# Dissociation constant

A dissociation constant is an equilibrium constant that quantifies how readily a complex or acid separates into its components: for a binding reaction it is the ratio of free partners to complex at equilibrium, and for an acid it is the equilibrium constant for losing a proton in water. In both uses a smaller value means the association is stronger or the dissociation weaker. The two most common symbols are \( K_{\mathrm{d}} \) for binding equilibria and \( K_{\mathrm{a}} \) for acid dissociation, both with units of mol L−1 in practice, although equilibrium constants are formally dimensionless.<sup>[1](https://goldbook.iupac.org/terms/view/14132)</sup><sup> • </sup><sup>[2](https://goldbook.iupac.org/terms/view/15441)</sup><sup> • </sup><sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC3623632/)</sup>

| Key fact | Detail |
|---|---|
| Binding definition | \( K_{\mathrm{d}} = [\mathrm{A}][\mathrm{B}]/[\mathrm{AB}] \) for A + B ⇌ AB; reciprocal of the association constant, equal to \( k_{\mathrm{off}}/k_{\mathrm{on}} \)<sup>[1](https://goldbook.iupac.org/terms/view/14132)</sup> |
| Operational meaning | \( K_{\mathrm{d}} \) is the ligand concentration at which half the receptors are occupied at equilibrium<sup>[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC13380234/)</sup> |
| Acid definition | For \( \mathrm{HB} \rightleftharpoons \mathrm{H^+} + \mathrm{B^-} \), \( K_{\mathrm{a}} = [\mathrm{H^+}][\mathrm{B^-}]/[\mathrm{HB}] \cdot c^{\mathrm{o}} \) with \( c^{\mathrm{o}} = 1 \ \mathrm{mol\ L^{-1}} \)<sup>[2](https://goldbook.iupac.org/terms/view/15441)</sup> |
| Logarithmic form | \( \mathrm{p}K_{\mathrm{a}} = -\log_{10} K_{\mathrm{a}} \); a larger \( K_{\mathrm{a}} \) (smaller \( \mathrm{p}K_{\mathrm{a}} \)) means a stronger acid<sup>[2](https://goldbook.iupac.org/terms/view/15441)</sup> |
| Water benchmark | \( K_{\mathrm{w}} = [\mathrm{H^+}][\mathrm{OH^-}] = 10^{-14} \) at 25 °C, so \( \mathrm{p}K_{\mathrm{a}} + \mathrm{p}K_{\mathrm{b}} = 14.00 \) for a conjugate pair<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chemical_Principles_%28Dickerson%29/05%253A_Solution_Equilibria%253A_Acids_and_Bases/5.01%253A_Hydrogen_Oxygen_and_Water)</sup><sup> • </sup><sup>[6](https://chem.libretexts.org/Courses/University_of_California_Davis/Chem_4B%3A_General_Chemistry_for_Majors_II_%28Larsen%29/Chem_4B_Textbook/Unit_III%3A_Chemical_Equilibria/VI%3A_AcidBase_Equilibria/6.03%3A_Acid_and_Base_Strength)</sup> |
| Typical measurement ranges | ITC is accurate for about \( 10^{-8} \) to \( 10^{-4} \ \mathrm{M} \); fluorescence methods suit \( 10^{-4} \) to \( 10^{-8} \ \mathrm{M} \)<sup>[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC3294027/)</sup><sup> • </sup><sup>[8](https://www.sciencedirect.com/science/article/abs/pii/S0076687904790102)</sup> |

## The binding dissociation constant

For a reversible interaction in which molecules A and B form a complex AB, the equilibrium dissociation constant is defined by IUPAC as the ratio \( [\mathrm{A}][\mathrm{B}]/[\mathrm{AB}] \) at equilibrium, the reciprocal of the association constant \( K_{\mathrm{a}} \), and equal to the ratio of the dissociation and association rate constants, \( k_{\mathrm{off}}/k_{\mathrm{on}} \).<sup>[1](https://goldbook.iupac.org/terms/view/14132)</sup> The association constant \( K_{\mathrm{a}} \) has units of M−1 and is proportional to affinity; biologists commonly prefer its reciprocal \( K_{\mathrm{d}} \), whose units of molar concentration make it directly comparable to the concentrations used in experiments.<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC3623632/)</sup>

