# Diffusion with a Buffer

Homework Problem 27 in Chapter 4 of *Intermediate Physics for Medicine and Biology* examines diffusion in the presence of a buffer. The problem shows that the buffer slows diffusion and introduces the idea of an effective diffusion constant. I like this problem but I admit it’s rather long-winded. Recently, when thumbing through John Crank’s book (doesn’t everyone thumb through *The Mathematics of Diffusion* on occasion?), I found an easier way to present the same basic idea. Below is a simplified version of Problem 27.

Section 4.8

Problem 27½.Calciumionswith concentrationCdiffuse inside cells. Assume that this free calcium is in instantaneous local equilibrium with calcium of concentrationSthat is bound to an immobile buffer, such thatS = RC , (1)

whereRis a dimensionless constant. Calcium released from the buffer acts as a source term in the diffusion equation∂C/∂

t= D ∂^2C/∂x^2 − ∂S/∂t . (2)

(a) Explain in words why ∂S/∂tis the correct source term. Be sure to address why there is a minus sign.

(b) Substitute Eq. (1) into Eq. (2), derive an equation of the form∂

C/∂t= D_eff ∂^2C/∂x^2 , (3)

and obtain an expression for the effective diffusion constantD_eff.

(c) IfRis much greater than one, describe the physical affect the buffer has on diffusion.

(d) Show that this problem corresponds to the case of Problem 27 when [B] is much greater than [CaB]. Explain physically what this means.

To do part (d), you will need to look at the problem in *IPMB*.

The bottom line: an immobile buffer hinders diffusion. The stronger the buffer (the larger the value of *R*), the slower the calcium diffuses. The beauty of the homework problem is that it illustrates this property with only a little mathematics.

Enjoy!

*Originally published at **http://hobbieroth.blogspot.com**.*