Molar conductivity

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The molar conductivity of an electrolyte solution is defined as its conductivity divided by its molar concentration. [1] [2]

Contents

where:

κ is the measured conductivity (formerly known as specific conductance), [3]
c is the molar concentration of the electrolyte.

The SI unit of molar conductivity is siemens metres squared per mole (S m2 mol−1). [2] However, values are often quoted in S cm2 mol−1. [4] In these last units, the value of Λm may be understood as the conductance of a volume of solution between parallel plate electrodes one centimeter apart and of sufficient area so that the solution contains exactly one mole of electrolyte. [5]

Variation of molar conductivity with dilution

There are two types of electrolytes: strong and weak. Strong electrolytes usually undergo complete ionization, and therefore they have higher conductivity than weak electrolytes, which undergo only partial ionization. For strong electrolytes, such as salts, strong acids and strong bases, the molar conductivity depends only weakly on concentration. On dilution there is a regular increase in the molar conductivity of strong electrolyte, due to the decrease in solute–solute interaction. Based on experimental data Friedrich Kohlrausch (around the year 1900) proposed the non-linear law for strong electrolytes:

where

Λ
m
is the molar conductivity at infinite dilution (or limiting molar conductivity), which can be determined by extrapolation of Λm as a function of c,
K is the Kohlrausch coefficient, which depends mainly on the stoichiometry of the specific salt in solution,
α is the dissociation degree even for strong concentrated electrolytes,
fλ is the lambda factor for concentrated solutions.

This law is valid for low electrolyte concentrations only; it fits into the Debye–Hückel–Onsager equation. [6]

For weak electrolytes (i.e. incompletely dissociated electrolytes), however, the molar conductivity strongly depends on concentration: The more dilute a solution, the greater its molar conductivity, due to increased ionic dissociation. For example, acetic acid has a higher molar conductivity in dilute aqueous acetic acid than in concentrated acetic acid.

Kohlrausch's law of independent migration of ions

Friedrich Kohlrausch in 1875–1879 established that to a high accuracy in dilute solutions, molar conductivity can be decomposed into contributions of the individual ions. This is known as Kohlrausch's law of independent ionic migration. [7] For any electrolyte AxBy, the limiting molar conductivity is expressed as x times the limiting molar conductivity of Ay+ and y times the limiting molar conductivity of Bx.

where:

λi is the limiting molar ionic conductivity of ion i,
νi is the number of ions i in the formula unit of the electrolyte (e.g. 2 and 1 for Na+ and SO2−
4
in Na2SO4).

Kohlrausch's evidence for this law was that the limiting molar conductivities of two electrolytes with two different cations and a common anion differ by an amount which is independent of the nature of the anion. For example, Λ0(KX) − Λ0(NaX) = 23.4 S cm2 mol−1 for X = Cl, I and 1/2 SO2−
4
. This difference is ascribed to a difference in ionic conductivities between K+ and Na+. Similar regularities are found for two electrolytes with a common anion and two cations. [8]

Molar ionic conductivity

The molar ionic conductivity of each ionic species is proportional to its electrical mobility (μ), or drift velocity per unit electric field, according to the equation

where z is the ionic charge, and F is the Faraday constant. [9]

The limiting molar conductivity of a weak electrolyte cannot be determined reliably by extrapolation. Instead it can be expressed as a sum of ionic contributions, which can be evaluated from the limiting molar conductivities of strong electrolytes containing the same ions. For aqueous acetic acid as an example, [4]

Values for each ion may be determined using measured ion transport numbers. For the cation:

and for the anion:

Most monovalent ions in water have limiting molar ionic conductivities in the range of 40–80 S cm2 mol−1. For example: [4]

Cationλ, S cm2 mol−1
Li+ 38.6
Na+ 50.1
K+ 73.5
Ag+ 61.9
Anionλ, S cm2 mol−1
F 55.4
Cl 76.4
Br 78.1
CH3COO 40.9

The order of the values for alkali metals is surprising, since it shows that the smallest cation Li+ moves more slowly in a given electric field than Na+, which in turn moves more slowly than K+. This occurs because of the effect of solvation of water molecules: the smaller Li+ binds most strongly to about four water molecules so that the moving cation species is effectively Li(H
2
O)+
4
. The solvation is weaker for Na+ and still weaker for K+. [4] The increase in halogen ion mobility from F to Cl to Br is also due to decreasing solvation.

Exceptionally high values are found for H+ (349.8 S cm2 mol−1) and OH (198.6 S cm2 mol−1), which are explained by the Grotthuss proton-hopping mechanism for the movement of these ions. [4] The H+ also has a larger conductivity than other ions in alcohols, which have a hydroxyl group, but behaves more normally in other solvents, including liquid ammonia and nitrobenzene. [4]

For multivalent ions, it is usual to consider the conductivity divided by the equivalent ion concentration in terms of equivalents per litre, where 1 equivalent is the quantity of ions that have the same amount of electric charge as 1 mol of a monovalent ion: 1/2 mol Ca2+, 1/2 mol SO2−
4
, 1/3 mol Al3+, 1/4 mol Fe(CN)4−
6
, etc. This quotient can be called the equivalent conductivity, although IUPAC has recommended that use of this term be discontinued and the term molar conductivity be used for the values of conductivity divided by equivalent concentration. [10] If this convention is used, then the values are in the same range as monovalent ions, e.g. 59.5 S cm2 mol−1 for 1/2 Ca2+ and 80.0 S cm2 mol−1 for 1/2 SO2−
4
. [4]

From the ionic molar conductivities of cations and anions, effective ionic radii can be calculated using the concept of Stokes radius. The values obtained for an ionic radius in solution calculated this way can be quite different from the ionic radius for the same ion in crystals, due to the effect of hydration in solution.