The operational meaning follows from mass action. When the free ligand concentration equals \( K_{\mathrm{d}} \), half of the binding sites are occupied, so \( K_{\mathrm{d}} \) is the concentration required to fill 50 percent of the target molecules at equilibrium.<sup>[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC13380234/)</sup><sup> • </sup><sup>[9](https://pmc.ncbi.nlm.nih.gov/articles/PMC6067677/)</sup> The occupancy curve is hyperbolic when plotted against linear concentration and sigmoidal against the logarithm of concentration; at \( [\mathrm{L}] = 10 \times K_{\mathrm{d}} \) occupancy reaches 91 percent of its maximum, and a typical binding curve spans more than two orders of magnitude of ligand concentration.<sup>[10](https://pmc.ncbi.nlm.nih.gov/articles/PMC3000649/)</sup>

**Two routes to a low value.** Because \( K_{\mathrm{d}} = k_{\mathrm{off}}/k_{\mathrm{on}} \), a tight interaction can arise either from a slow off-rate, meaning the complex rarely falls apart, or from a fast on-rate, meaning encounters convert to binding efficiently; both parameters can be modulated when engineering a stronger binder.<sup>[1](https://goldbook.iupac.org/terms/view/14132)</sup><sup> • </sup><sup>[9](https://pmc.ncbi.nlm.nih.gov/articles/PMC6067677/)</sup> The dissociation rate constant itself reflects the probability that the complex falls apart per unit time, independent of concentration, in the same way a radioactive decay constant describes an atom's probability of decaying.<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC3623632/)</sup>

In pharmacology, \( K_{\mathrm{d}} \) describes the drug concentration needed to occupy 50 percent of its target receptors at equilibrium, and \( \mathrm{p}K_{\mathrm{d}} \), defined as \( -\log_{10} K_{\mathrm{d}} \), is the parameter used to compare ligand affinities and selectivities and to predict efficacious in vivo drug concentrations, which are often about \( 3 \times K_{\mathrm{d}} \).<sup>[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC6406023/)</sup><sup> • </sup><sup>[10](https://pmc.ncbi.nlm.nih.gov/articles/PMC3000649/)</sup> Experimental algorithms are usually formulated to report \( \mathrm{p}K_{\mathrm{d}} \) rather than \( K_{\mathrm{d}} \) because \( \mathrm{p}K_{\mathrm{d}} \) is normally distributed while \( K_{\mathrm{d}} \) is not.<sup>[10](https://pmc.ncbi.nlm.nih.gov/articles/PMC3000649/)</sup> Antibody affinity constants for antigen span roughly \( 5 \times 10^{4} \) to \( 10^{11} \) liters per mole, equivalent to \( K_{\mathrm{d}} \) values from about \( 10^{-11} \) to \( 2 \times 10^{-5} \ \mathrm{M} \).<sup>[12](https://www.ncbi.nlm.nih.gov/books/NBK26884/)</sup>

## The acid dissociation constant

For an acid HB in water, IUPAC defines the [acid dissociation constant](https://www.edgepedia.org/acid-dissociation-constant) as the equilibrium constant for \( \mathrm{HB(aq)} \rightleftharpoons \mathrm{H^+(aq)} + \mathrm{B^-(aq)} \), written \( K_{\mathrm{a}} = [\mathrm{H^+}][\mathrm{B^-}]/[\mathrm{HB}] \cdot c^{\mathrm{o}} \), where \( c^{\mathrm{o}} = 1 \ \mathrm{mol\ L^{-1}} \) is the standard amount concentration and activity coefficients are neglected.<sup>[2](https://goldbook.iupac.org/terms/view/15441)</sup> Because activity coefficients are neglected, the constant is valid at a specified ionic strength; the thermodynamic dissociation constant is obtained by extrapolating the conditional constant to zero ionic strength.<sup>[2](https://goldbook.iupac.org/terms/view/15441)</sup> The change of the apparent constant with ionic strength is small for most monobasic acids.<sup>[13](https://nvlpubs.nist.gov/nistpubs/jres/30/jresv30n5p347_A1b.pdf)</sup>

Since \( K_{\mathrm{a}} \) values differ for each acid and vary over many orders of magnitude, they are usually reported as \( \mathrm{p}K_{\mathrm{a}} = -\log_{10} K_{\mathrm{a}} \). A larger \( K_{\mathrm{a}} \), equivalently a smaller \( \mathrm{p}K_{\mathrm{a}} \), indicates a stronger acid, because the extent of dissociation is larger at the same concentration.<sup>[2](https://goldbook.iupac.org/terms/view/15441)</sup> [Acetic acid](https://www.edgepedia.org/acetic-acid), for example, has \( K_{\mathrm{a}} = 1.76 \times 10^{-5} \).<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chemical_Principles_%28Dickerson%29/05%253A_Solution_Equilibria%253A_Acids_and_Bases/5.01%253A_Hydrogen_Oxygen_and_Water)</sup>

