In the chapter, we solved the problem of diffusion to...

In the chapter, we solved the problem of diffusion to capture using physical arguments to bypass explicitly solving the diffusion equation. In this problem, we do the math. (a) Write the diffusion equation for the perfect-absorber case in spherical coordinates. Use the method of separation of variables and reproduce the solution given in the chapter. (b) Use the flux to compute the number of molecules absorbed per unit time and find the corresponding $k_{\text {on }}$ implied by this solution. Plug in reasonable numbers to compute the diffusive speed limit for the case of oxygen binding to hemoglobin.

In the chapter, we solved the problem of diffusion to capture using physical arguments to bypass explicitly solving the diffusion equation. In this problem, we do the math.
(a) Write the diffusion equation for the perfect-absorber case in spherical coordinates. Use the method of separation of variables and reproduce the solution given in the chapter.
(b) Use the flux to compute the number of molecules absorbed per unit time and find the corresponding $k_{\text {on }}$ implied by this solution. Plug in reasonable numbers to compute the diffusive speed limit for the case of oxygen binding to hemoglobin.

Key Concepts

Fick’s Law and Flux Calculation

Fick’s law relates the diffusive flux of particles to the gradient of the concentration. In the context of the spherical absorber, the flux is computed as the derivative of the concentration profile at the absorber’s surface. Integrating the flux over the absorber’s surface yields the total number of molecules absorbed per unit time. This flux calculation is essential for linking the microscopic picture of diffusion with macroscopic reaction rates.

Boundary Conditions for a Perfect Absorber

In problems involving a perfect absorber, the boundary condition typically requires that the concentration of diffusing molecules goes to zero at the surface of the absorber. This condition is crucial since it models the absorber as an ideal sink, meaning that any molecule reaching the surface is immediately absorbed and does not contribute to the concentration in the system. These conditions directly influence the solution of the diffusion equation and dictate the form of the separation constants.

Diffusive Speed Limit and k_on

The diffusive speed limit defines the maximum rate at which molecules can be delivered to a reactive surface solely by diffusion. By calculating the flux and thereby the number of molecules absorbed per unit time, one can derive an expression for the rate constant k_on. This limit, sometimes referred to as the diffusion-limited rate, sets an upper bound on reaction rates for processes like oxygen binding to hemoglobin, and is quantified by plugging in reasonable physical parameters such as diffusion coefficients and molecular sizes.

Diffusion Equation in Spherical Coordinates

The diffusion equation describes how the concentration of diffusing particles changes over time due to random motion. In spherical coordinates, the equation accounts for the radial geometry of the problem and includes terms that reflect how the concentration gradient varies with respect to the radial distance. This form of the equation is particularly useful when dealing with systems that are radially symmetric, as in the case of a spherical absorber or emitter.

Separation of Variables Method

Separation of variables is a mathematical technique used to solve partial differential equations by assuming that the solution can be expressed as a product of functions, each dependent on a single coordinate. In the context of the diffusion equation, this method transforms a complex, multivariable problem into simpler, single-variable ordinary differential equations. Each of these can then be solved individually and combined to form the complete solution.

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