Re: Copper sulphate solutions resistance with distance

Hello, Chris.

That’s a very good question (previous MadSci Answer). You’ve done a great job of starting with a hypothesis and checking it against an actual experiment. Speaking experimentalist to experimentalist, I’ll acknowledge right off the bat that I may not have the right answer. What I do have is an idea that you can test. If you find that my idea doesn’t fit new observations then feel free to write again, or come up with further experiments to test new ideas.

What I think you are seeing is the effect of your container. I’ll explain what I mean in greater detail below, but the end result is that you should get different resistance-versus-distance measurements if you do your experiment in differently-shaped water tanks. I could be way off base on this, but it sparked a whole series of ideas that sound fun ‘on paper’!

When you put two probes in a large tank, the paths taken by the moving ions are not confined to a straight line between the probes. There are also ions which circle around and approach the far probe from different angles. If your tank were infinitely large, you should get the simple linear relationship of resistance versus distance: the only thing that matters is the distance between the probes. The shapes of the non- straight paths remain the same, they are merely scaled up or down by the distance between the probes.

On the other hand, when your tank shrinks down to the point where you start cutting off those alternate paths, either in depth or width, your resistance should change as the boundaries affect the shape of the current flow.

Of course, another way of exploring the same thing is to use a fixed-size tank but change the position of your probes from almost touching to far apart in the tank. When the probes are almost touching, a tank much bigger than the probes is going to be no different than an infinite-sized one. As you separate your probes to where the distance between them approaches the size of the tank, the tank is no longer going to act like an infinite one.

Somewhere between those positions, the resistance curve should go through a transition from the infinite-looking-tank behavior to the small-looking- tank behavior. I think this is what you’re seeing. The question of whether the resistance should go up or down at these boundaries would depend on whether current flows more easily or less easily along the edges. Different types of boundary (water-air, water-glass, water- plastic, water-aluminum foil!, etc) may well behave differently.

How can we test this?

Here’s one possibility: shrink the width (but not length) of your tank and do your measurement again. If I’m right, your transition point (the point where the curve gets ‘wavy’) should occur at smaller separations as the walls of your tank close in on the probes.

How can you do the experiment?

You don’t need a bunch of different tanks. If you can get two pieces of plastic or glass that just fit along the length of your existing tank, you can move those pieces closer and closer to simulate the walls of the tank closing in on your probes. By keeping the length of your tank the same, you can still take data for exactly the same probe distances (along the length of the tank). You may even be able to demonstrate transitions with different shapes by using different materials for your walls.

How else could you do the same measurement a little differently? How about by filling the tank first, and then removing liquid after each series of measurements. Then you can get a different curve for each depth, down to where your tank is very shallow. There could be some problem with this if your probe has any length to it, as a wire, for example. You’d have a problem accounting for the different lengths of wire immersed at shallow depths.

Lastly, I’ll propose one more, similar possibility. If you really do have wires for probes then they’ll go through another transition as the distance between the probes approaches the wire length. Close in the probes will look like long wires to each other. For example, the resistance should change when they’re parallel versus perpendicular to each other. If they are now moved many wire-lengths away, they will begin to act like points sources. Their mutual orientation won’t matter anymore. So, somewhere along the way they should go through another transition. The difference would probably be subtle, but would likely be made clear if you took a series of measurements for crossed probes and compared it to a similar series of measurements for parallel probes.

I had a lot of fun thinking about all this – I just hope I’m right. That’s the tough thing about real-world science: it doesn’t matter how ‘fun’ a theory might be if the ensuing experiments disprove it! Good luck.