Transport in an Infinite Medium

Abstract

This chapter develops some general ideas about the movement of solutes in solution. Two important mechanisms are drift, when solute molecules are dragged along by flowing solvent, and diffusion, the random motion of solute particles. Fick’s first and second laws of diffusion are derived, and the diffusion constant is related to the viscosity of the fluid. Solutions to the diffusion equation are analyzed in several cases of biological importance, including steady-state diffusion through pores in a membrane. We present a simple model illustrating what happens when drift and diffusion are both important. The chapter ends with a microscopic view of diffusion.

Problems

4.1.1 Section  4.1

Problem 1.

A cylindrical pipe with a cross-sectional area $S=1~$cm$^{2}$ and length $0.1~$cm has $j_{s}(0)S=200~$s$^{-1}$ and $j_{s}(0.1)S=150~$s$^{-1}$.

• (a) What is the total rate of buildup of particles in the pipe?

• (b) What is the average rate of change of concentration in the section of pipe?

Problem 2.

Write the continuity equation in cylindrical coordinates if $j_{\phi }=0$ but $j_{r}$ and $j_{z}$ can be nonzero.

Problem 3.

Consider two concentric spheres of radii $R$ and $r+dr$. If the particle fluence rate points radially and depends only on $R$, and the number of particles between $R$ and $r+dr$ is not changing, show that $d(r^{2}j)/dr=0$.

Problem 4.

Integrate Eq. 4.8 over a volume and subtract the result from Eq. 4.4. The resulting relationship is called the divergence theorem.

4.1.2 Section  4.2

Problem 5.

Suppose that the total blood flow through a region is $F$ (m$^{3}~$s$^{-1}$). A chemically inert substance is carried into the region in the blood. The total number of molecules of the substance in the region is $N$. The amount of blood in the region is not changing. Show that $dN/dt=(C_{A}-C_{V})F$, where $C_{A}$ and $C_{V}$ are the concentrations of substance in the arterial and venous blood. This is known as the Fick principle or the Fick tracer method. It is often used with radioactive tracers.

4.1.3 Section  4.3

Problem 6.

Allen et al. (1982) report seeing regular movement of particles in the axoplasm of a squid axon. At a temperature of $21\operatorname {{}^{\circ }\textrm {C}}$, the following mean drift speeds were observed:

$$ \begin{tabular} [c]{ll}Particle size ($\upmu$m) & Typical speed ($\upmu$m s$^{-1}$)\\ $0.8-5.0$ & $0.8$\\ $0.2-0.6$ & $2$ \end{tabular} \ $$

How do these values compare to thermal speeds? (Make a reasonable assumption about the density of particles and assume that they are spherical.)

Problem 7.

This problem looks at the original observations of Robert Brown that established Brownian motion .

• (a) Combine Eqs. 4.23 and 4.71 to determine an expression for the average distance a particle of radius $a$ will diffuse through a fluid of viscosity $\eta$ in time $t$.

• (b) Assume you observe a pollen grain with a radius of 50 microns in water at room temperature, and that your visual perception is particularly sensitive to motions occurring over a time of about one second. What is the average distance you observe the grain to move?

• (c) Now assume your eye cannot see movements that occur over angles of less than 1 min of arc, or $3\times 10^{-4}$ radians (In Chap. 14, we estimate 3 min of arc, but use 1 min here to be conservative). Most eyes cannot focus on objects closer than 25 cm. Determine the smallest displacement you can observe with the naked eye.

• (d) Robert Brown had a microscope that could magnify objects by a factor of about 370. What is the smallest displacement he could observe with his microscope? Is this larger or smaller than the displacement of a pollen grain in one second?

In fact, Brown did not observe the motion of entire pollen grains. He observed fat and starch particles about 2 $\upmu$m in diameter that are released by pollen. For more on Brown’s original observations, see Pearle et al. (2010).

4.1.4 Section  4.4

Problem 8.

• (a) Use the ideal gas law, $pV=Nk_{B}T=nRT$ to compute the volume of 1 mole of gas at $T=30\operatorname {{}^{\circ }\textrm {C}}$ and $p=1\operatorname {atm}.$ Express your answer in liters. Show that this is equivalent to a concentration of $2.4\times 10^{25}$ molecule m$^{-3}.$

• (b) Find the concentration of liquid water molecules at room temperature.

Problem 9.

Using the information on the mean free path in the atmosphere and assuming that all molecules have a molecular weight of 30, find the height at which the mean free path is 1 cm. Assume the atmosphere has a constant temperature.

4.1.5 Section  4.6

Problem 10.

Suppose $C(x,t)=\left ( N/\sqrt {4\uppi Dt}\right ) e^{{-x^2}/{4Dt}}.$ Find an expression for $j_{s}(x,t)$.

Problem 11.

Show that the momentum flux density, $j_p$, in Table 4.3 has the same units as force per unit area. Compare the equation to Eq. 1.33 and interpret $\eta$ physically.

