THE NATURE OF THE EQUILIBRIUM CONSTANT

Abstract

The thermodynamic equilibrium constant K is necessarily dimensionless, yet it is frequently treated as if it carried units. This paper traces the origin of the equilibrium constant and then derives the modern definition of K from the Gibbs free energy, emphasizing that K is dimensionless because it is the equilibrium value of the reaction quotient expressed in activities. Making the standard-state reference explicit shows why pure solids and pure liquids do not appear in equilibrium expressions and why the solvent term can be omitted in dilute solutions. The analysis also distinguishes the dimensionless practical constants from the unit-dependent variants and provides an explicit nondimensionalization linking them. As an application, the autoprotolysis of water is examined to show that pK_a (H₂O) ≈ 14.0 on the usual aqueous scale; the commonly quoted 15.7 is traced to mixing solvent and solute reference conventions. The same considerations clarify the concentration shorthand used for pH.

Keywords:
equilibrium constant; activity; standard state; water autoprotolysis; pH.

INTRODUCTION

Equilibrium calculations in chemistry are routinely expressed in terms of an equilibrium constant, yet confusion remains common about what the thermodynamic constant K represents and, in particular, why it is necessarily dimensionless.¹ This confusion is exacerbated by the practical use of the dimensionless forms K_c and K_p alongside commonly used dimensional variants - denoted here as K_c’ and K_p’ - which require explicit nondimensionalization to serve as approximations to K.¹,²

Much of the ambiguity surrounding the dimensionless character of K arises not from its definition but from the representational step that replaces activities by pressureor concentration-based proxies while leaving the standard-state reference implicit. Suppressing the factors p^ꝋ and c^ꝋ under-specifies the reference convention, so that dimensional expressions can masquerade as equilibrium constants, leading to the spurious assignment of units and even to logarithms applied to quantities with physical dimension.¹ In the thermodynamic definition, K is a product of activities (ratios to specified standard states) and is therefore necessarily dimensionless.²,³ The persistence of these representational shortcuts, and the interpretative errors they provoke, has been documented in the literature.¹,⁴ More broadly, interpretative difficulties in equilibrium reasoning are well documented in chemical education research.⁵-⁸

This article presents a unified framework that connects formal thermodynamic principles to equilibrium calculations. The discussion derives K from thermodynamic first principles, establishes the role of activities in distinguishing dimensionless from dimensional practical forms, and uses the pK_a of water as an integrative example. The same activity-based perspective is then applied to pH, clarifying why its logarithmic definition is necessarily dimensionless and when concentration-based approximations are justified. This approach makes the reference conventions explicit at the point of calculation and treats K as a thermodynamic quantity whose meaning and invariances are fixed by those conventions rather than by notation alone.

To maintain rigor within this framework, the notation employed distinguishes explicitly between thermodynamic definitions and practical approximations. While the International Union of Pure and Applied Chemistry (IUPAC) Green Book lists K or K^ꝋ for the standard equilibrium constant,²,⁹ K is used exclusively throughout this text. Furthermore, K_c and K_p are reserved for the dimensionless concentrationand pressure-based approximations, whereas the dimensional forms - typically labeled K_c and K_p in standard literature - are denoted here as K_c’ and K_p’.

Because the discussion that follows relies on several interrelated thermodynamic concepts, it is useful to establish the key concepts that will be developed rigorously in subsequent sections. In broad terms, the equilibrium constant K is the quantity that characterizes the composition of a reaction mixture at thermodynamic equilibrium for a given temperature; it is defined as a product of activities.²,³ The activity of a species i, denoted a_i, is a dimensionless number that expresses the effective contribution of that species to the thermodynamic properties of the system. It is the ratio of a measure of composition, such as molar concentration c_i, molality m_i, or partial pressure p_i, to its value in the adopted standard state, corrected by a coefficient that accounts for deviations from ideal behavior.²,¹⁰ The standard state is a conventional thermodynamic reference defined for each species and composition scale (e.g., molarity, molality), and is denoted by the superscript ^ꝋ throughout this text; both this symbol and the superscript circle ^o are accepted by IUPAC for standard-state quantities.²,⁹ Conventional standard-state values are c^ꝋ = 1 mol L^-1 (L ≡ dm³) for molar concentrations, m^ꝋ = 1 mol kg^-1 for molalities, and p^ꝋ = 1 bar for gas-phase partial pressures.² The subscript “r” appearing throughout this text (as in Δ_rG) identifies a reaction quantity; its precise meaning is established in due course.²,¹¹

THE ORIGIN OF THE EQUILIBRIUM CONSTANT: FROM CHEMICAL FORCE TO THE LAW OF MASS ACTION

The idea that chemical reactions tend toward a state of equilibrium is longstanding, but its quantification was a central challenge in the nineteenth century. A general mathematical formulation of what later became known as the law of mass action, for reversible reactions at equilibrium, was developed by Guldberg and Waage in a series of works beginning in 1864.¹²-¹⁵ Starting from the then-current concept of “chemical force” (affinity), they proposed that, in reversible reactions, two such opposing forces act simultaneously, one promoting the formation of products and the other acting toward the regeneration of reactants; equilibrium being the state in which these forces balance.¹⁶ The very notion of “chemical force” reflected a broader Newtonian aspiration to explain many natural phenomena in terms of forces;¹⁷,¹⁸ for many chemists, especially those seeking quantitative laws, a central challenge was to define affinity with the same precision that Newton had given to mechanical force.¹⁰,¹⁷,¹⁸ In this approach, Guldberg and Waage expressed the equilibrium condition in algebraic form by equating the opposing “forces”, thereby giving the first explicit mathematical formulation of chemical equilibrium.¹⁶

Their distinctive contribution was to quantify the “chemical force” in each reaction direction as a product of the “active masses”, i.e., the amount of each substance per unit volume (effectively concentrations in the solution systems they studied), each raised to an empirical exponent and multiplied by a proportionality coefficient characteristic of that direction (k for the forward and k’ for the reverse).¹⁵,¹⁶ For the simplified case A + B ⇌ C + D (that is, with unit exponents), the forward force is k [A][B] and the reverse force is k’[C][D].¹⁶ At equilibrium, the forces are equal,

(1) $$k[\mathrm{A}][\mathrm{B}]=k^{\prime }[\mathrm{C}][\mathrm{D}]$$

which leads to

(2) $$\frac{k}{k^{\prime }}=\frac{[\mathrm{C}][\mathrm{D}]}{[\mathrm{A}][\mathrm{B}]}$$

