# Current in a Concentration Cell

I would like to know if there is a simple function that gives the current as a function of time in a concentration cell. The current is measured in the wire connecting the two terminals of the cell. The two half cells are isolated i.e. not connected by a salt bridge and the electrolyte is a liquid.

I am looking at a very simple case without involving the factors that affect the current and the internal resistance of the cell.

From my workings, I've got $\displaystyle I=C\ln\left( \frac { A-q }{ B+q } \right)$ or more specifically

$\displaystyle \frac { dq }{ dt } =\frac { RT }{ nrF }\ln\left( \frac { VnF{ C }_\pu{ cathode }-q }{ VnF{ C }_\pu{ anode }+q } \right)$

where $R$ represents the gas constant

$T$ represents the absolute temperature

$n$ represents the number of moles of electrons exchanged

$r$ represents the resistance of the wire

$F$ represents the Faraday's constant

$V$ represents the Volume of the solution (assuming both electrolytes' to be same)

$C$ represents the initial concentration of the electrolyte in each half cell.

Now I've tried integrating this function using WolframAlpha and then differentiating it to obtain the current explicitly as a function of time. Seems this function is too complicated for Wolfram to evaluate.(I've used the expressions with A,B,C)

Is my derived expression correct?

Also, shouldn't a simple dynamic process have a simpler equation?

Thanks.

Yes, your derived expression looks correct to me.

As to the closed form for the integral, WolframAlpha is most likely unable to find one because it is simply not there among the elementary and special functions. This has nothing to do with the equation being simple. Of course it is simple; nature "solves" it quite easily. It is just that the nature's definition of "simple" does not necessarily coincide with ours. Helium atom is simple, so good luck trying to solve the Schrodinger equation for it. Would you argue an atom is not actually that simple? Well, your concentration cell is made up of quite a few atoms.

So it goes.