Conductivity (electrolytic)

Last updated April 17, 2024

Conductivity or specific conductance of an electrolyte solution is a measure of its ability to conduct electricity. The SI unit of conductivity is siemens per meter (S/m).

Conductivity measurements are used routinely in many industrial and environmental applications as a fast, inexpensive and reliable way of measuring the ionic content in a solution.^[1] For example, the measurement of product conductivity is a typical way to monitor and continuously trend the performance of water purification systems.

In many cases, conductivity is linked directly to the total dissolved solids (TDS).

High quality deionized water has a conductivity of
$\mathrm {\kappa \;=\;0.05501\,\pm \,0.0001\,{\frac {\mu S}{cm}}}$ at 25 °C.
This corresponds to a specific resistivity of
$\rho \,=\,18.18\pm 0.03\;\mathrm {M\Omega \cdot cm}$ .^[2]
The preparation of salt solutions often takes place in unsealed beakers. In this case the conductivity of purified water often is 10 to 20 times higher. A discussion can be found below.

Typical drinking water is in the range of 200–800 μS/cm, while sea water is about 50 mS/cm^[3] (or 0.05 S/cm).

Conductivity is traditionally determined by connecting the electrolyte in a Wheatstone bridge. Dilute solutions follow Kohlrausch's law of concentration dependence and additivity of ionic contributions. Lars Onsager gave a theoretical explanation of Kohlrausch's law by extending Debye–Hückel theory.

Units

The SI unit of conductivity is S/m and, unless otherwise qualified, it refers to 25 °C. More generally encountered is the traditional unit of μS/cm.

The commonly used standard cell has a width of 1 cm, and thus for very pure water in equilibrium with air would have a resistance of about 10⁶ ohms, known as a megohm. Ultra-pure water could achieve 18 megohms or more. Thus in the past, megohm-cm was used, sometimes abbreviated to "megohm". Sometimes, conductivity is given in "microsiemens" (omitting the distance term in the unit). While this is an error, it can often be assumed to be equal to the traditional μS/cm. Often, by typographic limitations μS/cm is expressed as uS/cm.

The conversion of conductivity to the total dissolved solids depends on the chemical composition of the sample and can vary between 0.54 and 0.96. Typically, the conversion is done assuming that the solid is sodium chloride; 1 μS/cm is then equivalent to about 0.64 mg of NaCl per kg of water.

Molar conductivity has the SI unit S m² mol⁻¹. Older publications use the unit Ω⁻¹ cm² mol⁻¹.

Measurement

The electrical conductivity of a solution of an electrolyte is measured by determining the resistance of the solution between two flat or cylindrical electrodes separated by a fixed distance.^[4] An alternating voltage is generally used in order to minimize water electrolysis.^{[ citation needed ]} The resistance is measured by a conductivity meter. Typical frequencies used are in the range 1–3 kHz. The dependence on the frequency is usually small,^[5] but may become appreciable at very high frequencies, an effect known as the Debye–Falkenhagen effect.

A wide variety of instrumentation is commercially available.^[6] Most commonly, two types of electrode sensors are used, electrode-based sensors and inductive sensors. Electrode sensors with a static design are suitable for low and moderate conductivities, and exist in various types, having either two or four electrodes, where electrodes can be arrange oppositely, flat or in a cylinder.^[7] Electrode cells with a flexible design, where the distance between two oppositely arranged electrodes can be varied, offer high accuracy and can also be used for the measurement of highly conductive media.^[8] Inductive sensors are suitable for harsh chemical conditions but require larger sample volumes than electrode sensors.^[9] Conductivity sensors are typically calibrated with KCl solutions of known conductivity. Electrolytic conductivity is highly temperature dependent but many commercial systems offer automatic temperature correction. Tables of reference conductivities are available for many common solutions.^[10]

Definitions

Resistance, $R$ , is proportional to the distance, $l$ , between the electrodes and is inversely proportional to the cross-sectional area of the sample, $A$ (noted $S$ on the Figure above). Writing $ρ$ (rho) for the specific resistance, or resistivity.

R=\rho {\frac {l}{A}}

In practice the conductivity cell is calibrated by using solutions of known specific resistance, $ρ*$ , so the individual quantities $l$ and $A$ need not be known precisely, but only their ratio.^[11] If the resistance of the calibration solution is $R *$ , a cell-constant, defined as the ratio of $l$ and $A$ ( $C$ = l⁄A), is derived.