Problem 12.

Jean Perrin measured the distribution of gamboge particles in water as a function of height, to determine Avagadro’s number (Perrin 1910). The radius of the spherical particles was $0.212~\upmu$m, the density of water was $1~\text {g}~\text {cm}^{-3}$, the density of the particles was $1.207~\text {g}~\text {cm}^{-3}$, and the temperature was $20\operatorname {{}^{\circ }\textrm {C}}$. He counted 13,000 particles, and found their relative number, $N$, as a function of height, $z$, to be (data normalized so $N$ is $100$ at $z=5~\upmu$m)

Table 6

• (a) Fit these data to a Boltzmann distribution, and determine a value for Boltzmann’s constant. Include the effect of buoyancy in your calculation. Fitting techniques are discussed in Chap. 11.

• (b) In Perrin’s time, the gas constant was known approximately: $R=8.32~$J K$^{-1}~$mol$^{-1}$. Use this value and your result from part (a) to calculate Avogadro’s number.

4.1.6 Section  4.7

Problem 13.

If all macromolecules have the same density, derive the expression for $D$ versus the molecular weight that was used to draw the line in Fig. 4.12.

Problem 14.

For diagnostic studies of the lung, it would be convenient to have radioactive particles that tag the air and that are small enough to penetrate all the way to the alveoli. It is possible to make the isotope $^{99m}$Tc into a “pseudogas” by burning a flammable aerosol containing it. The resulting particles have a radius of about 60 nm (Burch et al. 1984). Estimate the mean free path for these particles. If it is small compared to the molecular diameter, then Stokes’ law applies, and you can use Eq. 4.23 to obtain the diffusion constant. (The viscosity of air at body temperature is about 1.8$\times$10$^{-5}~$Pa s.)

Problem 15.

Figure 4.12 shows that $D$ for O$_{2}$ in water at 298 K is $1.2\times 10^{-9}~$m$^{2}~$s$^{-1}$ and that the molecular radius of O$_{2}$ is 0.2 nm. The diffusion constant of a dilute gas (where the mean free path is larger than the molecular diameter) is $D=\lambda ^{2}/2t_{c}$, where the collision time is given by Eq. 4.15.

• (a) Find a numeric value for the diffusion constant for O$_{2}$ in O  at 1 atm and 298 K and its ratio to $D$ for O$_{2}$ in water. The molecular weight of oxygen is 32.

• (b) Assuming that this equation for a dilute gas is valid in water, estimate the mean free path of an oxygen molecule in water.

4.1.7 Section  4.8

Problem 16.

(a) The three-dimensional normalized analog of Eq. 4.25 is

$$ C(x,y,z,t)=\frac{N}{\left[ 2\pi\,\sigma^{2}(t)\right] ^{3/2}}\exp\left( -\frac{x^{2}+y^{2}+z^{2}}{2\sigma^{2}(t)}\right) . $$

Find the three-dimensional analog of Eq. 4.27.

(b) Show that $\sigma ^{2}=\overline {x^{2}}+\overline {y^{2}}+\overline {z^{2}}=6Dt$.

Problem 17.

A crude approximation to the Gaussian probability distribution is a rectangle of height $P_{0}$ and width $2L$. It gives a constant probability for a distance $L$ either side of the mean.

• (a) Determine the value of $P_{0}$ and $L$ so that the distribution has the same value of $\sigma$ as a Gaussian.

• (b) Plot $P(x,t)$ if $\sigma$ is given by Eq. 4.27 and the mean remains centered at the origin for times of 1, 5, 50, 100, and 500 ms. Use $D$ for oxygen diffusing in water at body temperature.

• (c) How long does it take for the oxygen to have a reasonable probability of diffusing a distance of 8$~\upmu$m, the diameter of a capillary?

• (d) For $t=100~$ms, plot both the accurate Gaussian and the rectangular approximation.

Problem 18.

Write an equation for Fick’s second law in three-dimensional Cartesian coordinates when the diffusion constant depends on position: $D=D(x,y,z)$.

Problem 19.

The heat-flow equation in one dimension is

$$ j_{H}=-\kappa\,\left( \frac{\partial T}{\partial x}\right) , $$

where $\kappa$ is the thermal conductivity in W m$^{-1}~$K$^{-1}.$ One often finds an equation for the “diffusion” of energy by heat flow:

$$ \frac{\partial T}{\partial t}=D_{H}\left( \frac{\partial^{2}T}{\partial x^{2}}\right) . $$

The units of $j_{H}$ are J m$^{-2}~$s$^{-1}$. The internal energy per unit volume is given by $u=\rho cT$, where $C$ is the heat capacity per unit mass and $\rho$ is the density of the material. Derive the second equation from the first and show how $D_{H}$ depends on $\kappa$, $C$, and $\rho$.

Problem 20.

