Jones–Dole equation¶

An empirical dilute-solution relation separating a square-root and a linear concentration contribution to relative viscosity.

Version: v1 · 2026-10-17 · History
Domain-specific #: 14204
Domain group: Natural Sciences
Origin domain: Chemistry & Materials Science
Subdomains: Solution Chemistry, Electrolyte Viscosity → Chemistry & Materials Science
Aliases: Jones–Dole relation

Core Idea¶

The Jones–Dole equation describes how adding a solute changes a solvent's viscosity over a tested dilute range. At fixed temperature and pressure, it writes the relative viscosity as

\[ \eta_r=\frac{\eta}{\eta_0}=1+A\sqrt{c}+Bc, \]

where \(\eta\) is the solution viscosity, \(\eta_0\) is the matched reference-solvent viscosity, and \(c\) is concentration under a stated convention.^[1] Studies report different \(A\) and \(B\) values across solutes, solvents, and conditions.^[2]^[3]^[1] The square-root and linear powers separate contributions that change differently with concentration. Their fitted values do not by themselves prove two uniquely identified microscopic causes; a competing account of \(B\) trends illustrates why.^[4]

The name travels to new solution systems. Original measurements report Jones–Dole coefficients for alkali chlorides in water.^[2] Another study applies the equation to chlorides in propan-1-ol–water mixtures and reports cases requiring a further \(Dc^2\) term.^[3] It is the reusable form plus its conditions of application that is the abstraction. A particular salt, a fitted pair of coefficients, or one viscosity curve is an instance. A required higher-order term marks the simple equation's boundary in that regime.

Structural Signature¶

Matched viscosity ratio: solution viscosity \(\eta\) is compared with the viscosity \(\eta_0\) of the reference solvent under corresponding conditions.^[1] The ratio makes the response dimensionless; without a specified reference, the fitted coefficients are not interpretable across systems.
Concentration carrier: a solute concentration \(c\), with its unit and convention stated, indexes a series of solutions.^[1] The coefficients have units tied to that convention; changing it without converting them changes the numerical relation.
Square-root contribution: \(A\sqrt{c}\) supplies the lower-power term. In electrolyte applications \(A\) is associated with interionic interactions and may be calculated from other measured solvent and ion properties under a specified theory.^[1]
Linear contribution: \(Bc\) supplies a separately fitted concentration contribution. Reported \(B\) values differ by salt and are used to compare solute–solvent behavior.^[2] Their value alone does not establish one microscopic account of water structure, as an alternative interpretation of the trends shows.^[4]
Bounded applicability: the two-term form describes a measured regime, not every concentration or solvent composition. Studies report regimes needing a higher-order term or an ion-pair correction.^[3]^[1] Checking residuals is how an analyst tests the limit; it is not another term of the equation.

What It Is Not¶

The equation is not a universal viscosity law for all concentrations. The mixed-solvent study reports that its simple form was obeyed for the tested chlorides up to 30% propan-1-ol except LiCl, for which an added \(Dc^2\) term was required. At higher alcohol content the authors also had to account for ion association.^[3] A separate high-pressure KCl study explicitly fitted an extended Jones–Dole equation with \(Dc^2\).^[1] That extended fit cannot establish that the two-term equation alone was sufficient across the same range.

Nor is \(B\) a direct image of a hydrogen-bond network. It summarizes a viscosity contribution under the chosen fit. Feakins and colleagues revisited the common “structure-breaking” interpretation and proposed transition-state ion–solvent coordination as an alternative account of observed trends.^[4] The empirical coefficient can be useful while that causal question remains open.

The Jones–Dole equation is also distinct from a generic regression line. Fitting parameters is an operation used on it, but its identity includes a particular relative-viscosity response, a matched solvent reference, and the \(\sqrt{c}\) plus \(c\) dependence. An arbitrary linear fit to viscosity, or a lone reported \(B\) coefficient without its concentration and reference conventions, does not fully specify the named relation.

Scope of Application¶

Its principal setting is experimental solution chemistry: measure viscosity as solute concentration changes while temperature, pressure, and reference-solvent composition are held or explicitly tracked. The equation has been used for aqueous alkali-metal chlorides.^[2] It has also been used for alkali-metal chlorides in propan-1-ol–water mixtures, with composition-dependent limits on the simple form.^[3] A saccharide study uses the Jones–Dole name to obtain \(B\) coefficients in aqueous glycine and leucine media.^[5] Its accessible abstract does not show the full \(A\) treatment or a complete fit interval, so that study establishes broader use of the name without serving here as a full two-term example.