**Conjugate pairs and water.** For a conjugate acid–base pair, \( K_{\mathrm{a}} \times K_{\mathrm{b}} = K_{\mathrm{w}} \), the ion-product constant of water, equal to \( 1.0 \times 10^{-14} \) at room temperature, so \( \mathrm{p}K_{\mathrm{a}} + \mathrm{p}K_{\mathrm{b}} = 14.00 \) at 25 °C; knowing either constant gives the other.<sup>[6](https://chem.libretexts.org/Courses/University_of_California_Davis/Chem_4B%3A_General_Chemistry_for_Majors_II_%28Larsen%29/Chem_4B_Textbook/Unit_III%3A_Chemical_Equilibria/VI%3A_AcidBase_Equilibria/6.03%3A_Acid_and_Base_Strength)</sup><sup> • </sup><sup>[14](https://chem.libretexts.org/%40api/deki/pages/207322/pdf/3.4%253A%2BAcid%2BStrength%2Band%2Bthe%2BAcid%2BDissociation%2BConstant%2B%28Ka%29.pdf)</sup> Water itself dissociates endothermically, with \( \Delta H = +55.90 \ \mathrm{kJ\ mol^{-1}} \) for breaking one O–H bond, and \( K_{\mathrm{w}} \) varies with temperature.<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chemical_Principles_%28Dickerson%29/05%253A_Solution_Equilibria%253A_Acids_and_Bases/5.01%253A_Hydrogen_Oxygen_and_Water)</sup>

## Measuring dissociation constants

Binding affinities are measured by methods that detect complex formation directly or through a label. [Isothermal titration calorimetry](https://www.edgepedia.org/isothermal-titration-calorimetry) (ITC) titrates ligand into a protein solution and fits the heat released to yield stoichiometry, the association constant \( K_{\mathrm{a}} = 1/K_{\mathrm{d}} \), and the binding enthalpy, from which \( \Delta S \) and \( \Delta G \) are calculated.<sup>[15](https://pmc.ncbi.nlm.nih.gov/articles/PMC6570549/)</sup> The practical range for accurate direct measurement on most commercial ITC instruments is approximately \( 10^{-8} \) to \( 10^{-4} \ \mathrm{M} \); the displacement method, in which a strongly binding ligand displaces a weakly bound one, extends this to \( 10^{-12} \ \mathrm{M} \) and even to \( 10^{-3} \ \mathrm{M} \).<sup>[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC3294027/)</sup> For single-digit nanomolar or tighter affinities, direct ITC titration curves become too steep to fit accurately, which is why displacement or alternative assays are used.<sup>[16](https://pmc.ncbi.nlm.nih.gov/articles/PMC2705510/)</sup> The reliability of a direct ITC fit also depends on the Wiseman c value, the ratio of total macromolecule concentration in the cell to \( K_{\mathrm{d}} \), which must fall in an appropriate range for a sigmoidal curve.<sup>[17](https://link.springer.com/article/10.1007/s12551-025-01280-3)</sup>

Fluorescence-based methods, including fluorescence polarization, fluorescence correlation spectroscopy, and FRET, are widely used because they suit protein–ligand interactions with dissociation constants in the range of \( 10^{-4} \) to \( 10^{-8} \ \mathrm{M} \), and they lack a separation step during which fast dissociation could perturb the equilibrium.<sup>[8](https://www.sciencedirect.com/science/article/abs/pii/S0076687904790102)</sup><sup> • </sup><sup>[18](https://pmc.ncbi.nlm.nih.gov/articles/PMC3160472/)</sup> Other approaches include surface plasmon resonance, enzyme inhibition, NMR, and the thermal shift assay, which monitors ligand-induced changes in protein melting behavior and has been combined with ITC to measure nanomolar affinities.<sup>[19](https://www.sciencedirect.com/science/article/pii/S0165993623005046)</sup><sup> • </sup><sup>[16](https://pmc.ncbi.nlm.nih.gov/articles/PMC2705510/)</sup>