The dimensionless Lewis number is defined as the ratio of the diffusion constant for molecules and the diffusion constant for heat flow (see Problem 193). If the Lewis number is large, molecular diffusion occurs much more rapidly than the diffusion of energy by heat flow. If the Lewis number is small, energy diffuses more rapidly than molecules. Use the following parameters:

$$ \begin{tabular} [c]{lll} & Air & Water\\ $D$ (m$^{2}~$s$^{-1}$) & $2\times10^{-5}$ & $2\times10^{-9}$\\ $\kappa$ (W$~$m$^{-1}~$K$^{-1}$) & $0.03$ & $0.6$\\ $c$ (J$~$kg$^{-1}~$K$^{-1}$) & $1000$ & $4000$\\ $\rho$ (kg$~$m$^{-3}$) & $1.2$ & $1000.$ \end{tabular} \ \ $$

• (a) Calculate the Lewis number for oxygen in air and in water.

• (b) Is it possible using either air or water to design a system in which oxygen is transported by diffusion with almost no transfer of heat?

Problem 21.

A sheet of labeled water molecules starts at the origin in a one-dimensional problem and diffuses in the $x$ direction.

• (a) Plot $\sigma$ vs $t$ for diffusion of water in water.

• (b) Deduce a “velocity” versus time.

• (c) How long does it take for the water to have a reasonable chance of traveling $1~\upmu$m? $10~\upmu$m? $100~\upmu$m? 1 mm? 1 cm? 10 cm?

Problem 22.

In three dimensions the root-mean-square diffusion distance is $\sigma =\sqrt {6Dt},$ where $t$ is the diffusion time. Consider the diffusion of oxygen from air to the blood in the lungs. The terminal air sacs in the lungs, the alveoli, have a radius of about $100~\upmu$m. The radius of a capillary is about 4$~\upmu$m. Estimate the time for an oxygen molecule to diffuse from the center to the edge of an alveolus , and the time to diffuse from the edge to the center of a capillary. Which is greater? From the data in Table 1.4 estimate how long blood remains in a capillary. Is it long enough for diffusion of oxygen to occur? Assume the diffusion constant of oxygen in air is $2\times 10^{-5}~$m$^{2}~$s$^{-1}$ and in water is $2\times 10^{-9}~$m$^{2}~$s$^{-1}$.

Problem 23.

Why breathe? Estimate the time required for oxygen to diffuse from our nose to our lungs. Assume the diffusion constant of oxygen in air is $2\times 10^{-5}~$m$^{2}~$s$^{-1}.$

Problem 24.

At a nerve-muscle junction, the signal from the nerve is transmitted to the muscle by a chemical junction or synapse. Molecules of acetylcholine (ACh) must diffuse from the end of the nerve cell across an extracellular gap about 20 nm wide, to the muscle cell in order to activate the muscle. Assuming one-dimensional diffusion, estimate the signal delay caused by the time needed for ACh to diffuse. The delay of the signal at the nerve-muscle junction is about 0.5 ms. How does this compare to the diffusion time? Use a diffusion constant of $5\times 10^{-10}~$m$^{2}~$s$^{-1}$.

Problem 25.

A substance has diffusion constant $D$, and its concentration is distributed in space according to $C(x,t)=A(t)\sin (2\pi x/L),$ where $L$ is the wavelength and $A(t)$ is the amplitude of the distribution. Use the one-dimensional diffusion equation, Eq. 4.26, to show that the concentration decays exponentially with time, $A(t)\propto e^{-t/\tau }.$ Determine an expression for the time constant $\tau$ in terms of $L$ and $D$. Which decays faster: a long-wavelength (diffuse) distribution, or a short-wavelength (localized) distribution? This result can be used with the Fourier methods developed in Chap. 11 to derive very general solutions to the diffusion equation.

Problem 26.

Some tissues, such as skeletal muscle, are anisotropic: the rate of diffusion depends on direction. In these tissues, Fick’s first law in two dimensions has the form

$$ \left( \begin{array}[c]{c} j_{x}\\ j_{y} \end{array} \right) =-\left( \begin{array}[c]{cc} D_{xx} & D_{xy}\\ D_{yx} & D_{yy} \end{array} \right) \left( \begin{array}[c]{c} \partial C/\partial x\\ \partial C/\partial y \end{array} \right) . $$

The $2\times 2$ matrix is called the diffusion tensor. It is always symmetric, so $D_{xy}=D_{yx}.$

• (a) Derive the two-dimensional diffusion equation for anisotropic tissue. Assume the diffusion tensor depends on direction but not on position.