This ratio constitutes a formal precursor to the modern equilibrium constant, though it is grounded in (chemical) force-balance rather than free energy and operates with concentrations rather than thermodynamic activities.¹⁵,¹⁶ The interpretation of the equilibrium ratio k/k’ as set by opposing proportionality coefficients is nonetheless reminiscent of later kinetic treatments of elementary reversible steps, in which the ratio of forward and reverse rate coefficients is related to the equilibrium constant.¹⁵,¹⁶

In Guldberg and Waage’s original formulation, the exponents in their mass-action expressions were treated as empirical parameters, not necessarily coinciding with the stoichiometric coefficients. The pair’s initial publications, in Norwegian (1864)¹² and in French (1867),¹³ had little impact,¹⁵ but the law gained wider recognition after Julius Thomsen’s application in 1869 made their equilibrium equation generally known.¹⁵,¹⁶ The first explicit appearance of these exponents as equal to the stoichiometric coefficients in an equilibrium equation appears to be due to August Horstmann in 1877.¹⁶,¹⁹ Wilhelm Ostwald’s detailed discussion then encouraged Guldberg and Waage to restate their results for a more widely circulated German journal in 1879.¹⁴-¹⁶ In this way, their algebraic relation between active masses at equilibrium provided a calculational criterion for equilibrium composition.

The later thermodynamic reconstruction of chemical equilibrium, via the second law and, decisively, Gibbs’s formulation of chemical potentials and phase equilibria, placed the equilibrium constant on rigorous foundations,¹⁶,²⁰ with the full apparatus of standard states and activity conventions emerging in subsequent decades.

FROM GIBBS FREE ENERGY TO THE EQUILIBRIUM CONSTANT

Criterion of spontaneity and the nature of equilibrium

While the historical formulation of the law of mass action by Guldberg and Waage provided the initial algebraic framework, the rigorous modern definition of K is derived from the Gibbs free energy. Under conditions of constant temperature (T) and pressure (p), a spontaneous change in a closed system performing only pV-work proceeds in the direction that decreases the Gibbs free energy, G, until a minimum is reached.

For an isothermal change,

(3) $$\mathrm{\Delta }G=\mathrm{\Delta }H-T\mathrm{\Delta }S$$

where ΔH is the enthalpy change and TΔS is the entropic contribution. Many reactions do not consume reactants completely because G reaches its minimum at an intermediate extent of reaction. This minimum corresponds to chemical equilibrium; at that point, the reaction Gibbs energy satisfies Δ_rG = 0, and no further macroscopic change in composition occurs. The absence of macroscopic change does not imply microscopic stasis: at equilibrium the opposing molecular processes persist, but their net effect on composition vanishes.

The quantitative relationship between ∆_rG^ꝋ and K

The qualitative description of equilibrium as a minimum in free energy is formalized by the fundamental relation that governs the spontaneity of a reaction:

(4) $${\mathrm{\Delta }}_{\mathrm{r}}G={\mathrm{\Delta }}_{\mathrm{r}}{G}^{\mathrm{e}}+RT\ln Q$$

In this equation, each term plays a distinct role. Δ_rG is the Gibbs free energy of reaction; it may be understood as the instantaneous thermodynamic “driving force”, in the sense of a directional tendency rather than a mechanical force. Mathematically, it is defined as the partial derivative of the total Gibbs free energy of the system with respect to the extent of reaction²¹ at constant temperature and pressure,²

(5) $${\mathrm{\Delta }}_{\mathrm{r}}G={\left(\frac{\partial G}{\partial \xi }\right)}_{p,T}$$

A notational clarification is warranted. In ordinary usage, the symbol Δ denotes a finite difference between two states, as in ΔG = G_final - G_initial in Equation 3. In the symbol Δ_rG, however, the subscript r identifies a reaction quantity; for Gibbs energy, that quantity is defined by Equation 5. Thus, Δ_rG is not a finite change in the Gibbs free energy of the system, but the derivative of G with respect to the extent of reaction at fixed temperature and pressure, with units of energy per mole of reaction (e.g., kJ mol^-1). The potential for confusion between finite changes such as ΔG and the reaction quantity Δ_rG has been noted in the literature,²²-²⁴ and is one reason why the explicit form given in Equation 5 should always be kept in mind. Its sign determines the spontaneous direction of reaction. Δ_rG^ꝋ is the standard Gibbs free energy of reaction; for a given reaction at a fixed temperature, it is a characteristic value representing the “driving force” under specific reference conditions. Finally, Q is the reaction quotient, a dimensionless quantity expressing the instantaneous composition of the system in terms of activities.

Formally, for the generic reaction wW + xX ⇌ yY + zZ, the reaction quotient is defined as

(6) $$Q=\prod _i{a}_i^{v_i}=\frac{a_{\mathrm{Y}}^y{a}_{\mathrm{Z}}^z}{a_{\mathrm{W}}^w{a}_{\mathrm{x}}^x}$$

that is, as the ratio between the product of the activities of the products and the product of the activities of the reactants, each raised to its respective stoichiometric number, ν_i, where ν_i > 0 for products and ν_i < 0 for reactants; in this example, ν_W = -w, ν_X = -x, ν_Y = +y, and ν_Z = +z.

The equilibrium condition is defined as the point at which the “driving force” becomes zero (Δ_rG = 0).²²,²⁴ At this point, the quotient Q assumes its characteristic equilibrium value, the constant K. Substituting these conditions into Equation 4 gives

(7) $$0={\mathrm{\Delta }}_{\mathrm{r}}{G}^{\mathrm{e}}+RT\ln K$$

which leads to the central relation connecting thermodynamics to the equilibrium composition:²

(8) $${\mathrm{\Delta }}_{\mathrm{r}}{G}^{\mathrm{e}}=-RT\ln K$$

Equation 8 not only provides a bridge between energetic data (Δ_rG^ꝋ) and macroscopic composition, since K is the equilibrium value of the reaction quotient Q, and the latter is a known function of the activities (and hence of the concentrations or partial pressures) of all species, but also reveals a fundamental mathematical requirement: the equilibrium constant K must be dimensionless. Transcendental functions, such as the logarithm, by definition require arguments that are pure numbers.¹,²⁵ Thermodynamics satisfies this requirement elegantly, because K is not defined directly in terms of concentrations or pressures, but rather through the dimensionless quantity called activity, as will be detailed next.