The solvent and concentration conventions are substantive. A coefficient from water at one temperature cannot simply be transferred to an alcohol–water mixture or a different temperature. A KCl study reports changing \(B\) across pressure and temperature.^[1] Any comparison must retain the conditions under which each coefficient was obtained.

Clarity¶

Writing \(\eta/\eta_0\) makes the baseline explicit: the question is how the solute-bearing solution differs from its reference solvent. Writing the two concentration powers separately makes a further distinction visible. A fitted \(B\) is not “the viscosity”; it is the coefficient of the linear contribution after the constant baseline and square-root contribution have been accounted for. In an extended fit, Nakai and colleagues rearranged the relation so that \((\eta_r-1-A\sqrt c)/c=B+Dc\); the intercept as \(c\) approaches zero identifies \(B\), while a nonzero slope reveals the extra \(Dc^2\) contribution.^[1]

This also clarifies what an empirical success establishes. The 1986 authors report a low-alcohol mixed-solvent range in which the simple equation was obeyed by several salts, and a LiCl case where it was not.^[3] Those observations test a relation for specified systems. They do not show that one coefficient pair or one microscopic explanation must apply to every salt and solvent.

Manages Complexity¶

Without a compact relation, each concentration has a separate measured relative viscosity. The published form has a fixed baseline and two concentration terms.^[3]^[1] Within an adequate range it summarizes the series and lets a chemist compare contributions and fitted coefficients under matched conditions.

The compact form buys tractability by leaving out effects that may matter elsewhere. In the mixed-solvent data, LiCl required a quadratic term even where other tested chlorides followed the simpler form; higher-alcohol mixtures required attention to association.^[3] In the KCl pressure study, the extended equation made \(B\) a zero-concentration intercept.^[1] Recording the usable range alongside \(A\) and \(B\) prevents the compression from hiding these distinctions.

Abstract Reasoning¶

The operative sequence is: define the solvent reference and concentration convention; measure a concentration series at stated conditions; compare the two-term relation with those observations; then interpret coefficients only at the supported level. The 1986 LiCl and 1997 KCl studies each use an additional \(Dc^2\) term in their specified regimes.^[3]^[1] If residual behavior calls for that term, the analyst changes the model rather than renaming \(D\) as part of \(B\).

The two powers provide a diagnostic question: does the low-concentration behavior require a square-root contribution separately from a linear one? The coefficient comparison then asks how solute, solvent composition, temperature, or pressure changes the fitted response. Feakins and colleagues present a competing account of \(B\) trends.^[4] That comparison shows why the empirical inference cannot settle a unique cause without separate tests.

Knowledge Transfer¶

The equation transfers literally within solution-viscosity work when the roles can be filled anew: a different solute, a different reference solvent, a new concentration series, and newly estimated coefficients. The aqueous and mixed-solvent chloride studies use the same mathematical form on different physical mixtures.^[2]^[3] A saccharide study also uses the named equation to obtain \(B\) coefficients.^[5] The accessible abstract does not establish the full two-term fit there.

Outside that setting, a “baseline plus square-root and linear response” can be an analogy or a convenient curve shape. Without solution viscosity, a matched solvent reference, and solute concentration, it is not the Jones–Dole relation as named. General lessons about fitting and range-checking belong to broader modeling practice; the equation's particular powers and chemical variables keep this entry domain-specific.

Examples¶

Canonical¶

Crudden and colleagues measured alkali-metal chloride solutions in propan-1-ol–water solvents at 25 and 35 °C. They report the simple \(1+A\sqrt c+Bc\) form as obeyed through 30% alcohol for the tested electrolytes except LiCl.^[3] Thus NaCl, KCl, and CsCl in that reported regime are concrete instances; their own \(A\) and \(B\) values are not universal constants. The accessible abstract does not enumerate each salt's concentration interval, so no more precise bound is implied here.

Mapped back: The alcohol–water mixture supplies the reference-solvent role, and adding one of the named salts creates the measured solution. Its concentration is \(c\); the reported fit has a square-root \(A\) term and linear \(B\) term.^[3] Using the salt-free solvent at the same composition as \(\eta_0\) is the appropriate role inference for the reported relative viscosity, not a definition quoted from the accessible abstract. The study reports the simple fit within its low-alcohol regime and a required \(Dc^2\) for LiCl.^[3] The latter is a nearby departure, not another clean simple-form case.