• (b) If the coordinate system is rotated from $(x,y)$ to $(x^{\prime },y^{\prime })$ by

  $$ \left(*{-5pt} \begin{array}[c]{c} x^{\prime}\\ y^{\prime} \end{array} *{-5pt}\right) =\left(*{-5pt} \begin{array}[c]{cc} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{array} *{-5pt}\right) \left(*{-5pt} \begin{array}[c]{c} x\\ y \end{array} *{-5pt}\right) , $$

  the diffusion tensor changes by

  $$ \begin{aligned} & \left(*{-5pt} \begin{array}[c]{cc} D_{x^{\prime}x^{\prime}} & D_{x^{\prime}y^{\prime}}\\ D_{x^{\prime}y^{\prime}} & D_{y^{\prime}y^{\prime}} \end{array} \!\!\right) \\ & =\left(*{-5pt} \begin{array}[c]{cc} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{array} *{-5pt}\right) \left(*{-5pt} \begin{array}[c]{cc} D_{xx} & D_{xy}\\ D_{xy} & D_{yy} \end{array} *{-5pt}\right) \left(*{-5pt} \begin{array}[c]{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{array} *{-5pt}\right) . \end{aligned} $$

  Find the angle $\theta$ such that the tensor is diagonal ($D_{x^{\prime }y^{\prime }}=0).$ Typically, this direction is parallel to a special direction in the tissue, such as the direction of fibers in a muscle.

• (c) Show that the trace of the diffusion tensor (the sum of the diagonal terms) is the same in any coordinate system ($D_{xx}+D_{yy}=D_{x^{\prime }x^{\prime }}+D_{y^{\prime }y^{\prime }}$ for any $\theta$). Basser et al. (1994) invented a way to measure the diffusion tensor using Magnetic Resonance Imaging (Chap. 18). From the diffusion tensor, they can image the direction of the fiber tracts. When they want images that are independent of the fiber direction, they use the trace.

Problem 27.

Calcium ions diffuse inside cells. Their concentration is also controlled by a buffer:

$$ \text{Ca + B }\iff\text{ CaB.} $$

The concentrations of free calcium, unbound buffer, and bound buffer ([Ca], [B], and [CaB]) are governed, assuming the buffer is immobile, by the differential equations

$$ \begin{aligned} \frac{\partial\lbrack\text{Ca}]}{\partial t} & =D\nabla^{2}[\text{Ca}]-k^{+}[\text{Ca}][\text{B}]+k^{-}[\text{CaB}],\\ \frac{\partial\lbrack\text{B}]}{\partial t} & =-k^{+}[\text{Ca}][\text{B}]+k^{-}[\text{CaB}],\\ \frac{\partial\lbrack\text{CaB}]}{\partial t} & =k^{+}[\text{Ca}][\text{B}]-k^{-}[\text{CaB}]. \end{aligned} $$

• (a) What are the dimensions (units) of $k^{+}$ and $k^{-}$ if the concentrations are measured in mol l$^{-1}$ and time in s?

• (b) Derive differential equations governing the total calcium and buffer concentrations, $\left [ \text {Ca}\right ] _{T}=\left [ \text {Ca}\right ] +\left [ \text {CaB}\right ]$ and $\left [ \text {B}\right ] _{T}=\left [ \text {B}\right ] +\left [ \text {CaB}\right ] .$ Show that $\left [ \text {B}\right ] _{T}$ is independent of time.

• (c) Assume calcium and buffer interact so rapidly that they are always in equilibrium:

  $$ \frac{\lbrack\text{Ca}][\text{B}]}{[\text{CaB}]}=K, $$

  where $K=k^{-}/k^{+}.$ Write $\left [ \text {Ca}\right ] _{T}$ in terms of $\left [ \text {Ca}\right ] ,$ $\left [ \text {B}\right ] _{T},$ and $K$ (eliminate $\left [ \text {B}\right ]$ and $\left [ \text {CaB}\right ]$).

• (d) Differentiate your expression in (c) with respect to time and use it in the differential equation for $\left [ \text {Ca}\right ] _{T}$ found in (b). Show that $\left [ \text {Ca}\right ]$ obeys a diffusion equation with an “effective” diffusion constant that depends on $\left [ \text {Ca}\right ]$:

  $$ D_{\text{eff}}=\frac{D}{1+\frac{K\left[ \text{B}\right] _{T}}{\left( K+\left[ \text{Ca}\right] \right) ^{2}}}. $$

• (e) If $\left [ \text {Ca}\right ] \ll K$ and $\left [ \text {B}\right ] _{T}=100K$ (typical for the endoplasmic reticulum), determine $D_{\text {eff}}/D.\index {Endoplasmic reticulum}$

For more about diffusion with buffers, see Wagner and Keizer (1994).

Problem 28.

Inside cells, calcium is stored in compartments, such as the sarcoplasmic reticulum. In some cells, a rise in calcium concentration, $C,$ triggers the release of this stored calcium. A model of such calcium-induced calcium release is

$$ \frac{dC}{dt}=-\frac{k}{C_{0}^{2}}C\left( 4C-C_{0}\right) \left( C-C_{0}\right) $$

(1)

• (a) Plot the rate of calcium release (the right-hand side of Eq. 1) vs $C.$ Identify points for which the calcium release is zero (steady-state solutions to Eq. 1). By qualitative reasoning, determine which of these points are stable and which are unstable. (Will a small change in $C$ from the steady-state value cause $C$ to return to the steady-state value or move farther away from it?)