Once the standard states are fixed, isolating K in Equation 8 gives

(9) $$K=\exp (-\frac{{\mathrm{\Delta }}_{\mathrm{r}}{G}^{\bullet }(T)}{RT})$$

showing that K = K(T); that is, K depends only on temperature. Figure 1 illustrates how the equilibrium constant K emerges from the formalism for the reaction H₂(g) + I₂(s) ⇌ 2HI(g) at 298.15 K and 1 bar, starting from 1 mol of H₂, 1 mol of I₂, and 0 mol of HI, assuming ideal-gas behavior for the gas phase, and taking = 1 for the pure solid. At the extent of reaction coordinate corresponding to equilibrium, ξ = ξ_e, Δ_rG(ξ_e) = 0 (lower panel) and therefore Q(ξ_e) = K; in the upper panel, the same point coincides with the minimum of G(ξ).²⁷ For ξ < ξ_e, Δ_rG < 0 and the reaction proceeds forward; for ξ > ξ_e, Δ_rG > 0 and the reaction proceeds in reverse.

Figure 1
Gibbs free energy profile for the formation of hydrogen iodide from hydrogen and solid iodine. Upper panel: G(ξ); the dashed line indicates the extent of reaction at equilibrium, ξ_e ≈ 0.244. Lower panel: ∆_rG(ξ); the dotted line marks ∆_rG = 0 at ξ_e. Curves computed from tabulated ∆_fG^ꝋ (298.15 K, 1 bar)²⁶

It is also observed that Δ_rG rises sharply as ξ → 1, a consequence of the RT ln Q term. For H₂(g) + I₂(s) ⇌ 2HI(g) (with = 1 and p^ꝋ = 1 bar), Q = (p_HI / p^ꝋ)² / (p_H2 / p^ꝋ), where the replacement of activities by pressure ratios p_i / p^ꝋ follows from the ideal-gas expression for the activity of a gaseous species, as justified in the next section. As p_H2 → 0 the reaction quotient diverges (Q → ∞) and the reverse reaction becomes strongly favored. Δ_rG^ꝋ = 3.4 kJ mol^-1 quantifies the “driving force” under standard conditions (all activities a_i = 1) and not the depth of the G(ξ) well. From this value, insertion into Equation 9 yields K ≈ 0.254 at 298.15 K. It is important to note that K is a property of the reaction (for the given temperature), whereas the equilibrium composition (ξ_e) depends on the initial amounts.

Thermodynamic activity as an “effective concentration”

The key to the dimensionless nature of K and to the generality of thermodynamic relations lies in the concept of activity, traditionally denoted by the symbol a.

Activity represents the effective contribution of a species to the Gibbs energy of the system and may be understood as an “effective concentration” in solutions. By definition, the activity of a species i (a_i) is a dimensionless number given by the ratio between the quantity that measures the presence of the species (for example, concentration, mole fraction, or partial pressure) and the corresponding value in the adopted standard state (such as c^ꝋ for molar concentrations or p^ꝋ for pressures).² In the ideal limit, this nondimensionalization coincides numerically with the corresponding measure; in real systems, the activity incorporates an activity coefficient (γ) that accounts for deviations from ideality. Furthermore, its numerical value depends on the composition scale and on the adopted standard state.

For solutions, in practice the activity may be taken as the molar concentration divided by the reference value (i.e., the standard state c^ꝋ) and corrected by an activity coefficient, γ:

(10) $$a_{\text{solute}}=\gamma \frac{[\text{solute}]}{c^{\star }}$$

In the ideal limit, γ = 1. The same principle applies across different composition scales: one uses the ratio of composition to the standard state of the chosen scale together with the corresponding activity coefficient. Accordingly, consistent use of activity requires explicit specification, from the outset, of both the composition scale and the adopted standard state. Although any value could in principle be chosen, it is conventional to adopt c^ꝋ = 1 mol L^-1 and m^ꝋ = 1 mol kg^-1.²

For solutes in solution, the activity takes scale-dependent forms: a_i = γ_c,i(c_i/c^ꝋ) in molarity and a_i = γ_m,i(m_i/m^ꝋ) in molality, each employing the activity coefficient corresponding to its respective scale. On the molarity and molality scales, the standard state of a solute is a hypothetical solution at the standard composition (c^ꝋ, m^ꝋ) referenced to ideal dilute behavior, behaving like the infinitely dilute solution.² Consistently with this convention, the corresponding activity coefficient is defined so that γ_i → 1 as the solute composition tends to zero.

Two aspects of this definition deserve emphasis. First, the standard state just described is not a physically realizable solution.²,¹¹,²⁸ A real solution at a concentration of 1 mol L^-1 exhibits solute-solute interactions (electrostatic, hydrophobic, etc.)²⁸,²⁹ that become negligible in the limit of infinite dilution.²⁸,³⁰ The standard state is therefore a deliberate idealization: a hypothetical solution at the standard composition (c^ꝋ, m^ꝋ) that behaves like the infinitely dilute solution,² that is, whose partial molar enthalpy, heat capacity, and volume coincide with their infinite-dilution values.²,²⁸ This dual character, namely a finite standard composition but ideal-dilute limiting properties, is precisely what makes the convention seem counterintuitive; yet it is indispensable because it provides a well defined, composition-independent reference point from which departures can be measured.²⁸ Second, the phrase “ideal behavior” has a concrete physical meaning that goes beyond the formal statement γ = 1. For a solute in solution, ideal behavior corresponds to the ideal-dilute limit, in which solute-solute interactions become negligible relative to solute-solvent interactions; this limit is approached as the solution becomes more dilute.¹¹,²⁸ For gases, ideal behavior corresponds to the ideal-gas limit, in which intermolecular forces and molecular volumes are negligible; this limit is approached at sufficiently low pressures and, in practice, is favored by higher temperatures.¹⁰,¹¹ Since a_i = γ_c,i(c_i/c^ꝋ),² the activity coefficient acts as a multiplicative correction that relates thermodynamic activity to nominal composition; in the ideal limit, γ tends to unity.¹¹,²⁸

For gases, a_i = f_i/p^ꝋ, with p^ꝋ = 1 bar and f_i = ϕ_ip_i, where f_i is the fugacity (an “effective pressure” that corrects deviations from ideality), p_i is the partial pressure, and ϕ_i is the fugacity coefficient. In the ideal-gas limit, ϕ_i = 1 and f_i = p_i; under these conditions, a_i = p_i/p^ꝋ.²

For pure solids and liquids, the standard state is the pure substance in its stable form at 1 bar. By convention, a = 1 in this standard state. Thus, in heterogeneous equilibria, these terms do not appear explicitly in Q or K, since they are unit factors.²

With these definitions, the following sources of confusion can be resolved. Pure solids and liquids do not “disappear” from the expression for K; by definition, the pure substance is its own standard state (at 1 bar), so its activity is 1. Their terms are omitted from Q or K only because they contribute unit factors; the corresponding activities remain implicit. The standard state of a solvent (for example, water in an aqueous solution) is the pure solvent at 1 bar. In dilute solutions, the mole fraction of the solvent is very close to 1, and its behavior approaches that of the pure state. Formally, for nearly ideal solutions with an ideal vapor phase, its activity is given by Raoult’s law:

(11) $$a_{\text{solvent}}=\frac{p_{\text{solvent}}}{p_{\text{solvent}}^{\ast }}$$

where p_solvent is the vapor pressure of the solvent in the solution and p^*_solvent is the vapor pressure of the pure solvent.²,¹¹ Under these conditions, a_solvent ≈ 1, and its explicit appearance in equilibrium expressions is often suppressed without practical loss of accuracy.

