Diffusion equation in 1D

In summary, the solution to the diffusion equation in 1D can be represented by n'(x,t) = N/sqrt(4piDt) * exp(-x^2/4DT), with n'(x,t) being the concentration of particles at position x and time t, N being the total number of particles, and D being the diffusion coefficient. When finding the expression for the number of particles in a slab of thickness dx at position x, the integral of the function between x and x+dx is simply f(x) dx.
  • #1
poiuy
11
0
The solution to the diffusion equation in 1D may be written as follows:
n'(x,t) = N/sqrt(4piDt) * exp(-x^2/4DT)

where n'(x,t) is the concentration of the particles at position x at time t, N is the total number of particles and D is the diffusion coefficient.

Write down an expression for the number of particles in a slab of thickness dx located at position x.

I assumed it would be the integral of the function between x and x+dx with respect to x.

However exp(-x^2/4Dt) can't be integrated between these values. I have a standard integral for exp(-ax^2) which is 0.5 sqrt (pi/a) but this only applies to integrating between zero and infinity.If anybody could point me in the right direction it would be greatly appreciated, I think I am missing something obvious here and this is a really simple question.

Thanks
 
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  • #2
poiuy said:
The solution to the diffusion equation in 1D may be written as follows:



n'(x,t) = N/sqrt(4piDt) * exp(-x^2/4DT)

where n'(x,t) is the concentration of the particles at position x at time t, N is the total number of particles and D is the diffusion coefficient.

Write down an expression for the number of particles in a slab of thickness dx located at position x.


I assumed it would be the integral of the function between x and x+dx with respect to x.

However exp(-x^2/4Dt) can't be integrated between these values. I have a standard integral for exp(-ax^2) which is 0.5 sqrt (pi/a) but this only applies to integrating between zero and infinity.


If anybody could point me in the right direction it would be greatly appreciated, I think I am missing something obvious here and this is a really simple question.

Thanks
Welcome to the forums!

Note that the integrale of any function f(x) between x and x +dx is simply f(x) dx!

[tex] \int_x^{x+dx} f(x') dx' \approx f(x) dx [/tex]
 
  • #3
Wow thanks, incredible that I could have had 14 years of education and never been taught that, thanks very much.
 
  • #4
Actually thinking about it, it's incredible that I couldn't work that out for myself.
 

What is the diffusion equation in 1D?

The diffusion equation in 1D is a mathematical model that describes the movement of particles from an area of high concentration to an area of low concentration in a one-dimensional system. It takes into account factors such as diffusion coefficient, initial concentration, and boundary conditions to predict the behavior of the system over time.

What is the significance of the diffusion equation in 1D?

The diffusion equation in 1D is used in many fields of science and engineering, including physics, chemistry, biology, and materials science. It allows us to understand and predict the behavior of diffusion processes in various systems, such as diffusion of molecules in a liquid, heat transfer in a solid, or the spread of pollutants in the environment.

What are the assumptions made in the diffusion equation in 1D?

The diffusion equation in 1D assumes that the particles are moving in a one-dimensional space and that the diffusion coefficient is constant throughout the system. It also assumes that there are no external forces acting on the particles and that the concentration of particles at the boundaries of the system remains constant over time.

How is the diffusion equation in 1D solved?

The diffusion equation in 1D can be solved using various numerical and analytical methods. The most common approach is to use the method of separation of variables, where the equation is split into two simpler equations that can be solved separately. Other methods include finite difference methods, Green's function method, and Laplace transform method.

What are some applications of the diffusion equation in 1D?

The diffusion equation in 1D has many practical applications. It is used in the design of chemical reactors, in the study of mass transfer in fluids, in the analysis of drug delivery systems, and in the modeling of diffusion processes in biological systems. It is also used in materials science to study the diffusion of atoms in solids and the formation of thin films.

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