Applied / In Practice¶

Dordick, Korson, and Drost-Hansen measured aqueous LiCl, NaCl, KCl, RbCl, and CsCl at 25 °C and reported distinct Jones–Dole \(A\) and \(B\) coefficients. They also reported systematic departures from the limiting-law behavior for the salts investigated, with LiCl an apparent exception.^[2] A cautious use of aqueous LiCl therefore illustrates a candidate simple-form case, while the accessible abstract does not license an exact fit window or a claim of zero residual.

Mapped back: Water is the solvent reference; LiCl is the solute; changing its concentration changes the solution viscosity. The study reports LiCl \(A\) and \(B\) and an apparent exception to the reported systematic deviations.^[2] These fill the square-root and linear slots, but do not prove that the pair describes all concentrations. The other salts show why fitted coefficients and residual behavior must be retained separately for each system.

Structural Tensions¶

T1: Two-coefficient simplicity versus fit beyond the simple regime. The 1986 mixed-solvent study and 1997 KCl study report cases where an added \(Dc^2\) or association correction matters.^[3]^[1] A short equation makes concentration dependence easy to summarize and compare, but omitting a needed correction can create a systematic error. Adding it improves the representation of that departure at the cost of another fitted quantity and a less direct interpretation of the original two coefficients. Diagnostic: In the stated concentration and solvent regime, do observations support the two-term relation, or is the extension needed?

Structural–Framed Character¶

On evaluative weight, the equation makes a quantitative prediction, not a judgment about whether a solute is desirable. On human practice, viscosity changes may occur independently of an analyst, but establishing a coefficient and a usable range depends on a reference, conditions, and measurements. Nakai and colleagues measured KCl viscosity across pressure and temperature and used an extended fit to extract \(B\).^[1] The physical response and the fitted description must be kept distinct.

On institutional origin, the name records a historically authored equation, while its possible fit in a new solution does not require membership in the authors' laboratory or a particular institution. On vocabulary travel, its variables remain chemical: solution viscosity, matched solvent viscosity, and solute concentration. On import versus recognition, applying the same equation to a new measured solution is recognition of the named relation; borrowing only its two-power curve for unrelated quantities would be analogy. The water and mixed-solvent studies demonstrate the literal transfer within solution chemistry.^[2]^[3] Feakins and colleagues' alternative account of \(B\) trends shows that recognizing a fit does not settle a microscopic explanation.^[4]

Its character: structural in its quantitative relation, framed by empirical solution chemistry. The algebra is neutral and repeatable, but the named identity remains tied to the viscosity ratio, concentration input, and tested regime. That combination explains both its reuse among solutions and its limited claim beyond them.

Structural Core vs. Domain Accent¶

The portable skeleton is a baseline-normalized response with two distinguishable input powers and an explicit regime of applicability. It supports a general reasoning move: compress a series, inspect departures, and avoid transferring fitted parameters beyond their conditions. That skeleton alone is too broad to identify this equation; many systems can have a baseline and nonlinear response.

The domain accent is constitutive here: \(\eta/\eta_0\) compares solution and solvent viscosity, \(c\) is solute concentration, and the special dependence is \(A\sqrt c+Bc\) in a dilute solution regime.^[1]^[3] Removing those variables leaves a generic empirical model, not a substrate-independent Jones–Dole Prime. A bounded future-Prime question is whether two unlike non-solution domains independently use the same baseline-normalized square-root-plus-linear response, with a tested range and the same residual-based decision about extension. Evidence of that full operative pattern could support a more portable abstraction under another name. Merely fitting a curve with the same exponents, without corresponding roles or diagnostic consequences, would defeat that lift. No such cross-domain evidence is established here.

This entry is a kind of Mathematical Relation.

The direct catalog genus is Mathematical Relation: this is an empirical equation constraining typed concentration and relative-viscosity quantities, with a stated satisfaction condition, fitted parameters, and a bounded scope. The named square-root-plus-linear viscosity relation is its differentia. Other mathematical relations, including equivalence relations and equations for other physical responses, need not have it.

