A question about permeability

Factual Questions
1

I have a test in a couple days for a physiology class, and I’m doing trying to do the review problems the professor suggested to us, and I’m quite stuck.

I’ve got some ions, and let’s say I want them to permeate a cell membrane. They have a permeability coeffiecient, P, and the cell has volume and area A and V, respectively. We can mash all those together into a constant called k, whereby k=P*(A/V), and then we can relate that constant to time and concentration gradients inside and outside the cell by:

ΔC(t)=ΔC[sub]0[/sub]*e[sup]kt[/sup], where C[sub]0[/sub] is the concentration gradient at time 0, equal to C[sub]in[/sub]-C[sub]out[/sub].

Now, what I want to know is, how can we solve for the time, t, when the concentration gradient is abolished, ie, the concentration inside equals the concentration outside, and ΔC(t) would be zero. If we just go ahead and put in zero for ΔC(t), then we end up trying to take the natural log of zero, and that ain’t happening.

What am I missing? I see in the book, they talk about the time constant, tau (τ), that is equal to 1/k, and that τ is the time it takes the concentration gradient to drop to 1/e (37%) of it’s initial value. So should I just sub in 0.37ΔC[sub]0[/sub] for ΔC(t) and then solve for t? That doesn’t seem to make sense, since 37% of it’s original value is not an aboloshed gradient, that would happen when C[sub]in[/sub] equals C[sub]out[/sub], and therefore ΔC(t)=0.

I am really not getting this stuff, and it makes my head hurt.

2

If ΔC(t)=0, then either ΔC[sub]0[/sub] has to equal 0 or e[sup]kt[/sup] has to equal 0 since two non-zero numbers multiplied together is non-zero. e to negative infinity is zero. That doesn’t make sense in the context of the problem. ΔC[sub]0[/sub] could be zero I suppose. Either that or it’s impossible for ΔC(t) to be 0.

That’s what I’m coming up with anyway. And if you take t=tau you get 0.37ΔC[sub]0[/sub]=ΔC[sub]0[/sub]**e[sup]tau/tau[/sup]*. Cancel the ΔC[sub]0[/sub] and e[sup]1[/sup] does not equal 0.37. I think perhaps you have your equation copied down wrong? Is it e[sup]-kt[/sup]? That would make more sense since it solves all the problems.

3

First of all, is there susposed to be a negative in the exponent? In other words, is the equation susposed to be:

ΔC(t)=ΔC[sub]0[/sub]*e[sup]-kt[/sup]

Assuming there is susposed to be a negative in the exponent…

It would appear ΔC(t) can never be zero in the strictest mathematical sense. Even if t = 100 years, ΔC(t) would not equal zero. Yes, it would be infinitesimally small, but it would not be zero.

So someone needs to define, for all practical purposes, what an “abolished” concentration gradient is. In electrical engineering, we usually say that, after 5 time constants, the system has reached steady state.

So if everyone agrees that, after 5 time constants, the concentration gradient is very close to zero, then simply plug in 5*τ for the time, and then solve for ΔC.

4

BTW: A more common inquiry may be the following:

How long does it take for ΔC to reach steady state?

Using my previous example, we would assume steady state is achieved after 5 time constants, i.e. when t = 5τ. Therefore, the answer is t = 5τ = 5/k = 5V/(PA).

5

:smack:
Yes, it is indeed e[sup]-kt[/sup], my bad, I wtyped it in wrong when I was making my OP.

And the actual question is worded:

The first half of the question I figured out just fine, the equation is J = -PΔC (P being the permeabilities given to me, and ΔC being the change in concentration gradients.) That second half has me stumped, though.

6

Well, they need to define what “abolished” means, because ΔC can never equal zero based on the equation you provided.

7

Simple - permeability is B/H.

Oh wait - cells, ions?

8

Yeah, they need to, but don’t.

9

Go back and look at your textbook. I bet somewhere it says “abolished” means 5 (or 7 or 9) time constants. Or they give it as a fraction or percentage.

10

You sir, would lose that bet. I have read the entire chapter on diffusion and permeability over and over again (small book, small chapters, so it doen’t take that long to read it,) and nowhere does it give a definition of when the concentration gradient is considered ‘abolished,’ and neither did the prof. tell us in his lecture notes. As I said, the only significant number given was when t = tau, and the concentration would be 0.37 of it’s original value, which is hardly abolished. I guess I’ll just go and chat with the prof. today and see what he is asking, cause I know there’s going to be a question like this on the test, he made that quite clear.

11

I can think of two possible interpretations. First, the answer they’re looking for may, in fact, be “never”. Second, maybe they expect you to realize that concentrations aren’t actually a continuum, and “abolished” means “only one molecule different”. In that case, you’d have to work out what concentration corresponds to a single molecule, and solve for that concentration.