# What is A in the following equation

This equation describes the total current in bulk solution in a battery caused by migration in the electrolyte.

Then the source says what A is: In the bulk solution (away from the electrode), concentration gradients are generally small, and the total current is carried mainly by migration. All charged species contribute. For species j in the bulk region of a linear mass transfer system having a cross sectional area, $A$...

$$i = \sum_j i_j = \frac{FA\,\Delta E}{l} \sum_j |z_j| u_j C_j$$

I am confused if $A$ is the electrode area or the solution cross sectional area.

I will describe what the rest of the variables are

1. The $i$ is the current in the bulk solution.
2. F is Faraday's Constant
3. ∆E is the change in potential (i.e. Voltage measurement of the battery)
4. l is the distance of how far the electrodes are separated (or at least that is what I think)
5. z is the charge of species j
6. u is mobility of species j
7. C is the concentration of species j
• Can you show us where this equation is from? – user37142 Mar 25 '17 at 8:17
• just giving some random equation doesnt seem such a good idea, could you describe its origin, and/or define what it gets you? ie what is in the LHS of the eqn? – SubZero Mar 25 '17 at 12:41
• sorry for that. I edited the question to hopefully make it more understandable – user510 Mar 25 '17 at 13:20

## 1 Answer

$A$ is the cross-sectional area in solution. The "bulk" is the relatively homogeneous "mess" of charged particles and solvent -- $A$ is not the area of the electrode.

What this equation tells you in terms of $A$, is that, the larger the area through which some number of charges are passing in a given moment, the greater the possible current, because more area means more room for more charges that can pass through in a moment. If you squeeze the area you start to make a bottleneck wherein less charged particles can pass in a given time. (This is why copper wires with larger diameter (or, smaller US gauge) have higher current capacity).

This makes sense, right? Current is measured as charge per unit time. The implication there, is that it is motion of charge through some point (1D) or cross sectional area (2D, $A$ here) per unit time.

• So when you say A is the cross-sectional area in solution, do you mean if I had a KOH solution in a beaker, then A would be the cross-section area of the beaker in which the solution is being held? – user510 Mar 26 '17 at 16:31
• Not quite. It's the cross sectional area of the actual solvent (water) between electrodes. Not the beaker. – khaverim Mar 26 '17 at 16:53
• okay so if I have a basic galvanic cell setup with two cylindrical beakers that contain the electrolytes connected by a salt bridge, can I approximate the A as the cross-sectional area of the beaker? If not, how can I find the area between the electrodes? – user510 Mar 26 '17 at 17:01
• The salt bridge is the bottleneck. That is where current will be determined. – khaverim Mar 26 '17 at 17:04
• sorry, but what do you mean when you say the salt bridge will be the bottleneck? – user510 Mar 26 '17 at 17:08