Finally, K is intrinsically dimensionless: it is a product of activities raised to their respective stoichiometric numbers. This definition ensures that the argument of ln K in Equation 8 is legitimate, that the numerical value of K remains invariant with respect to the choice of units for pressures or concentrations, and that the omission of pure solids and liquids does not alter K, since their activities are unity.

The need for a standard state

The concept of activity is inseparable from the definition of a standard state. Thermodynamic quantities such as the Gibbs free energy (G) have measurable physical meaning through differences (ΔG), and Equation 4 implies that the reaction quotient Q must be dimensionless. This is ensured when Q is expressed in terms of activities referred to standard states.

The standard state provides a reference level and fulfills two roles. It provides a common origin for tabulated data, fixing the reference level for standard quantities. A very important such quantity is the standard Gibbs energy of formation, Δ_fG^ꝋ, where the subscript “f” denotes formation of a substance from its constituent elements in their reference states. By convention, Δ_fG^ꝋ = 0 for elements in their reference states, which provides a self-consistent origin for tabulated thermodynamic data.²,¹⁰,²⁶ In addition, it serves as the scale for nondimensionalization of activities, providing the denominator that renders Q and K pure numbers, unit-invariant and compatible with transcendental functions such as the logarithm.¹,³,²⁵

Is the standard state the equilibrium state?

The equilibrium state and the standard state are frequently conflated. To clarify the distinction, it is useful to separate the respective roles of Δ_rG and Δ_rG^ꝋ.

The term Δ_rG is the Gibbs free energy of reaction at a given state of the system: a variable quantity, dependent on instantaneous composition through Q, whose sign indicates the spontaneous direction of the reaction. It should be emphasized that there is no time dependence; in other words, Δ_rG provides no information about reaction rates; such aspects are addressed by chemical kinetics. The reaction proceeds in the direction that decreases the Gibbs free energy of the system and ceases when this “driving force” is exhausted, i.e., when Δ_rG = 0.

On the other hand, the term Δ_rG^ꝋ represents the standard Gibbs free energy of reaction. Unlike Δ_rG, it is a fixed value for a given reaction at a specified temperature. It does not describe an arbitrary reaction mixture but rather answers a hypothetical and specific question: what would the value of Δ_rG be if all species (reactants and products) were in their respective standard states? Since, by definition, in this reference all activities are equal to 1, it follows that Q = 1.²,³ Hence, starting from Equation 4, if, by construction, the reaction is evaluated in the reference of the standard states, then

(12) $${{\mathrm{\Delta }}_{\mathrm{r}}G|}_{Q=1}={\mathrm{\Delta }}_{\mathrm{r}}{G}^e+RT\ln (1)={\mathrm{\Delta }}_{\mathrm{r}}{G}^e$$

For this point to also correspond to equilibrium, it is necessary that Δ_rG = 0, which implies Δ_rG^ꝋ = 0. From the relation Δ_rG^ꝋ = -RT ln K, this occurs only when K = 1. Thus, except for this hypothetical case, the composition corresponding to unit activities does not coincide with the equilibrium composition; in general, Δ_rG|_Q=1 = Δ_rG^ꝋ ≠ 0.

THE PRACTICAL CONSTANTS K_c AND K_p AS APPROXIMATIONS TO K

The concept of activity ensures the strictly dimensionless character of the thermodynamic equilibrium constant K. In practice, however, K is seldom used directly; the question is how it relates to the expressions K_c and K_p, more commonly found in textbook and reference treatments. The answer lies in the fact that K_c and K_p are numerical approximations to K, whose validity rests on assumptions of ideal behavior, along with a notational simplification.

From the definition of the reaction quotient in Equation 6 and the equilibrium condition Q = K (when Δ_rG = 0), the thermodynamic equilibrium constant is given by the same functional form as Q, but now evaluated exclusively at the equilibrium composition; from this point onward, the activities a_i appearing in expressions for K refer to their equilibrium values unless stated otherwise:

(13) $$K=\prod _i{a}_i^{{\nu }_i}=\frac{a_{\mathrm{Y}}^y{a}_{\mathrm{Z}}^z}{a_{\mathrm{W}}^w{a}_{\mathrm{X}}^x}$$

For solutions, adopting the molar concentration scale, the activity of each species is expressed as

(14) $$a_i={\gamma }_{c,i}\frac{c_i}{c^2}$$

Substituting this expression into the definition of K yields

(15) $$K=\underset{K_{\gamma }}{\underbrace{\frac{{\gamma }_{c,\mathrm{Y}}^y{\gamma }_{c,\mathrm{Z}}^z}{{\gamma }_{c,\mathrm{W}}^w{\gamma }_{c,\mathrm{X}}^x}}}\underset{K_c}{\underbrace{\frac{{\left(c_{\mathrm{Y}}/c^{\star }\right)}^y{\left(c_{\mathrm{Z}}/c^{\star }\right)}^z}{{\left(c_{\mathrm{W}}/c^{\star }\right)}^w{\left(c_{\mathrm{X}}/c^{\star }\right)}^x}}}$$

that is, K = K_γ K_c. In the ideal-solution limit, corresponding to sufficiently dilute systems, the activity coefficients approach unity, so that γ_c,i ≈ 1 implies K_γ ≈ 1 and, therefore, K ≈ K_c.³

For gases, the activity of each species is defined by a_i = f_i / p^ꝋ, where p^ꝋ is the standard pressure, and f_i = ϕ_ip_i denotes the fugacity. Analogously to the case of solutions, substitution into the definition of K leads to

(16) $$K=\frac{{\varphi }_{\mathrm{Y}}^y{\varphi }_{\mathrm{Z}}^z}{\underset{K_{\theta }}{\underbrace{{\varphi }_{\mathrm{W}}^w{\varphi }_{\mathrm{X}}^x}}}\underset{K_p}{\underbrace{\frac{{\left(p_{\mathrm{Y}}/p^{\ast }\right)}^y{\left(p_{\mathrm{Z}}/p^{\ast }\right)}^z}{{\left(p_{\mathrm{W}}/p^{\ast }\right)}^w{\left(p_{\mathrm{X}}/p^{\ast }\right)}^x}}}$$

that is, K = K_ϕK_p. At sufficiently low pressures, the fugacity coefficients approach unity, so that ϕ_i ≈ 1 implies K_ϕ ≈ 1 and, therefore, K ≈ K_p.