R^{*}=\rho ^{*}\times C

The specific conductance (conductivity), $κ$ (kappa) is the reciprocal of the specific resistance.

\kappa ={\frac {1}{\rho }}={\frac {C}{R}}

Conductivity is also temperature-dependent. Sometimes the conductance (reciprocical of the resistance) is denoted as $G$ = 1⁄R. Then the specific conductance $κ$ (kappa) is:

\kappa =C\times G

Theory

The specific conductance of a solution containing one electrolyte depends on the concentration of the electrolyte. Therefore, it is convenient to divide the specific conductance by concentration. This quotient, termed molar conductivity, is denoted by $Λ m$

\Lambda _{\mathrm {m} }={\frac {\kappa }{c}}

Strong electrolytes

Strong electrolytes are hypothesized to dissociate completely in solution. The conductivity of a solution of a strong electrolyte at low concentration follows Kohlrausch's Law

\Lambda _{\mathrm {m} }=\Lambda _{\mathrm {m} }^{0}-K{\sqrt {c}}

where $Λ 0 m$ is known as the limiting molar conductivity, $K$ is an empirical constant and $c$ is the electrolyte concentration. (Limiting here means "at the limit of the infinite dilution".) In effect, the observed conductivity of a strong electrolyte becomes directly proportional to concentration, at sufficiently low concentrations i.e. when

\Lambda _{\mathrm {m} }^{0}\gg K{\sqrt {c}}

As the concentration is increased however, the conductivity no longer rises in proportion. Moreover, Kohlrausch also found that the limiting conductivity of an electrolyte;

λ 0 +

and

λ 0 -

are the limiting molar conductivities of the individual ions.

The following table gives values for the limiting molar conductivities for some selected ions.^[12]

Table of limiting ion conductivity in water at 298 K (approx. 25 °C)^[12]
Cations	$λ 0 +$ / mS m² mol⁻¹	Cations	$λ 0 +$ / mS m² mol⁻¹	Anions	$λ 0 -$ / mS m² mol⁻¹	Anions	$λ 0 -$ / mS m² mol⁻¹
H⁺	34.982	Ba²⁺	12.728	OH⁻	19.8	SO²⁻ ₄	15.96
Li⁺	3.869	Mg²⁺	10.612	Cl⁻	7.634	C^₂O²⁻ ₄	7.4
Na⁺	5.011	La³⁺	20.88	Br⁻	7.84	HC^₂O⁻ ₄	4.306^[13]
K⁺	7.352	Rb⁺	7.64	I⁻	7.68	HCOO⁻	5.6
NH⁺ ₄	7.34	Cs⁺	7.68	NO⁻ ₃	7.144	CO²⁻ ₃	7.2
Ag⁺	6.192	Be²⁺	4.50	CH₃COO⁻	4.09	HSO²⁻ ₃	5.0
Ca²⁺	11.90			ClO⁻ ₄	6.80	SO²⁻ ₃	7.2
Co(NH^₃)³⁺ ₆	10.2			F⁻	5.50

An interpretation of these results was based on the theory of Debye and Hückel, yielding the Debye–Hückel–Onsager theory:^[14]

\Lambda _{\mathrm {m} }=\Lambda _{\mathrm {m} }^{0}-\left(A+B\Lambda _{\mathrm {m} }^{0}\right){\sqrt {c}}

where $A$ and $B$ are constants that depend only on known quantities such as temperature, the charges on the ions and the dielectric constant and viscosity of the solvent. As the name suggests, this is an extension of the Debye–Hückel theory, due to Onsager. It is very successful for solutions at low concentration.

Weak electrolytes

A weak electrolyte is one that is never fully dissociated (there are a mixture of ions and complete molecules in equilibrium). In this case there is no limit of dilution below which the relationship between conductivity and concentration becomes linear. Instead, the solution becomes ever more fully dissociated at weaker concentrations, and for low concentrations of "well behaved" weak electrolytes, the degree of dissociation of the weak electrolyte becomes proportional to the inverse square root of the concentration.

Typical weak electrolytes are weak acids and weak bases. The concentration of ions in a solution of a weak electrolyte is less than the concentration of the electrolyte itself. For acids and bases the concentrations can be calculated when the value or values of the acid dissociation constant are known.

