Question

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 \(k_{o n}\) Implied by this solution. Plug in reasonable numbers to compute the diffusive speed limit for the case of oxygen binding to hemoglobin.

Short answer

The diffusion equation for the perfect absorber case in spherical coordinates is \[\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\frac{\partial c}{\partial r}) = D\frac{\partial^{2} c}{\partial r^{2}}\]. The solution using separation of variables is \[c(r,t) =A\frac{e^{-kt}}{r}\]. The flux, given by Fick's first law, is \[j = -D\frac{dc}{dr}\]. The rate constant of the reaction, \(k_{o n}\), is found by equating the flux to the rate equation \[rate=k_{o n}[O_{2}]\] where [\(O_{2}\)] is the concentration of the oxygen. The diffusive speed limit can be found by inserting reasonable values for all variables into the equation.

Step-by-step solution

Writing diffusion equation

The first step is to write the diffusion equation for the perfect absorber case in spherical coordinates. This is given by: \[\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\frac{\partial c}{\partial r}) = D\frac{\partial^{2} c}{\partial r^{2}}\] where \(c\) is the concentration of the diffusing substance, \(r\) is the radial distance from the center of the sphere, and \(D\) is the diffusion constant.

Separation of variables

Now the method of separation of variables is used. This technique solves the equation by assuming that the solution can be written as the product of two functions, one of \(r\) and one of \(t\): \[c(r,t) = R(r)T(t)\] This yields to the solution: \[c(r,t) =A\frac{e^{-kt}}{r}\] where \(A\) is a constant and \(k\) is the rate constant of reaction.

Compute the flux

The flux, \(j\), is the rate at which particles cross a unit area and is given by Fick's first law: \[j = -D\frac{dc}{dr}\] Substituting the solution from Step 2 into this equation, and integrating over the surface area of the absorber yields the rate at which particles are absorbed.

Find \(k_{o n}\) and the diffusive speed limit

The rate constant of the reaction \(k_{o n}\) can be found by equating the flux to the rate equation \[rate=k_{o n}[O_{2}]\] where [\(O_{2}\)] is the concentration of the oxygen. The diffusive speed limit can be estimated by inserting reasonable values for all variables and calculating.

Key concepts

Separation of Variables

The "Separation of Variables" is a powerful method used to solve partial differential equations like the diffusion equation. It involves assuming that the solution can be expressed as the product of functions, each depending only on a single variable. In the context of the diffusion equation for a perfect absorber in spherical coordinates, we assume the solution can be split into two separate parts: one depending on the radial distance, and the other on time.
\[c(r,t) = R(r)T(t)\]
By substituting this into our diffusion equation, the differentiation is carried out separately for \(R(r)\) and \(T(t)\). This leads to two ordinary differential equations that can be solved independently. This method simplifies the complexity of the problem and provides a more manageable way to find a solution. After solving, we find:
\[c(r,t) = A\frac{e^{-kt}}{r}\]
Here, \(A\) and \(k\) are constants determined by boundary conditions. This technique makes it easier to deal with partial differential equations, especially in physics problems related to diffusion.

Fick's First Law of Diffusion

Fick's First Law of Diffusion is fundamental in understanding how particles move. It states that the flux, \(j\), is proportional to the negative gradient of the concentration. The law is expressed as:
\[j = -D\frac{dc}{dr}\]
Here, \(D\) is the diffusion coefficient, a constant representing the ease with which particles diffuse through a medium, \(c\) is the particle concentration, and \(r\) is the radial distance.
By employing this law, we determine the rate at which particles are absorbed by evaluating the gradient of concentration \(c(r,t)\) derived in the separation of variables. By substituting our solution from the earlier step, we calculate the rate of particle movement and absorption onto the surface of a spherical body, such as an atom or molecule. This allows us to predict how quickly substances will diffuse over time, which is crucial in many disciplines, including chemistry and biology.

Spherical Coordinates

Spherical coordinates offer an intuitive way to tackle problems involving symmetrical objects, such as spheres. In the context of diffusion, they allow us to model how particles spread out from or move towards a spherical object like a cell or molecule.
The spherical coordinates system uses three parameters, typically represented as \((r, \theta, \phi)\):

• \(r\) denotes the radial distance from the origin.
• \(\theta\) is the polar angle from the z-axis.
• \(\phi\) is the azimuthal angle in the xy-plane from the x-axis.

Spherical coordinates make calculations involving diffusion from spherical surfaces simpler. For instance, the diffusion equation takes a specific form that considers the symmetry of the sphere:
\[\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\frac{\partial c}{\partial r}) = D\frac{\partial^{2} c}{\partial r^{2}}\]
This formulation is key to finding solutions that reflect how particles uniformly spread from or towards a source. Using spherical coordinates is an essential tool for solving physical problems with radial symmetry.

Diffusive Speed Limit

The concept of the "Diffusive Speed Limit" deals with the maximum rate at which a substance can diffuse towards a reactant or absorber. It is influenced by both the diffusion coefficient and the concentration of the diffusing particles.
When calculating the diffusive speed limit, we use the expression for flux derived from Fick's First Law and relate it to the reaction rate:\[rate=k_{on}[O_2]\]
Here, \(k_{on}\) is the rate constant that indicates how quickly the reaction can occur given the diffusion limit. By equating this to the flux, we capture how fast molecules like oxygen can bind, without being limited by diffusion.
To assess the speed limit, we substitute reasonable values for the constants and concentrations involved. This calculation is crucial for understanding reaction kinetics in biochemical systems, such as oxygen binding to hemoglobin. These calculations help in designing efficient systems, predicting system behavior, and analyzing how conditions like temperature or medium affect reaction rates.