The practical forms commonly found in textbooks often omit the divisors c^ꝋ and p^ꝋ, writing K_c and K_p as products of raw concentrations or pressures. To avoid the mistaken impression that K would possess physical dimensions, it is useful to make the dimensionless forms explicit:

(17) $$K_c=\prod _i{\left(\frac{c_i}{c^{\theta }}\right)}^{v_i},K_p=\prod _i{\left(\frac{p_i}{p^{\theta }}\right)}^{v_i}$$

Revisiting the generic reaction considered earlier, these expressions can be expanded as

(18) $$K_c=\frac{{\left(c_{\mathrm{Y}}/c^{\theta }\right)}^y{\left(c_{\mathrm{Z}}/c^{\theta }\right)}^z}{{\left(c_{\mathrm{W}}/c^{\theta }\right)}^w{\left(c_{\mathrm{X}}/c^{\theta }\right)}^x}$$

(19) $$K_p=\frac{{\left(p_{\mathrm{Y}}/p^{\theta }\right)}^y{\left(p_{\mathrm{Z}}/p^{\theta }\right)}^z}{{\left(p_{\mathrm{W}}/p^{\theta }\right)}^w{\left(p_{\mathrm{X}}/p^{\theta }\right)}^x}$$

Although these explicitly dimensionless forms are conceptually appropriate, it is useful to distinguish them from a dimensional variant that is frequently employed. For solutions, on the molarity scale, it is defined as

(20) $$K_c^{\prime }=\frac{c_{\mathrm{Y}}^y{c}_{\mathrm{Z}}^z}{c_{\mathrm{W}}^w{c}_{\mathrm{X}}^x}$$

which has dimensions of (mol L^-1)^∆ν, where , in contrast with the dimensionless form K_c.² Nondimensionalization is obtained directly from the relation

(21) $$K_c=\frac{K_c^{\prime }}{{\left(c^{\ast }\right)}^{\mathrm{\Delta }v}}$$

Analogously, the dimensional variant for gases is given by

(22) $$K_p^{\prime }=\frac{p_{\mathrm{Y}}^y{p}_{\mathrm{Z}}^z}{p_{\mathrm{w}}^w{p}_{\mathrm{X}}^x}$$

with dimensions of (bar)^∆ν and related to the dimensionless form by K_p = K_p’/(p^ꝋ)^Δν.

A brief example illustrates how the nondimensionalization is carried out. For the Haber process, N₂(g) + 3H₂(g) ⇌ 2NH₃, the dimensional constant K_p’ is defined by the ratio of raw partial pressures:

(23) $$K_p^{\prime }=\frac{{\left(p_{{\mathrm{NH}}_3}\right)}^2}{\left(p_{{\mathrm{N}}_2}\right){\left(p_{{\mathrm{H}}_2}\right)}^3}$$

If pressures are expressed in bar, the numerical value of K_p’ carries the units (bar)^-2. For this reaction, the change in stoichiometric coefficients is Δν = 2 - (1 + 3) = -2. The standard state for gases is p^ꝋ = 1 bar. Applying the nondimensionalization relation K_p = K’_p/(p^ꝋ)^∆ν gives:

(24) $$K_p=\frac{K_p^{\prime }}{{\left(p^{\ast }\right)}^{-2}}$$

Thus, dividing the dimensional quantity K_p’ (with units of [bar]^-2) by (p^ꝋ)^-2 (which has units of [bar]^-2) explicitly cancels the dimensions, yielding the dimensionless constant K_p. It should be noted that, when Δν = 0, the dimensional constant exhibits a fortuitous cancellation of units that can obscure the conceptual need for the standard state reference. For instance, consider the formation of hydrogen iodide, H₂(g) + I₂(g) ⇌ 2HI(g). The dimensional form K_p’ is:

(25) $$K_p^{\prime }=\frac{{\left(p_{\mathrm{HI}}\right)}^2}{\left(p_{{\mathrm{H}}_2}\right)\left(p_{{\mathrm{I}}_2}\right)}$$

Here, the units (e.g., (bar)²/(bar·bar)) cancel, making K_p’ appear dimensionless. This aligns with the change in stoichiometric coefficients, Δν = 2 - (1 + 1) = 0. Applying the formal nondimensionalization relation K_p = K_p’ / (p^ꝋ)^Δν confirms this numerical equality:

(26) $$K_p=\frac{K_p^{\prime }}{{\left(p^{\ast }\right)}^0}=\frac{K_p^{\prime }}{1}=K_p^{\prime }$$

Although K_p and K_p’ are numerically identical, it is the division by (p^ꝋ)^Δν that formally renders K_p dimensionless, a conceptual step obscured by the apparent unit cancellation in K_p’.

Therefore, K_c and K_p are dimensionless quantities: the units cancel term by term because each ratio employs the same unit in the numerator and in the denominator. When, for notational convenience, the divisors c^ꝋ or p^ꝋ are omitted from the expression, this corresponds merely to an implicit nondimensionalization rather than to a conceptual modification. The dimensional variants K_c’ and K_p’, although still frequently encountered in the literature, possess physical dimensions and do not constitute direct approximations to the thermodynamic equilibrium constant K. Their use, therefore, obscures the fundamentally dimensionless nature of K and introduces dimensional dependencies of a purely notational character.

Arguments regarding the utility of dimensional constants, however, merit mention. Helfferich highlighted their engineering convenience in encapsulating stoichiometric details,³¹ while Laidler defended the pedagogical convenience of equilibrium constants written directly in terms of concentration or pressure ratios carrying units, noting that the IUPAC Green Book permits the use of K_c and K_p with units alongside a dimensionless thermodynamic constant.²,³² Nonetheless, these dimensional forms are unit-dependent parameters rather than equilibrium constants in the thermodynamic sense; after explicit nondimensionalization they reduce to K_c and K_p, the usual approximations to K.

In summary, K_c and K_p are not fundamental constants but useful numerical approximations to K: they become equal to K when deviations from ideality are negligible (γ_c,i ≈ 1 or ϕ_i ≈ 1) and remain strictly dimensionless when expressed as ratios relative to the standard states. The dimensionless forms preserve both conceptual correctness and the direct connection with rigorous thermodynamics, thus constituting the most appropriate choice for the quantitative treatment of chemical equilibria.

