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I don't know where to start with the following problem:

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Can anyone give me any pointers?

(For maximum assistance, please adapt your responses and solutions to be understood by a beginner, prefacing and explaining what you are doing so I can follow along.)

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    Voted for migration to physics.SE .2011-11-27
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    The result can be obtained by fixing the time $t$ and integrating both sides of the conservation equation over an $x$-interval. The lefthand side, after exchanging the order of integration and partial differentiation, yields the LHS of the "Fick's Principle" equation (with $m$ being the mass in the interval at time $t$), and similarly for the RHS.2011-11-27
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    @r.e.s. If you feel up to it, perhaps you could kindly suggest a complete answer below?2011-11-27

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The left- and right-hand sides of the conservation-of-mass equation

$$\frac{\partial c}{\partial t} = - \frac{\partial q}{\partial x}$$

are assumed to be everywhere-continuous functions of spatial coordinate $x$ and time $t$, with $c$ being the solute mass concentration (density).

The equation expressing "Fick's Principle" can be derived by fixing $t$ and integrating both sides of the above equation with respect to $x$ over a finite interval, say $[x_1, x_2]$:

$$\int_{x_1}^{x_2}\frac{\partial c}{\partial t}\, dx = - \int_{x_1}^{x_2}\frac{\partial q}{\partial x}\, dx$$

$$\frac{\partial}{\partial t} \int_{x_1}^{x_2}c\, dx = -(q(x_2,t) - q(x_1,t))$$

$$\frac{\partial }{\partial t}m = q_{in} - q_{out}$$

where $m$ is the solute mass in the interval at time $t$.


For similar derivations in the rather more-complicated case of three spatial dimensions, see continuity equation.

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    Thank you very much r.e.s.! I didn't know you could reverse the order of integration and partial differentiation, and I had no idea that $m(x,t)=\int c(x,t) \quad dx$. How did you know about this relation, that the integral of $c(x,t)$ in terms of $x$ was equivalent to $m(x,t)$?? Is this supposed to be common knowledge? Considering this is a maths course and not a science course I am taking, I thought they should've explicitly provided this information in the question.2011-11-28
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    @ptrcao: The integral of a well-behaved mass-density over a simple finite spatial region will just be the total mass in that region. In this problem, $c$ is linear mass-density (mass per length), and the region is a finite x-interval. (I deduced from the problem statement that $c$ has to be linear mass-density.) **NB**: I think this was a borderline case of a physics problem poorly presented as a math problem, and I hesitated to answer it here for that reason (especially given the vote for migration to physics.SE). I hope it wasn't too out-of-order for me to have done so after waiting 9 hours.2011-11-28
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    As far as I'm concerned, it wasn't out of order at all; I'm glad you answered it rather than waiting around! For me it was purely a math problem, and since I know so little about physics it was best to treat it as such.2011-11-28