• (b) If $C\ll C_{0}/4,$ what does Eq. 1 become, and what is its solution?

• (c) Eq. 1 is difficult to solve analytically. To find a numerical solution, approximate it as

  $$ \frac{C(t+\Delta t)-C(t)}{\Delta t}=-\frac{k}{C_{0}^{2}}C(t)\left[ 4C(t)-C_{0}\right] \left[ C(t)-C_{0}\right] . \label{P2} $$

  (2)

  Write a computer program to determine $C(t)$ at times $t=n\Delta t,$ $n=1,2,3,\dots ,100,$ using $\Delta t=0.1\operatorname {s},$ $k=1\operatorname {s}^{-1},$ $C_{0}=1$ $\upmu$M, and $C(t=0)=C^{\prime }$. Find the threshold value of $C^{\prime },$ below which $C(t)$ goes to zero, and above which $C(t)$ goes to $C_{0}.$

• (d) If we include diffusion of calcium in one dimension, Eq. 1 becomes

  $$ \frac{\partial C}{\partial t}=D\frac{\partial^{2}C}{\partial x^{2}}-\frac{k}{C_{0}^{2}}C\left( 4C-C_{0}\right) \left( C-C_{0}\right) . $$

  (3)

  This is a type of reaction–diffusion equation. To solve Eq. 3 numerically, divide the distance along the cell into discrete points, $x=m\Delta x,$ $m=0,1,2,\dots ,M.$ Approximate Eq. 3 as

  $$ \begin{aligned} & \frac{C(x,t+\Delta t)-C(x,t)}{\Delta t} \end{aligned} $$

  (4)
  $$ \begin{aligned} & =D\frac{C(x+\Delta x,t)-2C(x,t)+C(x-\Delta x,t)}{(\Delta x)^{2}}\nonumber \end{aligned} $$
  $$ \begin{aligned} & -\frac{k}{C_{0}^{2}}C(x,t)\left( 4C(x,t)-C_{0}\right) \left( C(x,t)-C_{0}\right) \nonumber \end{aligned} $$

  Assume the ends of the cell are sealed, so $C(0,t)=C(\Delta x,t)$ at one end and $C(M\Delta x,t)=C((M-1)\Delta x,t)$ at the other. Start with the cell at $C(x,0)=0$ for all points except at one end, where $C(0,0)=C_{0}.$ Calculate $C(x,t)$ using $\Delta x=5\mu m ,$ $\Delta t=0.1\;{s},$ $D=200\mu m^2 s^{-1}$ and $C_{0}=1$ $\upmu$M. You should get a wave of calcium propagating down the cell. What is its speed?

  Calcium waves play an important role in many cells. This simple model does not include a mechanism to return the calcium concentration to its originally low value after the wave has passed (a process called recovery). For a more realistic model, see Tang and Othmer (1994). For more information about numerical methods, see Press et al. (1992).

Problem 29.

The numerical approximation for the diffusion equation, derived as part of Problem 202, has a key limitation: it is unstable if the time step is too large. This problem can be avoided using the Crank–Nicolson method. Replace the first time derivative in the diffusion equation with a finite difference, as was done in Problem 202. Next, replace the second space derivative with the finite difference approximation from Problem 202, but instead of evaluating the second derivative at time $t$, use the average of the second derivative evaluated at times $t$ and $t+\Delta t$.

• (a) Write down this numerical approximation to the diffusion equation, analogous to Eq. 4 in Problem 202.

• (b) Explain why this expression is more difficult to compute than the expression given in the first two lines of Eq. 4. Hint: consider how you determine $C(t+\Delta t)$ once you know $C(t)$. The difficulty you discover in part (b) is offset by the advantage that the Crank–Nicolson method is stable for any time step. For more information about the Crank–Nicolson method , stability, and other numerical issues, see Press et al. (1992).

4.1.8 Section  4.9

Problem 30.

Consider steady-state diffusion through two plane layers as shown in the figure. Show that the diffusion is the same as through a single plane layer of thickness $\Delta x_{1}+\Delta x_{2}$, with diffusion constant

$$ D=\frac{D_{1}D_{2}}{\dfrac{\Delta x_{1}}{\Delta x_{1}+\Delta x_{2}}D_{2}+\dfrac{\Delta x_{2}}{\Delta x_{1}+\Delta x_{2}}D_{1}}. $$

Problem 31.

A fluid on the right of a membrane has different properties than the fluid on the left. Let the diffusion constants on left and right be $D_{1}$ and $D_{2}$, respectively, and let the pores in the membrane be filled by the fluid on the right a distance $xL$, where $L$ is the thickness of the membrane.

• (a) Use the results of Problem 204 to determine the effective diffusion constant $D$ for a membrane of thickness $L$ when $D_{2}=yD_{1}$, $\Delta x_{1}=(1-x)L$, and $\Delta x_{2}=xL$. Neglect end effects.

