If a wire is placed between the two terminals of a battery, current will flow. The amount of current which will flow depends on the resistance of the wire according to Ohm’s law, V = I R, where V is the voltage, I the current, and R the resistance of the wire. This resistance will depend on the composition of the wire and on its geometry. For example, two copper wires of the same type and geometry will have the same resistance. However if a copper wire with double the cross sectional area and the same length is used, the resistance of that wire will decrease by one half. If the length, L, of a wire is doubled while its cross sectional area, A, remains the same, the resistance of that wire will double. It is possible to characterize a material, such as copper, with a measure of resistance that is an intrinsic property of the material called its resistivity, r. This resistivity can depend on other properties such as its purity and crystalline state, but is an intrinsic property of the material, as is the melting temperature. However, resistance to the passage of current depends on the geometry of the object formed from a material. The relationship between the resistance of a material and the geometric independent resistivity is:
r = R A/L
It is often convenient to discuss the conductivity, k, instead of the resistivity. Conductivity is the inverse of the resistivity.
The SI unit of resistance is the ohm (W ). The SI unit of conductance is the siemen (S) or W -1, though many people still refer to this unit as simply ohm written backwards, or mho. Conductivity is then measured in siemens m-1 (or Scm-1 is more convenient)
If electrodes are immersed in a solution of say sodium chloride in water, the conductivity can be measured using a conductivity meter. This conductivity will depend on the type and concentration of ions present, the solvent, the area of the electrode, and the distance between the electrodes. If the solvent is pure water, there are few charged species (ions) present, Kw =[OH-][H+]=10-14, the conductivity measured will be low. The addition of ions such as in a 0.10 N solution of NaCl (L = 106.8 W -1 cm2 mol-1) will increase the conductivity of the solution to106.8 W-1cm2mol-1x 0.01mol/1000 cm3 = 10.68 x 10-6W-1 cm-1 or 1068 x 10-6W-1 m-1.
The measurement of the conductivity of an electrolytic solution begins with the measurement of the resistance for that solution using a conductivity cell. The resistance measurements must be accomplished using alternating current (AC) to avoid changes in concentrations and product build up on the electrodes which in turn would effect the measurement. Usually, Platinum electrodes, coated with colloidal Pt to absorb any gas products, are used to in order to reverse changes occurring during phases with reverse polarity. After the resistance is determined, the resistivity is determined using equation (1). Since the geometry of the electrodes can change during their use through physically deforming, the ratio L/A could change, therefore it is necessary to determine the value of L/A by using a solution with known conductivity to calibrate the conductivity cell by determining a cell constant.
Specific conductance s is the reciprocal of the specific resistance. If a cell could be constructed with electrodes of exactly 1 cm2 area exactly 1 cm apart, the reciprocal of the cell resistance in reciprocal ohms would be numerically equal to the specific conductance in reciprocal ohms per centimeter since L/A = 1cm-1. Two sample conductance cells are given in Figure 3. These conductance cells do not satisfy these conditions, and so it is convenient to define a constant factor k determined by the cell geometry and called the cell constant, such that
k = k/R
where R is the measured resistance of the actual cell. The numerical value of k for a particular cell is determined experimentally by use of a standard solution of known specific conductance.
The specific conductance will clearly depend on the concentration of the electrolyte. In general, if the concentration doubles, the conductivity will double. The measurement of conductance is put on an per ion basis by defining the equivalent conductance L:
L = k V
where V is the volume of solution containing 1 g equiv. of solute. In the cgs system, V is in cubic centimeters and L is in square centimeters per ohm per equivalent. For ions with a single charge, the equivalent conductivity is equal to the molar conductivity, Lm, obtained by dividing the conductivity by its molar concentration, m.
Lm = k /m
Before proceeding, let’s stop for a moment and think about the conductivity of ions on a simple qualitative basis. Suppose the compound AB dissolves in water to produce the ions A+ and B-. The conductance of ionic solutions is the result of the movement of ions through the solution to the electrodes. When two electrodes in the solution are made part of a complete electrical circuit, the cations (+) are attracted to the negative pole (cathode) and the anions (-) are attracted to the positive pole (anode). Changes in the equivalent conductance of an electrolyte solution with changes in concentration may result from changes in both the number and the mobility of the ions present. If both anion and cation are monovalent, the overall electrical neutrality of the soluble solution assures that equal numbers of the two ions are present. They will not in generally have the same mobility, however, and thus do not share equally in the conduction of current.
The solution conducts electricity through motion of the ions under the effect of an electric field. At high concentrations, each ion is surrounded by other ions, both positive and negative. The field affecting any particular ion changes slightly because of these surrounding ions. At infinite dilution, the distance between nearest neighbor ions is large, and only the effect of the applied electric field is felt by individual ions. This is the reason for extrapolating the data to infinite dilution.
The conductivity of any particular ion will also be affected by the ease with which the ion can more through the water. Hence different ions should contribute differently to the total measured conductivity. The ease with which any ion moves through the solution depends on considerations such as the total charge and the size of the ion; large ions offer greater resistance to motion through the water than small ions.