## Why the same constant appears in different fields

Chemistry, biochemistry, and pharmacology all use dissociation constants, but the quantities they compare differ. An acid's \( K_{\mathrm{a}} \) measures the tendency to lose a proton to water and is reported as \( \mathrm{p}K_{\mathrm{a}} \), where larger numbers mean weaker acids. A binding \( K_{\mathrm{d}} \) measures the tendency of a complex to fall apart and is reported directly in concentration units, where smaller numbers mean tighter binding. The two are formally the same kind of object, an equilibrium constant for a dissociation written in the direction of separation, and the binding \( K_{\mathrm{d}} \) is the reciprocal of the association constant just as the acid \( K_{\mathrm{a}} \) is itself a dissociation constant for the acid–water pair.<sup>[1](https://goldbook.iupac.org/terms/view/14132)</sup><sup> • </sup><sup>[2](https://goldbook.iupac.org/terms/view/15441)</sup><sup> • </sup><sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC3623632/)</sup> The half-occupancy interpretation gives \( K_{\mathrm{d}} \) its practical appeal: it is a concentration a researcher can prepare in the laboratory, and in drug development it connects directly to the doses needed for effect.<sup>[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC13380234/)</sup><sup> • </sup><sup>[10](https://pmc.ncbi.nlm.nih.gov/articles/PMC3000649/)</sup>

## References

1. [Equilibrium dissociation constant, IUPAC Compendium of Chemical Terminology (Gold Book), entry 14132](https://goldbook.iupac.org/terms/view/14132)
2. [Acid dissociation constant, IUPAC Compendium of Chemical Terminology (Gold Book), entry 15441](https://goldbook.iupac.org/terms/view/15441)
3. [Take advantage of time in your experiments: a guide to simple, informative kinetics assays, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC3623632/)
4. [Two Routes to the Same Equation: Kinetic and Statistical-Thermodynamic Derivations of the Classical Drug-Receptor Occupancy Relationship, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC13380234/)
5. [5.1: Hydrogen, Oxygen, and Water, Chemistry LibreTexts](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chemical_Principles_%28Dickerson%29/05%253A_Solution_Equilibria%253A_Acids_and_Bases/5.01%253A_Hydrogen_Oxygen_and_Water)
6. [6.3: Acid and Base Strength, Chemistry LibreTexts](https://chem.libretexts.org/Courses/University_of_California_Davis/Chem_4B%3A_General_Chemistry_for_Majors_II_%28Larsen%29/Chem_4B_Textbook/Unit_III%3A_Chemical_Equilibria/VI%3A_AcidBase_Equilibria/6.03%3A_Acid_and_Base_Strength)
7. [Combination of isothermal titration calorimetry and time-resolved luminescence for high affinity antibody-ligand interaction thermodynamics and kinetics, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC3294027/)
8. [Fluorescence Quenching Methods to Study Protein–Nucleic Acid Interactions, ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S0076687904790102)
9. [Cell-Binding Assays for Determining the Affinity of Protein–Protein Interactions, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC6067677/)
10. [Ligand binding assays at equilibrium: validation and interpretation, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC3000649/)
11. [Binding kinetics of ligands acting at GPCRs, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC6406023/)
12. [B Cells and Antibodies, Molecular Biology of the Cell, NCBI Bookshelf](https://www.ncbi.nlm.nih.gov/books/NBK26884/)
13. [Dissociation constants and pH-titration curves at constant ionic strength: titrations of formic acid and acetic acid, NIST](https://nvlpubs.nist.gov/nistpubs/jres/30/jresv30n5p347_A1b.pdf)
14. [3.4: Acid Strength and the Acid Dissociation Constant (Ka), Chemistry LibreTexts](https://chem.libretexts.org/%40api/deki/pages/207322/pdf/3.4%253A%2BAcid%2BStrength%2Band%2Bthe%2BAcid%2BDissociation%2BConstant%2B%28Ka%29.pdf)
15. [Quantifying the Interactions between Biomolecules: Guidelines for Assay Design and Data Analysis, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC6570549/)
16. [Measurement of Nanomolar Dissociation Constants by Titration Calorimetry and Thermal Shift Assay, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC2705510/)
17. [Differential scanning fluorimetry followed by microscale thermophoresis and/or isothermal titration calorimetry as an efficient tool for ligand screening, Biophysical Reviews](https://link.springer.com/article/10.1007/s12551-025-01280-3)
18. [Analysis of protein-ligand interactions by fluorescence polarization, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC3160472/)
19. [Thermal shift assay for protein–ligand dissociation constant determination, ScienceDirect](https://www.sciencedirect.com/science/article/pii/S0165993623005046)

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*Topic: Encyclopedia › Physical world and mathematics › Chemistry › Chemical principles and methods › Thermodynamics and equilibrium › Chemical equilibrium › Equilibrium constant*

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