ON A PERSISTENT INCONSISTENCY: WHY THE pK_a OF WATER IS 14.0 (AND NOT 15.7)

An inconsistent use of activities and standard states has historically led to the widespread appearance of the value pK_a(H₂O) = 15.7 in textbooks, particularly in organic chemistry, as documented by Silverstein and Heller.³³ Prominent recent examples include McMurry,³⁴ Bruice,³⁵ and Solomons.³⁶ This number does not represent a thermodynamic pK_a on the standard-state scale used for aqueous equilibria, but rather a concentration-based quantity obtained by mixing distinct reference conventions. This inconsistency, documented since the work of Brønsted in the 1920s,³³ persists in much of the current textbook literature despite its thermodynamic inadequacy.³³,³⁷,³⁸

To understand the origin and nature of this inconsistency, the starting point is the autoionization (autoprotolysis) reaction of water:

(27) $$2{\mathrm{H}}_2\mathrm{O}(\mathrm{l})\rightleftharpoons {\mathrm{H}}_3{\mathrm{O}}^{+}(\mathrm{aq})+{\mathrm{OH}}^{-}(\mathrm{aq})$$

where H₃O⁺(aq) is used as the conventional representation of the hydrated proton. More generally, the excess proton in aqueous media is better described as a fluctuating ensemble of hydrated structures spanning the limiting Zundel-like H₅O₂⁺ and Eigen-like H₉O₄⁺ motifs, rather than as an isolated H₃O⁺ ion.³⁹-⁴¹ The symbol H⁺(aq) is nevertheless an equally acceptable thermodynamic shorthand.² The choice of notation does not affect the thermodynamic treatment, since both H₃O⁺(aq) and H⁺(aq) refer to the same conventional aqueous proton species under the adopted standard-state convention.

The thermodynamic equilibrium constant for this reaction is:²⁶

(28) $$K=\frac{a_{{\mathrm{H}}_3{\mathrm{O}}^{+}}{a}_{{\mathrm{OH}}^{-}}}{{\left(a_{{\mathrm{H}}_2\mathrm{O}}\right)}^2}\equiv K_{\mathrm{w}}$$

The discrepancy originates in the standard state assigned to the solvent. For the solvent, the reference is the pure substance at the same T, p as the system; on the mole-fraction scale, the activity can be written as = γ_wx_w, where γ_w is the corresponding activity coefficient.² In self-ionized water at 25 °C, [H₃O⁺] = [OH^-] ≈ 10^-7 mol L^-1,²⁶ so that

(29) $$x_{\mathrm{w}}\approx 1-\frac{2\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right]}{c_{\text{water}}}\approx 1-\frac{2\times {10}^{-7}}{55.3}\approx 0.9999999964$$

where c_water = 55.3 mol L^-1 is the molar concentration of pure water (the total number of moles of H₂O per liter of solution).

The relative deviation in solvent composition due to autoprotolysis is ≈ 3.6 × 10^-9, so the approximation ≈ 1 is justified. This approximation does not deny autoprotolysis; it recognizes that, because the standard state of the solvent is the pure liquid, the activity of nearly pure water is essentially unity.² Thus, the expression for the equilibrium constant reduces to²⁶

(30) $$K\approx a_{{\mathrm{H}}_3{\mathrm{O}}^{+}}{a}_{{\mathrm{OH}}^{-}}=1.01\times {10}^{-14}\left({25}^{\circ }\mathrm{C}\right)$$

As a direct consequence of ≈ 1, the solvent term is omitted from the equilibrium expression. On this solvent-standard-state convention, the autoprotolysis constant coincides with the acid-dissociation constant of water in water, K_a(H₂O) ≡ K_w, which leads to the thermodynamically consistent value²⁶

(31) $$\mathrm{p}{K}_{\mathrm{a}}\left({\mathrm{H}}_2\mathrm{O}\right)=13.995\approx 14.0\left({25}^{\circ }\mathrm{C}\right)$$

The value pK_a(H₂O) = 15.7 arises from a fundamental inconsistency: the asymmetric treatment of the two water molecules that participate in autoprotolysis. In this approach, ≈ 1 is assigned to the molecule acting as the base, but the molar concentration of pure water (≈ 55.3 mol L^-1) is used for the molecule acting as the acid. This arbitrary choice defines a constant K_a^×:

(32) $$\begin{array}{rl}{K}_{\mathrm{a}}^{\times }\left({\mathrm{H}}_2\mathrm{O}\right)& =\frac{a_{{\mathrm{H}}_3{\mathrm{O}}^{+}}{a}_{{\mathrm{OH}}^{-}}}{{\left[{\mathrm{H}}_2\mathrm{O}\right]}_{\text{acid}}{a}_{{\mathrm{H}}_2\mathrm{O},\text{base}}}\\ & \approx \frac{\left(\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right]/c^{\star }\right)\left(\left[{\mathrm{OH}}^{-}\right]/c^{\star }\right)}{\left(\left[{\mathrm{H}}_2\mathrm{O}\right]/c^{\star }\right)\times 1}=1.81\times {10}^{-16}\end{array}$$

which leads to pK_a^×(H₂O) = 15.7. The conceptual shortcoming is clear: the equilibrium expression cannot consistently treat one H₂O factor with the solvent standard state ( = 1) and the other with a solute-style concentration reference without changing the underlying reference convention.³⁷ This improper mixing of standard states defines K_a^× on a reference convention that differs from the one underlying every other aqueous pK_a value, so that K_a^× and the corresponding pK_a^× do not represent a well-defined thermodynamic equilibrium constant on the usual aqueous pK_a scale. In applications to dilute aqueous chemistry, a thermodynamically consistent treatment thus takes = 1 (essentially pure solvent), uses K_w = 1.01 × 10^-14 at 25 °C (or the corresponding value at another temperature), and, consequently, arrives at pK_a(H₂O) = 13.995 ≈ 14.0.

The comparison between pK_a(H₂O) ≈ 14.0 and pK_a values of solutes in water is valid because all are expressed on the same thermodynamic scale. The molar concentration of water (~ 55.3 mol L^-1) is irrelevant: introducing it into the expression for K_a(H₂O) changes the solvent’s reference state and produces the scale-dependent quantity 15.7, which is not directly comparable with the usual aqueous pK_a scale.

The perpetuation of pK_a(H₂O) = 15.7 entails serious interpretative consequences. Because tabulated pK_a values for aqueous solutes are determined on the standard scale of 1 mol L^-1, replacing the correct 14.0 with 15.7 introduces a systematic error of 10¹.⁷ ≳ 50 fold in acidity comparisons and, for example, incorrectly suggests that methanol (pK_a = 15.5) is more acidic than water in aqueous solution.²⁶,³³,³⁷ Beyond these numerical discrepancies, the persistence of 15.7 reflects the use of incompatible standard-state conventions: while 14.0 follows the thermodynamic definition based on activities, 15.7 results from inserting the molar concentration of water into the equilibrium expression. Resolving the discrepancy therefore requires keeping standard states explicit and avoiding mixed conventions.

