# Charge as a function of time in electrolysis

According to Faraday's laws of electrolysis, $$\frac{Q}{Z}$$ = no. of moles of substance released (n), where Z is n-factor x 96500. Assuming the electrolyte is a strong electrolyte, specific conductivity (κ) = $$\frac{Λ(n_0-n)}{10 vol}$$ and resistance = $$\frac{l}{Aκ} = \frac{V}{i}$$, where n0 is initial moles of substance, κ is specific conductance and Λ is molar conductance.

Combining all these equations together, we get $$\frac{Λ(n_0-(Q/Z))}{10vol} = \frac{li}{AV}$$, which after some working out gives us this, Q = Z$$n_0$$(1 - $$e^{kt}$$), charge passed as a function of time, where k = $$\frac{ΛAV}{10Zlvol}$$. Here I'm assuming that the electrolyte is strong and with the given constant potential difference the electrolysis is spontaneous, even with the change in concentration occuring. Is this correct? How would I change the equation to adjust it for weak electrolytes as well?

• It is better to say amount of substance/substance amount than number of moles of substance, similarly as mass of substance/substance mass sounds better than number of kilograms of substance. Commented Dec 20, 2023 at 16:43
• Specific conductance = conductivity. Conductivity is just conductivity. Molar conductance should be called molar conductivity. // Why "vol" and not V ? What the coefficient 10 is about ? Commented Dec 20, 2023 at 16:53
• Your equation says if the substance amount is unchanged ( n0 - n = 0)then the conductivity is zero. that is obviously false. Commented Dec 20, 2023 at 16:57
• The title of the question mentions "a charge as a function of time". But the question itself after the title does not speak of time. Never. There are plenty of formula related to conductance, resistance, number of moles, conductance, etc. but no time nowhere. It is about the same for "charge". There are no charge in the text of the question. Commented Dec 20, 2023 at 19:25
• There is an important detail that at electrolysis, the current is not equal to product of voltage and conductance. Commented Dec 20, 2023 at 19:38

Solution conductance, conductivity and molar conductivity are measured by small AC voltage of sufficient frequency, typically $$\pu{1 kHz}$$. In such a scenario, most of effects involved in electrolysis do not manifest.

In context of DC electrolysis:

• The resulting effective resistance or conductance of the electrolytic cell, formally fitting the Ohm law, depends on many factors. The solution conductance or conductivity is just one of them.
• Each electrode has for the given cell its voltammetric characteristic $$I=f(E)$$, which in large extent resembles a shape of a series of smoothed steps (one or several, often overlapping).
• This characteristics dynamically changes under the current load and with change of composition of solution or electrode.
• When an external voltage is applied, each of the electrodes establishes a momentary potential (being referred to SHE), where:
• The electrode potential difference is equal to the external voltage.
• The absolute value of currents passing the electrode are equal.

The external voltage is equal to the sum of potential differences

$$U_\text{ext}=U_\text{electrodes} + U_\text{charge transfer} \\ + U_\text{local electrode concentration gradient} + U_\text{electromigration}$$

The particular terms are complicated, not exactly modelled functions of current and time.

As a huge simplification with limited accuracy, there is the approximate proportionality

$$I \propto (U - U_0) \cdot c,$$

where

• $$U_0$$ is the voltage threshold when the current starts passing the cell.
• $$c$$ is the concentration of limiting electroactive component, depending on the scenario.
• The proportionality constant is the subject of dynamic changes or drifts, and the proportionality is just approximation.

If you want to control the charge passed the electrolytic cell, do not use the power source of constant voltage, but of constant current, with the max voltage limitation. Such sources are less common, but could be bought or assembled by skilled persons.

They internally keep constant voltage difference on a small serial internal resistor, effectively keeping the passing current constant.

They can be formally modelled as power sources of a huge voltage with a huge internal resistance. They, within limits, keep on the attached circuit such a voltage that is needed to keep the chosen current.