• (b) In the case that oxygen is diffusing in air and water at 310 K, the diffusion constants are $D_{1}=2.2\times 10^{-5}~$m$^{2}~$s$^{-1}$, $D_{2}=1.6\times 10^{-9}~$m$^{2}~$s$^{-1}$. Plot $D/D_{1}$ vs $x$.

4.1.9 Section  4.10

Problem 32.

• (a) Derive Eq. 4.45.

• (b) Derive Eqs. 4.51 and 4.52 from Eqs. 4.48 and 4.49.

Problem 33.

We can estimate $B/B^{\prime }$ of Eqs. 4.49– 4.55 by noting that $B^{\prime }$ must be larger than $b$ because of two effects. First, it is larger by $\pi R_/4$ because of end effects. Second, the concentration varies near the pores and smooths out further away, so $B^{\prime }$ must also be larger by an amount roughly equal to $L$, the spacing of the pores. There are $N/4\uppi B^2$ pores per m$^{2}$, so $l\approx R_p(\uppi/f)^{1/2}$. Use the example in the text: $B=5~\upmu$m, $\Delta Z=5~$nm, $f=0.001,$ to estimate these two corrections. Assume that the pore radius, $R_{p}$, is smaller than $\Delta Z.$ Are these corrections important?

Problem 34.

Consider an impervious plane at $z=0$ containing a circular disk of radius $a$ having a concentration $C_{0}.$ The concentration at large $z$ goes to zero. Carslaw and Jaeger (1959) show that the steady-state solution to the diffusion equation is

$$ C(r,z)=\frac{2C_{0}}{\pi}\sin^{-1}\left[ \frac{2a}{\sqrt{(r-a)^{2}+z^{2}}+\sqrt{(r+a)^{2}+z^{2}}}\right] . $$

• (a) (optional) Verify that $C(r,z)$ satisfies $\nabla ^{2}C=0.$ The calculation is quite involved, and you may wish to use a computer algebra program such as Mathematica or Maple.

• (b) Show that for $z=0$, $C=C_{0}$ if $r<a.$

• (c) Show that for $z=0$, $dC/dz=0$ if $r>a$.

• (d) Integrate $j_{z}$ over the disk ($z=0,$ $0<r<a$) and show that $i=4DaC_{0}.$

Problem 35.

Apply the analysis of Sect. 4.10 to determine how the current $i_{vessel}$ depends on the fraction of surface area covered by pores, for a cylindrical vessel of radius $b$. Assume that the concentration reaches a value $C_5$ at some large finite radius $R$.

4.1.10 Section  4.11

Problem 36.

The processes of heat conduction and diffusion are similar: the concentration and temperature both obey the diffusion equation (Problem 193). Consider a spherical cow of radius $R$ having a specific metabolic rate $Q~$W kg$^{-1}$. Assume the temperature of the outer surface of the cow is the same as the surroundings, $T_{\text {sur}}.$ Assume that heat transfer within the cow is by heat conduction.

• (a) Calculate the steady state temperature distribution inside the animal and find the core temperature at the center of the sphere.

• (b) Consider a smaller (but still spherical) animal such as a rabbit. What is its core temperature?

• (c) Calculate the temperature distribution and core temperature in a rabbit covered with fur of thickness $D$.

Assume the bodies of the cow and rabbit have the thermal properties of water and that the fur has the thermal properties of air. Let $d=0.03~$m and $T_{\text {sur}}=20\operatorname {{}^{\circ }\textrm {C}}.$

$$ \begin{tabular} [c]{lll} & Water & Air\\ $\kappa$ (W$~$m$^{-1}~$K$^{-1}$) & $0.6$ & $0.03$\\ $c$ (J$~$kg$^{-1}~$K$^{-1}$) & $4000$ & $1000$\\ $\rho$ (kg$~$m$^{-3}$) & $1000$ & $1.2$\\ & Cow & Rabbit\\ $R$ (m) & $0.3$ & $0.05$\\ $Q$ (W$~$kg$^{-1})$ & $0.6$ & $1.6$\\ \end{tabular} $$

Problem 37.

The goal of this problem is to estimate how large a cell living in an oxygenated medium can be before it is limited by oxygen transport. Assume the extracellular space is well stirred with uniform oxygen concentration $C_{0}.$ The cell is a sphere of radius $R$. Inside the cell oxygen is consumed at a rate $Q$ molecule m$^{-3}~$s$^{-1}$.The diffusion constant for oxygen in the cell is $D$.

• (a) Calculate the concentration of oxygen in the cell in the steady state.

• (b) Assume that if the cell is to survive the oxygen concentration at the center of the cell cannot become negative. Use this constraint to estimate the maximum size of the cell.

• (c) Calculate the maximum size of a cell for $C_{0}=8~$mol m$^{-3}$, $D$ $=2\times 10^{-9}~$m$^{2}~$s$^{-1}$, $Q=0.1~$mol m$^{-3}~$s$^{-1}$. (This value of $Q$ is typical of protozoa; the value of $C_{0}$ is for air and is roughly the same as the oxygen concentration in blood.)