Suppose we now consider the compound CB, which dissociates on solution to produce C+ and B, where the ion B- is the same as the B- produced by the compound AB discussed above. One expects the contribution of the anion B- to the total conductivity of the solution to be independent of the nature of the cation at infinite dilution.
Friedrich Kohlrausch(1840-1910) found that the molar conductivity varied as the square root of the concentration for many solutions.
L m = Lmo - k c1/2
This basic relationship is one of Kohlrausch’s law where the limiting molar conductivity at infinite dilution, Lmo , is a constant which depends on the electrolyte. The constant, k, depends more on stoichiometry of the electrolyte than its nature. The molar conductivities of several electrolytes are plotted as a function of the square root of concentration to illustrate that concept.
The molar conductivity of NaCl, NaC2H3O2, and KCl as a function of concentration.
The molar conductivity of NaCl and NaC2H3O2, as a function of the square root of concentration.
The solid line is a linear least squares fit of the data using mentioned equation on the four most dilute solutions. The value of Lo for NaCl was found to be 126.0 W-1 cm2 mol-1, while Lo for NaC2H3O2 was 90.6 ± 0.1 W-1 cm2 mol-1.
These molar conductivities at infinite dilution can be used to determine a conductivity at infinite dilution for individual ions to yield the law of the independent migration of ions. That is, the limiting conductivity of NaCl could be determined by the relationship:
LoNaCl = L oNa+ + L oCl-
When a current flows in an electrolyte solution, charge is carried by the motion of both anions and cations.
Next Figure shows plots of the molar conductances of several substances vs the square root of the concentration. The curves in Figures 6 indicate that this extrapolation can be carried out quite simply for strong electrolytes, whereas the problems engendered by weak electrolytes such as HAc (acetic acid) are severe.
The increase of the equivalent conductance of solutions of strong electrolytes in the low-concentration range is not due to an increase in dissociation, because the dissociation is already complete, but to an increased mobility of the ions. In a concentrated solution of a highly ionized strong electrolyte, the ions are close enough to one another so that any one of them in moving is influenced not only by the electrical field impressed across the electrodes but also by the field of the surrounding ions. The ionic velocities are, then, dependent upon both forces. Arrhenius attempted to treat the electrolytic-conductance behavior of the strong electrolytes in the way in which he had successfully treated the weak electrolytes; such a treatment is, however, inconsistent with the experimental fact, discovered by Kohlrausch, that a plot of the equivalent
Molar conductivities of several electrolytes in aqueous solution at 298.15 K vs square root of the concentrations.
conductance of a strong electrolyte against the square root of the concentration is very nearly linear. Debye and Hückel and Onsager have been able to calculate the effect of the surrounding ions on the mobility of any given ion and, for dilute solutions, have obtained results entirely consistent with the experimental facts. Complete dissociation is here assumed.
The conductivity of a solution is then given by the conductivities of the anion and the cation:
Lo = L+o + L-o
Equation describes the true state of affairs and states Kohlrausch’s law of
independent mobilities of ions in infinitely dilute solutions. It is amenable to easy verification, since the difference in Lo for pairs of salts with a common ion should be the same regardless of the common ion.
THEORY: A factor which affects the equivalent conductance of a solution that was not considered above is the possible limited dissociation of the electrolyte. Some electrolytes, known as weak electrolytes, do not dissociate completely in solution. Instead, there is equilibrium between ions and associated electrolyte. Acetic acid is a typical weak electrolyte. The apparent equilibrium constant for dissociation may be calculated as
where a = degree of dissociation
c = concentration of the solute
According to the Arrhenius theory, the equivalent conductance at any concentration is related to the degree of dissociation by
a = L/Lo
where L = equivalent conductance at concentration c
Lo = equivalent conductance at infinite dilution
In the case of a weak electrolyte the value of Lo cannot be obtained by the extrapolation to infinite dilution of results obtained at finite concentration, because L is a rapidly varying and nonlinear function of Lc. Instead, Lo is obtained by the of the law of Kohlrausch:
LoHR = LoHCl + LoNaR - LoNaCl
This is equivalent to saying that for a simple, binary electrolyte like HR,
LoHR= LoH+ + LoR-.
The equivalent conductance at infinite dilution is a sum of ionic contribution, but LoH+, for example, is independent of the electrolyte from which the hydrogen ions are obtained. Since HCI and the salts NaR and NaCI are all strong electrolytes, the values on the right-hand side of can all be obtained by extrapolation, as discussed in previous section.