For a monoprotic acid, HA, obeying the inverse square root law, with a dissociation constant $K a$ , an explicit expression for the conductivity as a function of concentration, $c$ , known as Ostwald's dilution law, can be obtained.

{\frac {1}{\Lambda _{\mathrm {m} }}}={\frac {1}{\Lambda _{\mathrm {m} }^{0}}}+{\frac {\Lambda _{\mathrm {m} }c}{K_{\mathrm {a} }\left(\Lambda _{\mathrm {m} }^{0}\right)^{2}}}

Various solvents exhibit the same dissociation if the ratio of relative permittivities equals the ratio cubic roots of concentrations of the electrolytes (Walden's rule).

Higher concentrations

Both Kohlrausch's law and the Debye–Hückel–Onsager equation break down as the concentration of the electrolyte increases above a certain value. The reason for this is that as concentration increases the average distance between cation and anion decreases, so that there is more interactions between close ions. Whether this constitutes ion association is a moot point. However, it has often been assumed that cation and anion interact to form an ion pair. So, an "ion-association" constant $K$ , can be derived for the association equilibrium between ions A⁺ and B⁻:

A⁺ + B⁻⇌ A⁺B⁻ with

K

= [A⁺B⁻][A⁺] [B⁻]

Davies describes the results of such calculations in great detail, but states that $K$ should not necessarily be thought of as a true equilibrium constant, rather, the inclusion of an "ion-association" term is useful in extending the range of good agreement between theory and experimental conductivity data.^[15] Various attempts have been made to extend Onsager's treatment to more concentrated solutions.^[16]

The existence of a so-called conductance minimum in solvents having the relative permittivity under 60 has proved to be a controversial subject as regards interpretation. Fuoss and Kraus suggested that it is caused by the formation of ion triplets,^[17] and this suggestion has received some support recently.^[18]^[19]

Other developments on this topic have been done by Theodore Shedlovsky,^[20] E. Pitts,^[21] R. M. Fuoss,^[22]^[23] Fuoss and Shedlovsky,^[24] Fuoss and Onsager.^[25]^[26]

Mixed solvents systems

The limiting equivalent conductivity of solutions based on mixed solvents like water alcohol has minima depending on the nature of alcohol. For methanol the minimum is at 15 molar % water,^[20]^[27]^[28] and for the ethanol at 6 molar % water.^[29]

Conductivity versus temperature

Generally the conductivity of a solution increases with temperature, as the mobility of the ions increases. For comparison purposes reference values are reported at an agreed temperature, usually 298 K (≈ 25 °C or 77 °F), although occasionally 20 °C (68 °F) is used. So called 'compensated' measurements are made at a convenient temperature but the value reported is a calculated value of the expected value of conductivity of the solution, as if it had been measured at the reference temperature. Basic compensation is normally done by assuming a linear increase of conductivity versus temperature of typically 2% per kelvin.^[30]^[31] This value is broadly applicable for most salts at room temperature. Determination of the precise temperature coefficient for a specific solution is simple and instruments are typically capable of applying the derived coefficient (i.e. other than 2%).

Measurements of conductivity $\sigma$ versus temperature can be used to determine the activation energy $E_{A}$ , using the Arrhenius equation:^[32]

\sigma =\sigma _{0}e^{-E_{a}/RT}

where $\sigma _{0}$ is the exponential prefactor, $R$ the gas constant, and $T$ the absolute temperature in Kelvin.

Solvent isotopic effect

The change in conductivity due to the isotope effect for deuterated electrolytes is sizable.^[33]

Applications

Despite the difficulty of theoretical interpretation, measured conductivity is a good indicator of the presence or absence of conductive ions in solution, and measurements are used extensively in many industries.^[34] For example, conductivity measurements are used to monitor quality in public water supplies, in hospitals, in boiler water and industries that depend on water quality such as brewing. This type of measurement is not ion-specific; it can sometimes be used to determine the amount of total dissolved solids (TDS) if the composition of the solution and its conductivity behavior are known.^[1] Conductivity measurements made to determine water purity will not respond to non conductive contaminants (many organic compounds fall into this category), therefore additional purity tests may be required depending on application.