Problem 38.

A diffusing substance is being consumed by a chemical reaction at a rate $Q$ per unit volume per second. The reaction rate is limited by the concentration of some enzyme, so $Q$ is independent of the concentration of the diffusing substance. For a slab of tissue of thickness $b$ with concentration $C_{0}$ at both $x=0$ and $x=b$, solve the equation to find $C(x)$ in the steady state. This is known as the Warburg equation (Warburg 1923). It is a one-dimensional model for the consumption of oxygen in tissue: points $x=0$ and $x=b$ correspond to the walls of two capillaries side by side.

Problem 39.

Suppose that a diffusing substance disappears in a chemical reaction and that the rate at which it disappears is proportional to the concentration $-kC$. Write down the Fick’s second law in this case. Show what the equation becomes if one makes the substitution $C(x,y,z,t)=C^{\prime }(x,y,z,t)e^{-kt}$.

Problem 40.

A spherical cell has radius $R$. The flux density through the surface is given by $j_{s}=-D\,$grad$\,C$. Suppose that the substance in question has concentration $C(t)$ inside the cell and zero outside. The material outside is removed fast enough so that the concentration remains zero. Using spherical coordinates, find a differential equation for $C(t)$ inside the cell. The thickness of the cell membrane is $\Delta r\ll R$.

Problem 41.

The cornea of the eye must be transparent, so it can contain no blood vessels. (Blood absorbs light.) Oxygen needed by the cornea must diffuse from the surface into the corneal tissue. Model the cornea as a plane sheet of thickness $L=500\operatorname .$ The oxygen concentration, $C,$ is governed by a one-dimensional steady-state diffusion equation

$$ D\frac{d^{2}C}{dx^{2}}=Q. $$

Assume the cornea is consuming oxygen at a rate $Q=4\times 10^{22}~$molecule m$^{-3}~$s$^{-1}$ and has a diffusion constant $D=3\times 10^{-9}\operatorname {m}^{2}\operatorname {s}^{-1}.$ The rear surface of the cornea is in contact with the aqueous humor, which has a uniform oxygen concentration $C_{2}=1.8\times 10^{24}~$molecule m$^{-3}.$ Consider three cases for the front surface:

• (a) Solve the diffusion equation for $C(x)$ when the front surface is in contact with air, which has an oxygen concentration $C_{1}=5\times 10^{24}~$m$^{-3}.$

• (b) The eye is closed, but a layer of tears maintains the concentration at the front surface that is the same as the aqueous humor: $C_{1}=1.8\times 10^{24}~$m$^{-3}.$ Plot $C(x).$

• (c) The eye is covered by an oxygen-impermeable contact lens, so that at the front surface $dC/dx=0.$ Solve the diffusion equation and plot $C(x).$

Supplying oxygen to the cornea is a major concern for people who wear contact lenses. Often a tear layer between the contact and cornea, replenished by blinking, is sufficient to keep the cornea oxygenated. If you sleep wearing a contact lens, this tear layer may not be replenished, and the cornea will be deprived of oxygen. For a similar but somewhat more realistic model, see Fatt and Bieber (1968).

Problem 42.

The distance $L$ that oxygen can diffuse in the steady-state is approximately $L=\sqrt {CD/Q},$where $C$ is the oxygen concentration, $D$ is the diffusion constant, and $Q$ is the rate per unit volume that oxygen is used for metabolism.

• (a) Show that $L$ has dimensions of length.

• (b) The diffusion of oxygen in air is about 10,000 times larger than the diffusion of oxygen in water (Denny 1993). By how much will the diffusion distance $L$ change if oxygen diffuses through air instead of water, all other things being equal?

Insects deliver oxygen to their flight muscles by diffusion down air-filled tubes instead of by blood vessels, thereby taking advantage of the large diffusion constant of oxygen in air (Weiss-Fogh 1964).

4.1.11 Section  4.12

Problem 43.

Dimensionless numbers, like the Reynolds number of Chap. 1, are often useful for understanding physical phenomena. The Péclet number is the ratio of transport by drift to transport by diffusion. When the Péclet number is large, drift dominates. The solute fluence rate from drift is $Cv,$ where $C$ is the concentration and $v$ the solvent speed. The solute fluence rate from diffusion is $D$ times the concentration gradient (roughly $C/L,$ where $L$ is some characteristic distance over which the concentration varies).

• (a) Determine an expression for the Péclet number in terms of $C$, $L$, $v$, and $D$.

• (b) Verify that the Péclet number is dimensionless.

• (c) Which parameter in Sect. 4.12 is equivalent to the Péclet number?

• (d) Estimate the Péclet number for oxygen for a person walking.

• (e) Estimate the Péclet number for a swimming bacterium. For more about the Péclet number, see Denny (1993) and Purcell (1977) .

