrium, Sec. 15.7: Chemical Potential, Activity, and Equilibrium, Chemistry LibreTexts](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_%28Ellgen%29/15%3A_Chemical_Potential_Fugacity_Activity_and_Equilibrium/15.07%3A_Chemical_Potential_Activity_and_Equilibrium)
15. [Water Activity: One Quantity, Four Instruments, Chem.Academy](https://chem.academy/notes/water-activity.html)
16. [Baucke, Chemistry International, IUPAC pH document](https://publications.iupac.org/ci/2003/2504/pac3_baucke.html)
17. [The acidity of atmospheric particles and clouds, Atmospheric Chemistry and Physics 20, 4809 (2020)](https://acp.copernicus.org/articles/20/4809/2020/index.html)
18. [15.2: The Chemical Potential and Activity of a Gas](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/15:_Chemical_Potential_Fugacity_Activity_and_Equilibrium/15.02:_The_Chemical_Potential_and_Activity_of_a_Gas)
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*Topic: Encyclopedia › Physical world and mathematics › Chemistry › Chemical principles and methods › Thermodynamics and equilibrium › Chemical equilibrium › Non-ideal and perturbed equilibria*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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