# B Equation for Conductivity of Aqueous Solutions

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1. Jun 27, 2017

### neilparker62

I have found a lot of data and graphs showing conductivity (micro-mhos) vs % by weight of various salt solutions. The salts for which graphs appear particularly linear are those of alkali earth metals - eg sodium, potassium - KI, KBr, KCl etc.

Is there a theoretical equation for conductivity vs % by weight (concentration) of simple salt solutions such as those mentioned above ? Just covering the linear portion of the graphs which generally appear at more dilute concentrations.

2. Jun 28, 2017

### mjc123

There is the Kohlrausch equation. If κ is the conductivity of a solution of molar concentration c, the molar coductivity Λm = κ/c is given by
Λm(c) = Λm0 - Kc1/2
where Λm0 is the molar conductivity at infinite dilution and K is a constant dependent on the type of salt (MX, MX2 etc.).
Λm0 can be expressed in terms of the molar conductivities of the individual ions as
Λm0 = ν+λ+0 + ν-λ-0
where ν+ and ν- are the numbers of cations and anions in the molecular formula.
Then κ = Λm0c - Kc3/2
The second term is usually a small correction to the first. I'm sure you can work out for yourself how to apply this equation to mass concentrations.
By the way, group 1 elements like sodium and potassium are the alkali metals. Alkaline earth metals are the group 2 elements, Mg, Ca etc.

Last edited: Jun 28, 2017
3. Jun 28, 2017

### neilparker62

Many thanks for the above response to my query. Can I conclude that if I convert the % by mass concentrations to molar concentrations, then the gradient of a conductivity vs concentration graph should be the molar conductivity of (for eg) KCl ?

Then I need to understand how to calculate the theoretical molar conductivity from the formula you have given. Could you perhaps provide a numeric example using KCl - I presume the v+ and v- in the formula would both be 1 but I am not sure where λ+ and λ- come from - connected as far as I can make out with ion mobility and Faraday constant.

4. Jun 29, 2017

### mjc123

Careful. The molar conductivity κ/c has a value at each concentration. The gradient of the graph of κ vs. c at low c is the molar conductivity at infinite dilution, Λm0; but the best way of measuring this would be to plot Λm vs. c and extrapolate to zero concentration. Careful too about units. "Micro-mhos" is a unit of conductance, not conductivity, which is measured in Ω-1 cm-1 or similar units. Atkins: "If κ is expressed in Ω-1 cm-1 and c in mol dm-3, the ratio κ/c should be multiplied by 1000 to obtain Λm in Ω-1 cm2 mol-1."

You are right that ν+ and ν- are both 1 for KCl; Atkins gives values of 73.5 and 76.3 Ω-1 cm2 mol-1 for λ+ of K+ and λ- of Cl-.
λ± = u±|z±|F
where z is the charge on the ion and u is the mobility, given by s± = u±E, where s is the terminal speed (magnitude of the terminal velocity) of the ion moving in an electrical field of strength E.

5. Jun 29, 2017

### neilparker62

Ok - here is my data set. I have read the values from the first set of conductance graphs in this reference. As can be seen from the equation of the regression line and correlation coefficient there is a very strong linear fit. So now what exactly does the gradient represent ? There are similar graphs for Potassium Bromide and Potassium Iodide and these are also very linear.

"% by weight" does confuse me because if I attempt a conversion to molarity - say 25.5 grams per 100 grams total mass, this yields 25.5 grams of KCl (0.34 moles) dissolved in 74.5 grams (ml) water - an extremely high concentration and at such values I don't think we are supposed to expect a linear fit at all. Well if in similar fashion one converts all the values to molarity, the fit is a lot more like a sqrt(c) curve. What is it about graphing % by weight vs conductance that seems to straighten out the curve so effectively? Surely not mere co-incidence if it happens for all the KX salts.

Potassium Chloride

% by weight vs Conductance Micromhos 25C

x, y
25.5 , 375
21.5 , 315
20.2 , 300
15 , 228
10 , 155
5 , 78
0 , 0

y = 14.605x + 4.4893
R² = 0.9993

6. Jun 30, 2017

### mjc123

OK, first some points to note with the data. The reference talks of "conductance", when it clearly means what we usually call "conductivity". The introduction states that "conductance" is given in units of µmhos/cm, although the y axis on the graphs is wrongly labelled "micromhos". Then, your values are out by a factor of 1000, e.g. KCl at 25.5% is 375,000 µmhos/cm.

I think you may be calculating your molarities wrongly. Molarity is the number of moles of solute per litre of solution (not solvent). So 25.5% is not 0.34 mol/74.5 mL, but 0.34 mol/the volume of 100g of solution. For that you need to know the density as a function of concentration, tables of this are available. To a first approximation you could assume that the desity is 1 g/cm3. I think you will find your curve more nearly linear if you do this.

At these high concentrations the equations given above will not be valid.

7. Jun 30, 2017

### neilparker62

As I understand it a person preparing a 25% by weight solution will weigh out 25 grams of the salt (KCl) and dissolve it in 75ml (75 grams) of water. So that total mass of salt and water is 100 grams. This is 25/75 or 0.33 grams per ml or 330 grams per litre - close to (slightly exceeding in fact) the saturation solubility of KCl which I googled - 31 grams per 100 ml or 310 grams per litre at 10 degrees (C). Obviously a bit more at 25C.

The point is that the % by weight vs conductivity graphs are very linear even at such high concentrations and I'm trying to understand why that is so not just for KCl but also for KF, KI and KBr. We don't seem to have any type of linear conductivity models - only the Kohlrausch equation which you say is invalid at these concentrations.

8. Jun 30, 2017

### mjc123

Correct
This is not how we express concentration. Concentration is mass (or moles) of solute divided by the total volume of the solution - not the volume of the solvent alone. If we estimate the density as 1 g/cc, the volume of 100g is 100 mL, so the concentration is 250 g/L. To be more accurate, using density tables, the density of 24% KCl solution at 20°C is 1.16 g/cc, so 100 g occupies 86.2 mL, so the concentration is 290 g/L.

As for the linearity - perhaps that's just coincidence for the K salts. There are plenty of salts for which the curves are non-linear!

9. Jul 3, 2017

### neilparker62

Ok - I think I have got it now (finally!). I suppose working with fairly dilute concentrations one never even considered density so it is quite a revelation to discover what a difference it makes. Now I just need to find density tables at 25C - only ones I can find via google search to date are these at 20 Celsius.

http://molbiol.ru/eng/protocol/01_22.html

Perhaps I will just use them and hope the difference between 20 and 25 C is relatively minimal.

Many thanks again.

10. Jul 3, 2017

### mjc123

The density difference between 20 and 25°C will not be great.
As to the linearity, it appears that the most linear curves are those for 1:1 alkali metal salts of strong monobasic acids, e.g. the halides. These are the salts that give the smallest deviations from ideality. Di- or trivalent ions have stronger electric fields, and the solutions behave less like solutions of independent ions. For salts of weak acids, the degree of dissociation varies with concentration. Linearity seems to be better for higher molecular weight salts (e.g. KCl compared with NaCl), because for higher-MW salts, the same mass concentration corresponds to a lower molar concentration.