The Péclet number is sometimes known as the Sherwood number .

Problem 44.

Extend Fick’s second law in one dimension $\partial C/\partial t=D\,(\partial ^{2}C/\partial x^{2})$ to include solvent drag.

Problem 45.

Use Eqs. 4.63 and 4.64 to derive Eq. 4.66.

Problem 46.

Expand $e^{x}=1+x+x^{2}/2!+x^{3}/3!$ to derive Eq. 4.67 from Eq. 4.66.

Problem 47.

Use a Taylor’s series expansion to show that $G(\xi )$in Eq. 4.69 is equal to $\xi /12$ for small $\xi$.

Problem 48.

Consider Eq. 4.63 with $C_{0}=0$ and $C_{0}^{\prime }=1$.

• (a) If $v>0$, write an equation for $C(x)$. Plot $C(x)$ for $0<x/x_{1}<1$ for two cases: $x_{1}\ll \lambda$ and $x_{1}\gg \lambda$. Interpret these results physically.

• (b) Repeat the analysis for $v<0$.

4.1.12 Section  4.14

Problem 49.

We can use the microscopic model of a random walk to derive important information about diffusion without ever using the binomial probability distribution. Let $x_{i}(n)$ be the position of the $i$th particle after $N$ steps of a random walk. Then

$$ x_{i}(n)=x_{i}(n)\pm\lambda, $$

where half the time you take the $+$ sign and half the time the $-$ sign. Then $\overline {x}(n),$ the value of $x$ averaged over $N$ particles, is

$$ \overline{x}(n)=\frac{1}{N}\sum_{i=1}^{N}x_{i}(n). $$

• (a) Show that $\overline {x}(n)=\overline {x}(n-1)$ so that on average the particles go nowhere.

• (b) Show that $\overline {x^{2}}(n)=\overline {x^{2}}(n-1)+\lambda ^{2}$. Use this result to show that $\overline {x^{2}}(n)=n\lambda ^{2}.$

For a detailed discussion of this approach, see Denny (1993).

Problem 50.

We can write the diffusion constant, $D,$ and the thermal speed, $v_{\text {rms}}$, in terms of the step size, $\lambda ,$ and the collision time, $t_{c}$, as

$$ \begin{aligned} D & =\frac{\lambda^{2}}{2t_{c}},\\ v_{\text{rms}} & =\frac{\lambda}{t_{c}}. \end{aligned} $$

Solve for $\lambda$ and $t_{c}$ in terms of $D$ and $v_{\text {rms}}.$

Problem 51.

Using the definitions in Prob. 224, write the diffusion constant in terms of $\lambda$ and $v_{\text {rms}}.$ By how much do you expect the diffusion constant for heavy water (water in which the two hydrogen atoms are deuterium, $^{2}$H) to differ from the diffusion constant for water? Assume the mean free path is independent of mass.

Problem 52.

Write a computer program to model a two-dimensional random walk. Make several repetitions of a random walk of 3600 steps and plot histograms of the displacements in the $x$ and $y$ directions and mean square displacement.

Problem 53.

Write a program to display the motion of 100 particles in two dimensions.

Problem 54.

Particles are released from a point between two perfectly absorbing plates located at $x=0$ and $x=1$. The particles random walk in one dimension until they strike a plate. Find the probability of being captured by the right-hand plate as a function of the position of release, $x$. (Hint: The probability is related to the diffusive fluence rate to the right-hand plate if the concentration is $C_{0}$ at $x$ and is 0 at $x=0$ and $x=1$.)

Problem 55.

The text considered a one-dimensional random-walk problem. Suppose that in two dimensions the walk can occur with equal probability along $+x$, $+y$, $-x$, or $-y$. The total number of steps is $N=N_{x}+N_{y}$, where the number of steps along each axis is not always equal to $N/2$.

• (a) What is the probability that $N_{x}$ of the $N$ steps are parallel to the $x$ axis?

• (b) What is the probability that the net displacement along the $x$ axis is $m_{x}\lambda$?

• (c) Show that the probability of a particle being at $(m_{x}\lambda ,m_{y}\lambda )$ after $N$ steps is

  $$ \begin{aligned} & P^{\prime}(m_{x},m_{y})=\\ &\sum_{N_{x}}\left( \frac{N!}{N_{x}!\,(N-N_{x})!}\right) \left( \frac{1}{2}\right) ^{N}P(m_{x},N_{x})\,P(m_{y},N-N_{x}), \end{aligned} $$

  where $P(m,N)$ on the right-hand side of this equation is given by Eq. 4.76.

• (d) The factor $N!/N_{x}!(N-N_{x})!$ is proportional to a binomial probability. What probability? Where does this factor peak when $N$ is large?

• (e) Using the above result, show that $P^{\prime }(m_{x},m_{y})=P(m_{x},N/2)\,P(m_{y},N/2).$

• (f) Write a Gaussian approximation for two-dimensional diffusion.