Applications of TDS measurements are not limited to industrial use; many people use TDS as an indicator of the purity of their drinking water. Additionally, aquarium enthusiasts are concerned with TDS, both for freshwater and salt water aquariums. Many fish and invertebrates require quite narrow parameters for dissolved solids. Especially for successful breeding of some invertebrates normally kept in freshwater aquariums—snails and shrimp primarily—brackish water with higher TDS, specifically higher salinity, water is required. While the adults of a given species may thrive in freshwater, this is not always true for the young and some species will not breed at all in non-brackish water.

Sometimes, conductivity measurements are linked with other methods to increase the sensitivity of detection of specific types of ions. For example, in the boiler water technology, the boiler blowdown is continuously monitored for "cation conductivity", which is the conductivity of the water after it has been passed through a cation exchange resin. This is a sensitive method of monitoring anion impurities in the boiler water in the presence of excess cations (those of the alkalizing agent usually used for water treatment). The sensitivity of this method relies on the high mobility of H⁺ in comparison with the mobility of other cations or anions. Beyond cation conductivity, there are analytical instruments designed to measure Degas conductivity, where conductivity is measured after dissolved carbon dioxide has been removed from the sample, either through reboiling or dynamic degassing.

Conductivity detectors are commonly used with ion chromatography.^[35]

Conductivity of purified water in electrochemical experiments

The electronic conductivity of purified distilled water in electrochemical laboratory settings at room temperature is often between 0.05 and 1 μS/cm. Environmental influences during the peparation of salt solutions as gas absorption due to storing the water in an unsealed beaker may immediately increase the conductivity from $\mathrm {0.055\;\mu S/cm}$ and lead to values between 0.5 and 1 $\mathrm {\mu S/cm}$ .
When distilled water is heated during the preparation of salt solutions, the conductivity increases even without adding salt. This is often not taken into account.

In a typical experiment under the fume hood in an unsealed beaker the conductivity of purified water increases typically non linearly from values below 1 μS/cm to values close 3.5 μS/cm at $\mathrm {95^{0}C}$ . This temperature dependence has to be taken into account particularly in dilute salt solutions.

Related Research Articles

In a chemical reaction, chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the system. This state results when the forward reaction proceeds at the same rate as the reverse reaction. The reaction rates of the forward and backward reactions are generally not zero, but they are equal. Thus, there are no net changes in the concentrations of the reactants and products. Such a state is known as dynamic equilibrium.

Wilhelm Ostwald’s dilution law is a relationship proposed in 1888 between the dissociation constant $K d$ and the degree of dissociation $α$ of a weak electrolyte. The law takes the form

In chemistry, pH, also referred to as acidity or basicity, historically denotes "potential of hydrogen". It is a logarithmic scale used to specify the acidity or basicity of aqueous solutions. Acidic solutions are measured to have lower pH values than basic or alkaline solutions.

In chemical thermodynamics, activity is a measure of the "effective concentration" of a species in a mixture, in the sense that the species' chemical potential depends on the activity of a real solution in the same way that it would depend on concentration for an ideal solution. The term "activity" in this sense was coined by the American chemist Gilbert N. Lewis in 1907.

In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.

In plasmas and electrolytes, the Debye length, is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. With each Debye length the charges are increasingly electrically screened and the electric potential decreases in magnitude by 1/e. A Debye sphere is a volume whose radius is the Debye length. Debye length is an important parameter in plasma physics, electrolytes, and colloids. The corresponding Debye screening wave vector $for particles of density, charge at a temperature is given by in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures are known as the Thomas-Fermi length and the Thomas-Fermi wave vector. They are of interest in describing the behaviour of electrons in metals at room temperature.$

In thermodynamics, an activity coefficient is a factor used to account for deviation of a mixture of chemical substances from ideal behaviour. In an ideal mixture, the microscopic interactions between each pair of chemical species are the same and, as a result, properties of the mixtures can be expressed directly in terms of simple concentrations or partial pressures of the substances present e.g. Raoult's law. Deviations from ideality are accommodated by modifying the concentration by an activity coefficient. Analogously, expressions involving gases can be adjusted for non-ideality by scaling partial pressures by a fugacity coefficient.

A silver chloride electrode is a type of reference electrode, commonly used in electrochemical measurements. For environmental reasons it has widely replaced the saturated calomel electrode. For example, it is usually the internal reference electrode in pH meters and it is often used as reference in reduction potential measurements. As an example of the latter, the silver chloride electrode is the most commonly used reference electrode for testing cathodic protection corrosion control systems in sea water environments.