At fixed coefficients and conditions, the equation has a Function Mapping aspect: each admissible concentration receives one predicted relative viscosity. That formal mapping alone does not say whether the prediction fits observations. Measurement supplies the viscosity and concentration values used to assess it, but the equation is not itself an instrument or measurement procedure. Approximation is relevant when a two-term fit stands in for more complex behavior, provided the acceptable error has actually been established for the intended use; a fitted formula alone does not supply that guarantee.

Relationships to Other Abstractions¶

Current abstraction Jones–Dole equation Domain-specific

Parents (1) — more general patterns this builds on

Jones–Dole equation is a kind of Mathematical Relation Domain-specific

The Jones–Dole equation is a bounded empirical mathematical relation with typed concentration and relative-viscosity quantities, a stated satisfaction equation, and system-specific coefficients.

Hierarchy path (1) — routes to 1 parentless root

Jones–Dole equation → Mathematical Relation

Neighborhood in Abstraction Space¶

Jones–Dole equation sits in a sparse region of the domain-specific corpus (94^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Molecular Structure & Chemical Bonding (35 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-17

Not to Be Confused With¶

Extended Jones–Dole equation: a KCl study fits an additional \(Dc^2\) term.^[1] The extension shares lower-order terms but has an additional parameter and a different validity claim.
Dilution: changing concentration by adding solvent is an operation on a solution; the Jones–Dole relation predicts or organizes a viscosity response across concentrations. One does not imply the other.
Volume viscosity: a material's resistance to compression or expansion flow is a different viscosity concept from the shear viscosity compared across a solution concentration series.
A lone B coefficient: a reported coefficient is one parameter extracted under a concentration, solvent, and fitting convention. Feakins and colleagues give a competing interpretation of \(B\) trends.^[4] Thus the coefficient is neither the whole equation nor independent proof of one water-structure cause.

References¶

[1] Takehiro Nakai, Seiji Sawamura, Yoshihiro Taniguchi, and Takashi Kuboyama, “Effects of Pressure and Temperature on the Viscosity B Coefficient for Potassium Chloride in Water,” Materials Science Research International 3 (1997): 204–209, publisher PDF, PDF pp. 1–4, especially Eq. 1, §3.3 Eqs. 7–8 and Fig. 3. Full six-page original consulted; the text extraction damages Greek glyphs, so equation meaning was checked against the page context. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p

[2] Robert Dordick, Lawrence Korson, and W. Drost-Hansen, “High-precision viscosity measurements: I. Aqueous solutions of alkali chlorides,” Journal of Colloid and Interface Science 72 (1979): 206–214, DOI: 10.1016/0021-9797(79)90102-4. Publisher-indexed abstract consulted for tested salts, coefficients, and qualified deviations; direct page access was blocked and full article was not inspected. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h

[3] Joseph Crudden, Gerard M. Delaney, David Feakins, Philip J. O'Reilly, W. Earle Waghorne, and Kenneth G. Lawrence, “The viscosity and structure of solutions. Part 2.—Measurement of the B coefficient of viscosity for alkali-metal chlorides in propan-1-ol–water mixtures at 25 and 35 °C,” Journal of the Chemical Society, Faraday Transactions 1 82 (1986): 2195–2206, DOI: 10.1039/F19868202195. Publisher abstract consulted for formula, systems, and reported fit limits; full article not inspected. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p

[4] David Feakins, W. Earle Waghorne, and Kenneth G. Lawrence, “The viscosity and structure of solutions. Part 1.—A new theory of the Jones–Dole B-coefficient and the related activation parameters: application to aqueous solutions,” Journal of the Chemical Society, Faraday Transactions 1 82 (1986): 563–568, DOI: 10.1039/F19868200563. Publisher abstract consulted for the alternative account of \(B\) trends; full article not inspected. registry ↩a ↩b ↩c ↩d ↩e ↩f

[5] Kuldeep Kumar, Baljeet Singh Patial, and Suvarcha Chauhan, “Interactions of Saccharides in Aqueous Glycine and Leucine Solutions at Different Temperatures of (293.15 to 313.15) K: A Viscometric Study,” Journal of Chemical & Engineering Data 60 (2015): 47–56, DOI: 10.1021/je500647a. Publisher-indexed abstract consulted for saccharide \(B\)-coefficient use; direct page access was blocked and full \(A\)-term treatment and fit window were not inspected. registry ↩a ↩b