Applications

Ostwald's law of dilution, which gives the dissociation constant of a weak electrolyte as a function of concentration, can be written in terms of molar conductivity. Thus, the pKa values of acids can be calculated by measuring the molar conductivity and extrapolating to zero concentration. Namely, pKa = p(K/1 mol/L) at the zero-concentration limit, where K is the dissociation constant from Ostwald's law.

Related Research Articles

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

In chemistry, a salt is a chemical compound consisting of an ionic assembly of positively charged cations and negatively charged anions, which results in a compound with no net electric charge. A common example is table salt, with positively charged sodium ions and negatively charged chloride ions.

In chemistry, an acid dissociation constant is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for a chemical reaction

<span class="mw-page-title-main">Aqueous solution</span> Solution in which the solvent is water

An aqueous solution is a solution in which the solvent is water. It is mostly shown in chemical equations by appending (aq) to the relevant chemical formula. For example, a solution of table salt, or sodium chloride (NaCl), in water would be represented as Na+(aq) + Cl(aq). The word aqueous means pertaining to, related to, similar to, or dissolved in, water. As water is an excellent solvent and is also naturally abundant, it is a ubiquitous solvent in chemistry. Since water is frequently used as the solvent in experiments, the word solution refers to an aqueous solution, unless the solvent is specified.

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.

The self-ionization of water (also autoionization of water, and autodissociation of water) is an ionization reaction in pure water or in an aqueous solution, in which a water molecule, H2O, deprotonates (loses the nucleus of one of its hydrogen atoms) to become a hydroxide ion, OH. The hydrogen nucleus, H+, immediately protonates another water molecule to form a hydronium cation, H3O+. It is an example of autoprotolysis, and exemplifies the amphoteric nature of water.

The common-ion effect refers to the decrease in solubility of an ionic precipitate by the addition to the solution of a soluble compound with an ion in common with the precipitate. This behaviour is a consequence of Le Chatelier's principle for the equilibrium reaction of the ionic association/dissociation. The effect is commonly seen as an effect on the solubility of salts and other weak electrolytes. Adding an additional amount of one of the ions of the salt generally leads to increased precipitation of the salt, which reduces the concentration of both ions of the salt until the solubility equilibrium is reached. The effect is based on the fact that both the original salt and the other added chemical have one ion in common with each other.

In chemistry and biochemistry, the Henderson–Hasselbalch equation

The van 't Hoff factor i is a measure of the effect of a solute on colligative properties such as osmotic pressure, relative lowering in vapor pressure, boiling-point elevation and freezing-point depression. The van 't Hoff factor is the ratio between the actual concentration of particles produced when the substance is dissolved and the concentration of a substance as calculated from its mass. For most non-electrolytes dissolved in water, the van 't Hoff factor is essentially 1. For most ionic compounds dissolved in water, the van 't Hoff factor is equal to the number of discrete ions in a formula unit of the substance. This is true for ideal solutions only, as occasionally ion pairing occurs in solution. At a given instant a small percentage of the ions are paired and count as a single particle. Ion pairing occurs to some extent in all electrolyte solutions. This causes the measured van 't Hoff factor to be less than that predicted in an ideal solution. The deviation for the van 't Hoff factor tends to be greatest where the ions have multiple charges.

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.

<span class="mw-page-title-main">Dissociation (chemistry)</span> Separation of molecules or ionic compounds into smaller constituent entities

Dissociation in chemistry is a general process in which molecules (or ionic compounds such as salts, or complexes) separate or split into other things such as atoms, ions, or radicals, usually in a reversible manner. For instance, when an acid dissolves in water, a covalent bond between an electronegative atom and a hydrogen atom is broken by heterolytic fission, which gives a proton (H+) and a negative ion. Dissociation is the opposite of association or recombination.

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.

In chemistry, ion association is a chemical reaction whereby ions of opposite electric charge come together in solution to form a distinct chemical entity. Ion associates are classified, according to the number of ions that associate with each other, as ion pairs, ion triplets, etc. Ion pairs are also classified according to the nature of the interaction as contact, solvent-shared or solvent-separated. The most important factor to determine the extent of ion association is the dielectric constant of the solvent. Ion associates have been characterized by means of vibrational spectroscopy, as introduced by Niels Bjerrum, and dielectric-loss spectroscopy.

<span class="mw-page-title-main">Conductivity (electrolytic)</span> Measure of the ability of a solution containing electrolytes to conduct electricity

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

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.

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.

Conductometry is a measurement of electrolytic conductivity to monitor a progress of chemical reaction. Conductometry has notable application in analytical chemistry, where conductometric titration is a standard technique. In usual analytical chemistry practice, the term conductometry is used as a synonym of conductometric titration while the term conductimetry is used to describe non-titrative applications. Conductometry is often applied to determine the total conductance of a solution or to analyze the end point of titrations that include ions.

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:

Acid strength is the tendency of an acid, symbolised by the chemical formula , to dissociate into a proton, , and an anion, . The dissociation of a strong acid in solution is effectively complete, except in its most concentrated solutions.

References

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