The chemists Peter Debye and Erich Hückel noticed that solutions that contain ionic solutes do not behave ideally even at very low concentrations. So, while the concentration of the solutes is fundamental to the calculation of the dynamics of a solution, they theorized that an extra factor that they termed gamma is necessary to the calculation of the activities of the solution. Hence they developed the Debye–Hückel equation and Debye–Hückel limiting law. The activity is only proportional to the concentration and is altered by a factor known as the activity coefficient $. This factor takes into account the interaction energy of ions in solution.$

A surface charge is an electric charge present on a two-dimensional surface. These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m⁻²), is used to describe the charge distribution on the surface. The electric potential is continuous across a surface charge and the electric field is discontinuous, but not infinite; this is unless the surface charge consists of a dipole layer. In comparison, the potential and electric field both diverge at any point charge or linear charge.

The ionic strength of a solution is a measure of the concentration of ions in that solution. Ionic compounds, when dissolved in water, dissociate into ions. The total electrolyte concentration in solution will affect important properties such as the dissociation constant or the solubility of different salts. One of the main characteristics of a solution with dissolved ions is the ionic strength. Ionic strength can be molar or molal and to avoid confusion the units should be stated explicitly. The concept of ionic strength was first introduced by Lewis and Randall in 1921 while describing the activity coefficients of strong electrolytes.

The Debye–Hückel theory was proposed by Peter Debye and Erich Hückel as a theoretical explanation for departures from ideality in solutions of electrolytes and plasmas. It is a linearized Poisson–Boltzmann model, which assumes an extremely simplified model of electrolyte solution but nevertheless gave accurate predictions of mean activity coefficients for ions in dilute solution. The Debye–Hückel equation provides a starting point for modern treatments of non-ideality of electrolyte solutions.

The molar conductivity of an electrolyte solution is defined as its conductivity divided by its molar concentration.

An osmotic coefficient $is a quantity which characterises the deviation of a solvent from ideal behaviour, referenced to Raoult's law. It can be also applied to solutes. Its definition depends on the ways of expressing chemical composition of mixtures.$

In theoretical chemistry, Specific ion Interaction Theory is a theory used to estimate single-ion activity coefficients in electrolyte solutions at relatively high concentrations. It does so by taking into consideration interaction coefficients between the various ions present in solution. Interaction coefficients are determined from equilibrium constant values obtained with solutions at various ionic strengths. The determination of SIT interaction coefficients also yields the value of the equilibrium constant at infinite dilution.

The Bromley equation was developed in 1973 by Leroy A. Bromley with the objective of calculating activity coefficients for aqueous electrolyte solutions whose concentrations are above the range of validity of the Debye–Hückel equation. This equation, together with Specific ion interaction theory (SIT) and Pitzer equations is important for the understanding of the behaviour of ions dissolved in natural waters such as rivers, lakes and sea-water.

Nanofluidic circuitry is a nanotechnology aiming for control of fluids in nanometer scale. Due to the effect of an electrical double layer within the fluid channel, the behavior of nanofluid is observed to be significantly different compared with its microfluidic counterparts. Its typical characteristic dimensions fall within the range of 1–100 nm. At least one dimension of the structure is in nanoscopic scale. Phenomena of fluids in nano-scale structure are discovered to be of different properties in electrochemistry and fluid dynamics.

Pitzer equations are important for the understanding of the behaviour of ions dissolved in natural waters such as rivers, lakes and sea-water. They were first described by physical chemist Kenneth Pitzer. The parameters of the Pitzer equations are linear combinations of parameters, of a virial expansion of the excess Gibbs free energy, which characterise interactions amongst ions and solvent. The derivation is thermodynamically rigorous at a given level of expansion. The parameters may be derived from various experimental data such as the osmotic coefficient, mixed ion activity coefficients, and salt solubility. They can be used to calculate mixed ion activity coefficients and water activities in solutions of high ionic strength for which the Debye–Hückel theory is no longer adequate. They are more rigorous than the equations of specific ion interaction theory, but Pitzer parameters are more difficult to determine experimentally than SIT parameters.