The definition of pH and the role of activity

The case of pH directly illustrates the same conceptual subtleties. By definition,²

(33) $$\mathrm{pH}\equiv -{\log }_{10}{a}_{{\mathrm{H}}_3{\mathrm{O}}^{+}}$$

in which is the activity of the oxonium ion (H₃O⁺, the hydrated proton; the term “hydronium”, though widespread in textbook usage, is not recommended by IUPAC),⁴² hence a dimensionless number.

Strictly, the pH scale is referenced to the molality activity scale,⁴³

(34) $$a_{{\mathrm{H}}_3{\mathrm{O}}^{+}}={\gamma }_{m,{\mathrm{H}}_3{\mathrm{O}}^{+}}\frac{m_{{\mathrm{H}}_3{\mathrm{O}}^{+}}}{m^{\ast }},m^{\ast }=1\mathrm{mol}{\mathrm{kg}}^{-1}$$

For dilute aqueous solutions at 25 °C, the numerical difference between molality-based and molarity-based activities is small (the density of water being close to 1 kg L^-1), so the more familiar molarity-based activity expression is an excellent approximation:

(35) $$a_{{\mathrm{H}}_3{\mathrm{O}}^{+}}\approx {\gamma }_{c,{\mathrm{H}}_3{\mathrm{O}}^{+}}\frac{\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right]}{c^{\star }},c^{\star }=1\mathrm{mol}{\mathrm{L}}^{-1}$$

Accordingly, in sufficiently dilute solutions (where ≈ 1), the usual approximation is obtained:

(36) $$\mathrm{pH}\approx -{\log }_{10}\left(\frac{\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right]}{1\mathrm{mol}{\mathrm{L}}^{-1}}\right)$$

often shortened to pH ≈ -log₁₀[H₃O⁺] when the unit “1 mol L^-1” is left implicitly in the denominator.

The shorthand is valid because nondimensionalization with respect to c^ꝋ makes the logarithm’s argument a pure number; omitting this reference makes the expression appear to take a logarithm of a dimensional quantity. The framework developed in this manuscript, i.e., activities as dimensionless ratios referenced to standard states, the distinction between dimensionless and dimensional forms, and the role of activity coefficients, resolves this tension: it explains why pH is necessarily dimensionless, justifies the approximation by concentrations only in regimes of ideality, and anticipates when it fails (higher ionic strengths, concentrated strong acids, etc.), cases in which with γ ≠ 1 should be employed. Thus, the same reasoning that corrects “pK_a(H₂O) = 15.7” also clarifies why the notation pH ≈ -log₁₀[H₃O⁺] is, strictly speaking, an abbreviated form of -log₁₀([H₃O⁺]/c^ꝋ) and demonstrates why consistency of standard states is essential to avoid apparent contradictions.

It should be emphasized that the pH scale is inherently conventional. Since pH is defined in terms of a single-ion activity, its determination necessarily rests on extra-thermodynamic assumptions, and the numerical values assigned to pH standards therefore depend on the convention adopted, specifically the Bates-Guggenheim convention.⁴³ For historical background on the development of the pH scale, see Bates.⁴⁴

CONCLUSIONS

The thermodynamic equilibrium constant K is necessarily dimensionless, as required by its appearance in Δ_rG^ꝋ = -RT ln K. In thermodynamics, K is defined as a product of activities, with standard states providing the reference that renders each activity a pure number. This explains why pure solids, pure liquids, and the solvent are omitted from Q and K: their activities are unity by definition for pure substances and approximately so for the solvent in dilute solution, contributing only unit factors.

The practical constants K_c and K_p are dimensionless approximations to K obtained by replacing activities with concentrationor pressure-based expressions referenced to c^ꝋ and p^ꝋ. The commonly used dimensional variants K_c’ and K_p’ depend on the chosen units and acquire thermodynamic meaning only after explicit nondimensionalization; agreement with K further requires the usual ideality approximations (e.g., γ_i ≈ 1 or ϕ_i ≈ 1).

The pK_a of water provides a concrete illustration. Using a consistent solvent standard state gives and pK_a(H₂O) ≈ 14.0 at 25 °C; the frequently quoted value 15.7 results from mixing solvent and solute reference conventions by inserting [H₂O] into the equilibrium expression. The same point underlies , and the common concentration form is a shorthand for -log₁₀([H₃O⁺]/c^ꝋ) in the dilute limit.

In equilibrium calculations, keeping standard states explicit and treating K as activity-based avoids spurious units and prevents inconsistencies when comparing constants across systems and subdisciplines. More broadly, the persistence of errors such as pK_a(H₂O) = 15.7 illustrates an epistemological pitfall: when reference conventions are left implicit, notational shorthands can be mistaken for physical content.

ACKNOWLEDGMENTS

The author used Google Gemini 3 Pro for grammar and spelling, preparation of the graphical abstract, and literature search. The scientific content, analysis, and conclusions are the author’s own.

DATA AVAILABILITY STATEMENT

All data discussed in this work are available in the text and in the cited references. The MATLAB⁴⁵ code used to generate Figure 1 is available from Zenodo at https://doi.org/10.5281/zenodo.17930170.