The apparent equilibrium constant Ka, is equal to the true equilibrium constant, which is expressed in terms of activities, only for ideal solutes.
where g i is the activity coefficient of species i. Since g i= 1 for infinitely dilute solutions,
Lim Ka = K
Arrhenius noted that the molar conductivities of electrolytes decreased with increasing concentration. He considered a solution of the salt AB to consist partly of unionized AB molecules and partly of cations and anions. He attributed the decrease in conductivity to the decrease in the degree of ionization of the electrolyte. Suppose that a solution of AB is prepared with n molecules of AB per cubic centimeter of solution. The degree of ionization, a, is determined by the number of positive ions per cubic centimeter, n+ = a n, or the number of negative ions, n- = a n. Suppose that the velocities with which the ions move through the water are u+ and u-. The total current carried across a unit area is then
I = n+eu+ + n-(-e)(-u-) = ne(u+ + u-)a
where the negative sign for u_ arises because the two different ions move in opposite directions; the electron charge is denoted by e.
The mobility m of an ion is defined as the velocity per unit field strength, or
m+ = u+/E and is m -= u-/E. From Ohm’s law and the definition of conductivity, then k = I/E, where I is the current across a unit area and E is the electric field. The specific conductivity of the solution is
k = ne(m+ + m-)
The equivalent conductivity is then
L = Noe(u+ + u-)a
here No is the Avogadro number. Since Noe is the charge on a mole of electrons, or the Faraday constant F, can be written as
Lo = F(m+ + m-)
It is here that Arrhenius made the critical assumption that at infinite dilution
ionization is complete, that is, as m ~ 0, a ~ 1. In the limit,
L = F(m+ + m-)a
and by combining Equations we get the expression for the degrees of ionization,
a = L /Lo
This model accomplishes two things. First, it enables the calculation of the degree of ionization of electrolytes from conductivity data, and secondly it provides an explanation of Kohlrausch’s law of independent migrations. The mobilities of the ions are independent of the chemical constitution and the terms above can be interpreted by taking
L o+ = Fm + and L o- = Fm -
Suppose the dissociation of acetic acid at a concentration of c moles per dm3 is take as an example.
HAc = H+ + Ac-
c(1-a) ca ca
The concentrations of the various species at equilibrium have been written in terms of c and the degree of dissociation. Shortly after Arrhenius first proposed his ionic dissociation theory, Ostwald applied the law of mass action to a partially ionized substance. For HAc :
where the brackets indicate molar concentrations.
Measurements of Ka over a range of concentrations thus provide a way of estimating K and the activity coefficient factor gH+gA- /gHA at the various concentrations. To a good approximation g HA may be considered to be unity. The value of the product gH+ gA- is, by definition, g± 2 the square of the mean ionic activity coefficient for the weak electrolyte HA.
That is the limiting conductivity of a solution of ions is the sum of the limiting conductivities of the individual ions. This can be extended to determine the conductivity of the weak acid, acetic acid, at infinite dilution from the conductivities of hydrochloric acid and sodium acetate minus that of sodium chloride.
L HAco = L HClo + L NaAco - L NaClo
H+ + Ac- = H+ + Cl- + Na+ + Ac- - Na+ -Cl-
As the concentration of a weak acid increases, a new species, the undissociated acid (HAc) is present in the solution. This can be contrasted with the solubility of a salt. When a soluble salt such as sodium chloride dissolves in water, Na+ ions and Cl- ions are present in solution. As the solution becomes saturated the solid sodium chloride precipitates out of solution. However, for acetic acid, the HC2H3O2 species, will be in solution and in equilibrium with the ions H3O+ and C2H3O2-. The conductivity of the solution of a concentration, c, can be used to determine the degree of dissociation, a =L /Lo and then the equilibrium constant:
[H3O+] = ac, [C2H3O2-] = ac, [HC2H3O2] = (1 - a)c
If the normalities of the acids are not known accurately, they are determined by titration with standard base. The conductance of a 0.05 N solution of acetic acid
is determined, and a 0.025 N solution is then prepared by quantitative dilution with conductance water. In this fashion, conductance measurements are made on 0.05, 0.025, 0.0125, 0.00625, 0.00312, and 0.00156 N acetic acid solutions.
1. R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," 2d ed. (revised), Academic Press, Inc., New York, 1965.
2. T. Shedlovsky, Conductometry, in A. Weissberger (ed.), "Technique of Organic Chemistry," vol. 1, "Physical Methods of Organic Chemistry," 3d ed., pt. 4, chap. 45, Interscience Publishers, Inc., New York, 1960.
3. M. Spiro, Determination of Transference Numbers, in A. Weissberger (ed.), "Technique of Organic Chemistry," vol. 1, "Physical Methods of Organic Chemistry," 3d ed., pt. 4, chap 46, Interscience Publishers, Inc., New York, 1960.
4. D. A. MacInnes, "The Principles of Electrochemistry," Reinhold Publishing Corporation New York, 1939.
5. G. Jones and B. C. Bradshaw, J. Am. Chem. Soc., 55: 1780 (1933).
6. D. A. MacInnes and T. Shedlovsky, J. Am. Chem. Soc., 54: 1429 (1932).
7. "International Critical Tables," vol. VI, McGraw-Hill Book Company, New York,1928.