# 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.