[SkepticalPetrophysicist]

Dear Benefactors & Tutors

I have exhausted my ability to find anything definitive on the internet or in my personal library to help with the calculation below. I have formulas for the coefficients in the Debye-Hückel-Onsager version of the Kohlrausch “law”, and I have values for the coefficients found in the literature. And, I have tables of values for the constants that enter into the coefficients. My failing is that, try as I might, as many ways I can think of, to put the constants into the formulas to get the published results for the coefficients, I have not yet been successful. Pretty clearly, I do not understand something, but I don’t know what. Thus, it is time for me (for the second time in my personal experience) to seek folks on the internet who are wiser and more experienced than I. (I am not an electrochemist, and my physics background is not helping here.) I would be forever grateful for a reply showing how to plug the constants into the formulas below to get the published values for the coefficients.

Thanks in advance.

In Debye Huckel Onsager theory, a formula for molar conductivity litters the literature. Here is an example of the formula

[tex] \Lambda_m =\Lambda_m^0 - (A+B\Lambda_m^0)\sqrt(C)[/tex]

Other examples differ by the symbols used for A and B.  A and B are expressed in terms of combinations of physical constants:

 [tex]A =\frac{z^2 e F^2}{3 \pi \eta} \sqrt{ \frac{2}{\epsilon R T}}[/tex]

 [tex]B = 0.586 \frac{z^2 e F}{24 \pi \epsilon R T} \sqrt{ \frac{2}{\epsilon R T}}[/tex]

where e is the elementary charge (charge on electron), F is Faraday constant, R is gas constant, T is absolute temperature,  epsilon is electric permittivity of the medium, z is ion valence, and  eta is viscosity of the solvent.  These are measured in different units in SI and esu/cgs systems (i.e., some, but not all, of the numerical values are different). Values for these combinations also litter the literature. But the values are all very close to

[tex]A= 82.4 \frac{1}{\eta \sqrt{DT}} [/tex]

 [tex]B= 8.2 \times 10^{-5} \frac{1}{(DT)^{3/2}} [/tex]

(a notational quirk. The electric permittivity is given by the product of the permittivity of free space (epsilon naught) and relative permeability [itex] \epsilon =  \epsilon_r \epsilon_0 [/itex], which is the same thing as the dielectric constant, [itex]\epsilon_r = D [/itex], in the formulas above.)

My problem is that I cannot plug these physical constants into the formulas and get the published results (corroborated in many publications, and so presumed to be correct), no matter what system of units that I try. I am talking about matching the significant figures; I could probably figure out the powers of 10 if I could get the coefficient digits to come out right. I list the values that are not working for me:

In SI units

e = 1.6 x 10^-19 coulomb
F = 96485.3 coulomb/mole of charge or coulomb/equivalent
 = 8.85 x 10^-12 farad/meter
D = 80 (dielectric constant of H₂O, a pure number)
 = 0.0091 viscosity of water (in poise)
R = 8.314472 gas constant in SI units
k_B = 1.38 x 10^-23 Boltzmann constant in SI units (not used in the formulas above)
N_A = 6.02 x 10²³


In cgs / esu units

e = 4.8 x 10^-10
F = 2.89 x 10¹⁴  
 = 1
D = 80                 dielectric constant of H2O, a pure number
 = 0.0091         viscosity of water (in poise)
R = 8.314472 x 10⁷
k_B = 1.38 x 10^-16 Boltzman contant
N_A = 6.02 x 10²³ a pure number

The early literature (from 1930s) seems to use esu and/or cgs units. Since the units of molar conductivity are either S-cm²/g-mole or S-m²/kg-mole (the SAME number regardless of units), it would seem that the regardless of how the above constants are expressed (i.e., which unit system is used), the resulting combination will have to be the same number, at least to within a power of 10.

Nonetheless, I have not been able to find any combination of these numbers in any system of units that gives the two values needed; i.e., 82.4 and 8.2 x 10^-5.

Any help most appreciated by me, the chemist’s apprentice without a master.

davek

[mjc123]

This is quite complicated, I had to try it a few times before getting it right.
First, there are some errors in your expressions (for A at least, I haven't checked B). The numerator should contain e²FN_A instead of eF². (Do a dimensional analysis to check). [Edit: Actually F = eN_A, isn't it? D'oh!]
It is not true that S-cm²/g-mole or S-m²/kg-mole  give the same numbers, as 1 m² = 10000 cm² but 1 kg = 1000 g.
Then you have to convert from the unit system in which you did your calculations to that commonly used for molar conductivities. For example, if you calculate A in SI units you will get a value in S m² mol^-1/(mol m^-3)^1/2. To convert this to S cm² mol^-1/(mol dm^-3)^1/2 you must multiply by a factor of 3.16 x 10⁵.
Thus for the constant in the expression for A I got 2.59 x 10^-5 in SI units, which converts to 8.19 - or 81.9 if you use poise for η instead of Pa s. Using this, I got a value of A of 53 S cm² mol^-1/(mol dm^-3)^1/2 for water at 25°C; Atkins gives 60.2.

[mjc123]

For B, I think you need z³eF² instead of z²eF in the numerator. In the expression in Atkins (my 2nd ed), there is an extra factor of π in the denominator inside the square root, but I wonder if this is a mistake. Without this factor, I get 8.16 x 10⁵ (not 10^-5) for the constant in the simplified expression for B, and a value for water at 25°C of 0.222 (Atkins 0.229).

I've just checked; that extra factor of π has disappeared in a later edition of Atkins. But I think he's now (7th ed) missing a factor of F. Shows how difficult it is to get this right!