In chemistry, ion transport number, also called the transference number, is the fraction of the total electric current carried in an electrolyte by a given ionic species $i$ :

The Jones–Dole equation, or Jones–Dole expression, is an empirical expression that describes the relationship between the viscosity of a solution and the concentration of solute within the solution. The Jones–Dole equation is written as

References

1 2 Gray, James R. (2004). "Conductivity Analyzers and Their Application". In Down, R. D.; Lehr, J. H. (eds.). Environmental Instrumentation and Analysis Handbook. Wiley. pp. 491–510. ISBN 978-0-471-46354-2 . Retrieved 10 May 2009.
↑ Light, Truman; Licht, Stuart; Bevilaqua, Anthony; Morash, Kenneth (2004). "The Fundamental Conductivity and Resistivity of Water". Electrochemical and Solid-State Letters. 8 (1): E16–E19.
↑ "Water Conductivity". Lenntech. Retrieved 5 January 2013.
↑ Bockris, J. O'M.; Reddy, A.K.N; Gamboa-Aldeco, M. (1998). Modern Electrochemistry (2nd. ed.). Springer. ISBN 0-306-45555-2 . Retrieved 10 May 2009.
↑ Marija Bešter-Rogač and Dušan Habe, "Modern Advances in Electrical Conductivity Measurements of Solutions", Acta Chim. Slov. 2006, 53, 391–395 (pdf)
↑ Boyes, W. (2002). Instrumentation Reference Book (3rd. ed.). Butterworth-Heinemann. ISBN 0-7506-7123-8 . Retrieved 10 May 2009.
↑ Gray, p 495
↑ Doppelhammer, Nikolaus; Pellens, Nick; Martens, Johan; Kirschhock, Christine E. A.; Jakoby, Bernhard; Reichel, Erwin K. (27 October 2020). "Moving Electrode Impedance Spectroscopy for Accurate Conductivity Measurements of Corrosive Ionic Media". ACS Sensors. 5 (11): 3392–3397. doi: 10.1021/acssensors.0c01465 . PMC 7706010 . PMID 33107724.
↑ Ghosh, Arun K. (2013). Introduction to measurements and instrumentation (4th ed., Eastern economy ed.). Delhi: PH Learning. ISBN 978-81-203-4625-3. OCLC 900392417.
↑ "Conductivity ordering guide" (PDF). EXW Foxboro. 3 October 1999. Archived from the original (PDF) on 7 September 2012. Retrieved 5 January 2013.
↑ "ASTM D1125 - 95(2005) Standard Test Methods for Electrical Conductivity and Resistivity of Water" . Retrieved 12 May 2009.
1 2 Adamson, Arthur W. (1973). Textbook of Physical Chemistry. London: Academic Press inc. p. 512.
↑ Bešter-Rogač, M.; Tomšič, M.; Barthel, J.; Neueder, R.; Apelblat, A. (1 January 2002). "Conductivity Studies of Dilute Aqueous Solutions of Oxalic Acid and Neutral Oxalates of Sodium, Potassium, Cesium, and Ammonium from 5 to 35 °C". Journal of Solution Chemistry. 31 (1): 1–18. doi:10.1023/A:1014805417286. ISSN 1572-8927. S2CID 92641871.
↑ Wright, M.R. (2007). An Introduction to Aqueous Electrolyte Solutions. Wiley. ISBN 978-0-470-84293-5.
↑ Davies, C. W. (1962). Ion Association. London: Butterworths.
↑ Miyoshi, K. (1973). "Comparison of the Conductance Equations of Fuoss–Onsager, Fuoss–Hsia and Pitts with the Data of Bis(2,9-dimethyl-1,10-phenanthroline)Cu(I) Perchlorate". Bull. Chem. Soc. Jpn. 46 (2): 426–430. doi:10.1246/bcsj.46.426.
↑ Fuoss, R. M.; Kraus, C. A. (1935). "Properties of Electrolytic Solutions. XV. Thermodynamic Properties of Very Weak Electrolytes". J. Am. Chem. Soc. 57: 1–4. doi:10.1021/ja01304a001.
↑ Weingärtner, H.; Weiss, V. C.; Schröer, W. (2000). "Ion association and electrical conductance minimum in Debye–Hückel-based theories of the hard sphere ionic fluid". J. Chem. Phys. 113 (2): 762–. Bibcode:2000JChPh.113..762W. doi:10.1063/1.481822.
↑ Schröer, W.; Weingärtner, H. (2004). "Structure and criticality of ionic fluids" (PDF). Pure Appl. Chem. 76 (1): 19–27. doi:10.1351/pac200476010019. S2CID 39716065.
