Question

Consider the one-dimensional diffusion of radioactive tracer atoms initially absent from a crystalline solid but deposited uniformly at time $t=0$ on the surface $x=0$ of the semi-infinite solid. The number of radioactive atoms deposited at $t=0$ is $C$ per unit surface area. Show that the concentration of radioactive atoms $n$ (number per unit volume) in the solid must satisfy the diffusion equation $$\frac{\partial n}{\partial t}=D\frac{{\partial }^{2}n}{\partial x^2}$$Equation $(7-79)$ can be obtained by first deriving the equation of conservation of tracer atoms $$\frac{\partial n}{\partial t}=-\frac{\partial J}{\partial x}$$ where we assume that tracer atoms diffuse in the $x$ direction only. The actual decay of the tracer atoms has been ignored in formulating the mass balance. Solve Equation $(7-79)$ subject to the initial and boundary conditions $$\begin{array}{ll}n(x,t=0)=0& (7-81)\\ \\ {\int }_0^{\mathrm{\infty }}n(x,t)dx=C.& (7-82)\end{array}$$ Show that $n(x,t)$ is given by $$n(x,t)=\frac{C}{(\pi Dt{)}^{1/2}}\exp \left(\frac{-x^2}{4Dt}\right)$$

Short answer

The solution is a Gaussian function meeting the diffusion equation and both initial and boundary conditions.

Step-by-step solution

Introduce the Diffusion Equation

We start with the diffusion equation given by$$\frac{\partial n}{\partial t}=D\frac{{\partial }^{2}n}{\partial x^2}.$$This partial differential equation describes how the concentration of radioactive atoms $n(x,t)$ changes over time in a crystalline solid due to diffusion, assuming a constant diffusion coefficient $D$.

Apply the Initial Condition

The initial condition states that at time $t=0$, the concentration $n(x,t)$ is zero everywhere within the solid:$$n(x,t=0)=0.$$This implies that initially, no radioactive atoms are present inside the solid.

Apply the Boundary Condition

The boundary condition ensures that the total number of atoms deposited per unit area on the surface is $C$. This translates mathematically to:$${\int }_0^{\mathrm{\infty }}n(x,t)dx=C.$$This condition must hold at all times $t>0$.

Assume a Form for the Solution

We assume a solution of the form$$n(x,t)=\frac{C}{(\pi Dt{)}^{1/2}}\exp \left(\frac{-x^2}{4Dt}\right).$$This form ensures the Gaussian distribution, which is typical for diffusion problems.

Verify the Assumed Solution

Calculate$$\frac{\partial n}{\partial t}$$ and $$\frac{{\partial }^{2}n}{\partial x^2}$$for the assumed solution to verify it satisfies the diffusion equation. This involves differentiating the Gaussian function with respect to $t$ and $x$. Checking the partial derivatives, we find:$$\frac{\partial n}{\partial t}=\frac{C{x}^2}{4(D{t}^2)(\pi Dt{)}^{1/2}}\exp \left(\frac{-x^2}{4Dt}\right)-\frac{C}{2t(\pi Dt{)}^{1/2}}\exp \left(\frac{-x^2}{4Dt}\right),$$$$\frac{{\partial }^{2}n}{\partial x^2}=\frac{C}{(\pi Dt{)}^{1/2}}(\frac{x^2}{4Dt}-\frac{1}{2D})\exp \left(\frac{-x^2}{4Dt}\right).$$

Validate the Solution with Conditions

Substitute the expression for $n(x,t)$ into the conservation condition:$${\int }_0^{\mathrm{\infty }}\frac{C}{(\pi Dt{)}^{1/2}}\exp \left(\frac{-x^2}{4Dt}\right)dx=C.$$The integral of a Gaussian over its entire domain is well-known to equal $C$, meeting the solution's requirement.Thus, the solution satisfies both the diffusion equation and the boundary conditions.

Key concepts

Radioactive Tracer

Radioactive tracers are substances containing radioactive atoms used to investigate processes in scientific experiments. In the context of diffusion in a solid, these tracers help us track the movement of atoms within a material. At time $t=0$, radioactive tracer atoms are introduced at the surface $x=0$ of a crystalline solid. This initial setup allows scientists to study how the atoms move into the solid over time.
By tracing this movement, we can understand diffusion—a key process affecting many physical, chemical, and biological systems. These tracers are invaluable because they provide easy-to-measure signals that can be tracked precisely to study material behaviors under various conditions.

• They allow precise measurement of diffusion rates.
• Supporting a wide range of fields: medicine, environmental science, and materials science.

Gaussian Distribution

A Gaussian distribution, also known as a normal distribution, is a mathematical function often encountered in statistics and natural processes. It is symmetrical and bell-shaped, characterized by its mean and variance. In diffusion problems, it helps describe how substances spread out over time.
For our diffusion equation problem, the concentration of radioactive atoms $n(x,t)$ is given by a Gaussian distribution. The solution of the form $n(x,t)=\frac{C}{(\pi Dt{)}^{1/2}}\exp \left(\frac{-x^2}{4Dt}\right)$ highlights how diffusion leads to a symmetric spread around the initial point of tracer deposition.

• The form of Gaussian distribution accounts for the randomness of diffusion.
• It reveals the atoms spread more as time $t$ increases, which is a typical diffusion behavior.

The Gaussian shape indicates how quickly or slowly the substance is spreading, making it crucial for understanding the diffusion process.

Initial and Boundary Conditions

Initial and boundary conditions are key to solving partial differential equations like the diffusion equation. They set the parameters for how the solution behaves at the start and how it interacts with limits or borders of the system. For this problem, these conditions specify distribution at time $t=0$ and ensure model realism.
The given initial condition is $n(x,t=0)=0$, indicating no radioactive atoms are present at time zero inside the solid. The boundary condition, ${\int }_0^{\mathrm{\infty }}n(x,t)dx=C$, ensures that over time, the total number of atoms remains constant and equal to $C$, deposited on the surface.

• Initial condition: sets starting state of the system (no atoms initially in solid).
• Boundary condition: maintains the total atoms amount (deposited amount $C$).

These conditions are foundational for finding meaningful solutions that align with physical realities of atom diffusion.

Concentration of Atoms

In diffusion problems, concentration of atoms $n(x,t)$ is vital in understanding how a substance spreads within a medium over time and space. It refers to the number of tracer atoms per unit volume at a given position $x$ and time $t$. This measurement helps quantify how evenly or unevenly atoms disperse across the solid.
The concentration is governed by the diffusion equation $\frac{\partial n}{\partial t}=D\frac{{\partial }^{2}n}{\partial x^2}$, which describes changes in concentration over time due to the random movement of particles. The solution accounts for how quickly atoms leave a region and enter another.

• Higher concentration: More atoms in a volume area.
• Depends on time $t$ and position $x$, changing as diffusion occurs.

Monitoring concentration profiles like $n(x,t)=\frac{C}{(\pi Dt{)}^{1/2}}\exp \left(\frac{-x^2}{4Dt}\right)$ provides insights into the diffusion speed and pattern, critical for both theoretical analysis and practical application.