REFERENCES

• ¹ Özcan, M.; J. Phys. Chem. Lett. 2022, 13, 3507. [Crossref]
  » Crossref
• ² Brett, C. M. A.; Frey, J. G.; Hinde, R.; Kuroda, Y.; Marquardt, R.; Pavese, F.; Quack, M.; Stohner, J.; Thor, A. J.; Quantities, Units and Symbols in Physical Chemistry: Abridged Version, 4^th ed.; Royal Society of Chemistry: London, 2023.
• ³ Gordus, A. A.; J. Chem. Educ. 1991, 68, 138. [Crossref]
  » Crossref
• ⁴ Quílez-Pardo, J.; International Journal of Physics and Chemistry Education 2019, 11, 73. [Crossref]
  » Crossref
• ⁵ Hackling, M. W.; Garnett, P. J.; European Journal of Science Education 1985, 7, 205. [Crossref]
  » Crossref
• ⁶ Banerjee, A. C.; Int. J. Sci. Educ. 1991, 13, 487. [Crossref]
  » Crossref
• ⁷ van Driel, J. H.; Gräber, W. In Chemical Education: Towards Research-Based Practice; Gilbert, J. K.; de Jong, O.; Justi, R.; Treagust, D. F.; van Driel, J. H., eds.; Springer: Dordrecht, 2002, ch. 12.
• ⁸ Cheung, D.; J. Chem. Educ. 2009, 86, 514. [Crossref]
  » Crossref
• ⁹ Cohen, E. R.; Cvitaš, T.; Frey, J. G.; Holmström, B.; Kuchitsu, K.; Marquardt, R.; Mills, I.; Pavese, F.; Quack, M.; Stohner, J.; Strauss, H. L.; Takami, M.; Thor, A. J.; Grandezas, Unidades e Símbolos em Físico-Química, 3^a ed.; EditSBQ: São Paulo, 2018.
• ¹⁰ Kondepudi, D.; Prigogine, I.; Modern Thermodynamics, 2^nd ed.; John Wiley & Sons: Chichester, 2015.
• ¹¹ Atkins, P. W.; de Paula, J.; Keeler, J.; Atkins’ Physical Chemistry, 12^th ed.; Oxford University Press: Oxford, 2023.
• ¹² Guldberg, C. M.; Waage, P.; Forhandlinger i Videnskabs-Selskabet i Christiania 1864, 35. [Link] accessed in April 2026
  » Link
• ¹³ Guldberg, C. M.; Waage, P.; Études Sur Les Affinités Chimiques; Imprimerie de Brøgger & Christie: Christiania, 1867.
• ¹⁴ Guldberg, C. M.; Waage, P.; J. Prakt. Chem. 1879, 19, 69. [Crossref]
  » Crossref
• ¹⁵ Baird, J. K.; J. Chem. Educ. 1999, 76, 1146. [Crossref]
  » Crossref
• ¹⁶ Lund, E. W.; J. Chem. Educ. 1965, 42, 548. [Crossref]
  » Crossref
• ¹⁷ Thackray, A.; Atoms and Powers: An Essay on Newtonian Matter-Theory and the Development of Chemistry; Harvard University Press: Cambridge, 1970.
• ¹⁸ Levere, T. H.; Affinity and Matter: Elements of Chemical Philosophy, 1800-1865; Clarendon Press: Oxford, 1971.
• ¹⁹ Horstmann, A.; Justus Liebigs Ann. Chem. 1877, 187, 48. [Crossref]
  » Crossref
• ²⁰ Gibbs, J. W.; Trans. Connecticut Acad. Arts Sci. 1876, 3, 108. [Link] accessed in April 2026
  » Link
• ²¹ Garst, J. F.; J. Chem. Educ. 1974, 51, 194. [Crossref]
  » Crossref
• ²² Raff, L. M.; J. Chem. Educ. 2014, 91, 386. [Crossref]
  » Crossref
• ²³ Raff, L. M.; J. Chem. Educ. 2014, 91, 839. [Crossref]
  » Crossref
• ²⁴ Raff, L. M.; J. Chem. Educ. 2014, 91, 2128. [Crossref]
  » Crossref
• ²⁵ Matta, C. F.; Massa, L.; Gubskaya, A. V.; Knoll, E.; J. Chem. Educ. 2011, 88, 67. [Crossref]
  » Crossref
• ²⁶ Haynes, W. M.; CRC Handbook of Chemistry and Physics, 97^th ed.; CRC Press: Boca Raton, 2016.
• ²⁷ Anderson, G. M.; Found. Chem. 2025, 27, 59. [Crossref]
  » Crossref
• ²⁸ Robinson, R. A.; Stokes, R. H.; Electrolyte Solutions, 2^nd ed.; Butterworths & Co.: London, 1959.
• ²⁹ Israelachvili, J. N.; Intermolecular and Surface Forces, 3^rd ed.; Academic Press: Burlington, 2011.
• ³⁰ Silbey, R. J.; Alberty, R. A.; Bawendi, M. G.; Physical Chemistry, 4^th ed.; John Wiley & Sons: Hoboken, 2004.
• ³¹ Helfferich, F. G.; J. Chem. Educ. 1987, 64, 1069. [Crossref]
  » Crossref
• ³² Laidler, K. J.; J. Chem. Educ. 1990, 67, 88. [Crossref]
  » Crossref
• ³³ Silverstein, T. P.; Heller, S. T.; J. Chem. Educ. 2017, 94, 690. [Crossref]
  » Crossref
• ³⁴ McMurry, J. E.; Organic Chemistry: A Tenth Edition, 10^th ed.; OpenStax: Houston, 2023.
• ³⁵ Bruice, P. Y.; Organic Chemistry, 8^th ed.; Pearson: Upper Saddle River, 2016.
• ³⁶ Solomons, T. W. G.; Fryhle, C. B.; Snyder, S. A.; Organic Chemistry, 12^th ed.; John Wiley & Sons, Inc.: Hoboken, 2016.
• ³⁷ Neils, T. L.; Silverstein, T. P.; Schaertel, S.; J. Chem. Educ. 2023, 100, 1676. [Crossref]
  » Crossref
• ³⁸ Antonelli, L.; De Paula, R.; Cunha, S.; Quim. Nova 2024, 47, e-20230093. [Crossref]
  » Crossref
• ³⁹ Robertson, E. B.; Dunford, H. B.; J. Am. Chem. Soc. 1964, 86, 5080. [Crossref]
  » Crossref
• ⁴⁰ Fournier, J. A.; Carpenter, W. B.; Lewis, N. H. C.; Tokmakoff, A.; Nat. Chem. 2018, 10, 932. [Crossref]
  » Crossref
• ⁴¹ Calio, P. B.; Li, C.; Voth, G. A.; J. Am. Chem. Soc. 2021, 143, 18672. [Crossref]
  » Crossref
• ⁴² Connelly, N. G.; Damhus, T.; Hartshorn, R. M.; Hutton, A. T.; Nomenclature of Inorganic Chemistry; Royal Society of Chemistry: Cambridge, 2005.
• ⁴³ Buck, R. P.; Rondinini, S.; Covington, A. K.; Baucke, F. G. K.; Brett, C. M. A.; Camões, M. F.; Milton, M. J. T.; Mussini, T.; Naumann, R.; Pratt, K. W.; Spitzer, P.; Wilson, G. S.; Pure Appl. Chem. 2002, 74, 2169. [Crossref]
  » Crossref
• ⁴⁴ Bates, R. G.; Determination of pH: Theory and Practice, 2^nd ed.; John Wiley & Sons: New York, 1973.
• ⁴⁵ MATLAB, version R2025b; The MathWorks Inc., Natick, 2025.

Edited by

• Executive Editor handled this article:
  Gustavo F. S. Andrade

Publication Dates

• Publication in this collection
  26 June 2026
• Date of issue
  2026

History

• Received
  12 Feb 2026
• Accepted
  17 Apr 2026
• Published
  08 May 2026