1 2 Shedlovsky, Theodore (1932). "The Electrolytic Conductivity of some Uni-Univalent Electrolytes in Water at 25°". Journal of the American Chemical Society . 54 (4). American Chemical Society (ACS): 1411–1428. doi:10.1021/ja01343a020. ISSN 0002-7863.
↑ Pitts, E.; Coulson, Charles Alfred (1953). "An extension of the theory of the conductivity and viscosity of electrolyte solutions". Proc. R. Soc. A217 (1128): 43. Bibcode:1953RSPSA.217...43P. doi:10.1098/rspa.1953.0045. S2CID 123363978.
↑ Fuoss, Raymond M. (1958). "Conductance of Ionophores". Journal of the American Chemical Society. 80 (12). American Chemical Society (ACS): 3163. doi:10.1021/ja01545a064. ISSN 0002-7863.
↑ Fuoss, Raymond M. (1959). "Conductance of Dilute Solutions of 1-1 Electrolytes1". Journal of the American Chemical Society. 81 (11). American Chemical Society (ACS): 2659–2662. doi:10.1021/ja01520a016. ISSN 0002-7863.
↑ Fuoss, Raymond M.; Shedlovsky, Theodore. (1949). "Extrapolation of Conductance Data for Weak Electrolytes". Journal of the American Chemical Society. 71 (4). American Chemical Society (ACS): 1496–1498. doi:10.1021/ja01172a507. ISSN 0002-7863.
↑ Fuoss, Raymond M.; Onsager, Lars (1964). "The Conductance of Symmetrical Electrolytes.1aIV. Hydrodynamic and Osmotic Terms in the Relaxation Field". The Journal of Physical Chemistry. 68 (1). American Chemical Society (ACS): 1–8. doi:10.1021/j100783a001. ISSN 0022-3654.
↑ Fuoss, Raymond M.; Onsager, Lars; Skinner, James F. (1965). "The Conductance of Symmetrical Electrolytes. V. The Conductance Equation1,2". The Journal of Physical Chemistry. 69 (8). American Chemical Society (ACS): 2581–2594. doi:10.1021/j100892a017. ISSN 0022-3654.
↑ Shedlovsky, Theodore; Kay, Robert L. (1956). "The Ionization Constant of Acetic Acid in Water–Methanol Mixtures at 25° from Conductance Measurements". The Journal of Physical Chemistry. 60 (2). American Chemical Society (ACS): 151–155. doi:10.1021/j150536a003. ISSN 0022-3654.
↑ Strehlow, H. (1960). "Der Einfluß von Wasser auf die Äquivalentleitfähigkeit von HCl in Methanol". Zeitschrift für Physikalische Chemie . 24 (3–4). Walter de Gruyter GmbH: 240–248. doi:10.1524/zpch.1960.24.3_4.240. ISSN 0942-9352.
↑ Bezman, Irving I.; Verhoek, Frank H. (1945). "The Conductance of Hydrogen Chloride and Ammonium Chloride in Ethanol-Water Mixtures". Journal of the American Chemical Society. 67 (8). American Chemical Society (ACS): 1330–1334. doi:10.1021/ja01224a035. ISSN 0002-7863.
↑ "NIST Special Publication 260-142 Primary Standards and Standard Reference Materials for Electrolytic Conductivity" (PDF). U.S. DEPARTMENT OF COMMERCE, Technology Administration, National Institute of Standards and Technology. p. 5.
↑ R. A. Robinson and R. H. Stokes, Electrolyte Solutions, 3 ld ed., Butterworths, London (1959).
↑ Petrowsky, Matt; Frech, Roger (30 April 2009). "Temperature Dependence of Ion Transport: The Compensated Arrhenius Equation". The Journal of Physical Chemistry B. 113 (17): 5996–6000. doi:10.1021/jp810095g. ISSN 1520-6106. PMID 19338318.
↑ Biswas, Ranjit (1997). "Limiting Ionic Conductance of Symmetrical, Rigid Ions in Aqueous Solutions: Temperature Dependence and Solvent Isotope Effects". Journal of the American Chemical Society. 119 (25): 5946–5953. doi:10.1021/ja970118o.
↑ "Electrolytic conductivity measurement, Theory and practice" (PDF). Aquarius Technologies Pty Ltd. Archived from the original (PDF) on 12 September 2009.
↑ "Detectors for ion-exchange chromatography". Archived from the original on 20 August 2009. Retrieved 17 May 2009.