Chemical diffusion derivation from thermodynamic factor and self-diffusion

In summary, chemical diffusion is the movement of molecules from an area of high concentration to an area of low concentration, driven by random particle motion. It is related to the thermodynamic factor, which measures the deviation of a solution's behavior from an ideal solution, and can be affected by non-ideal interactions between molecules. Self-diffusion, or the movement of molecules of the same substance, is also important in understanding chemical diffusion. The derivation of chemical diffusion involves applying Fick's laws to a non-ideal solution, taking into account both the thermodynamic factor and self-diffusion. Factors such as temperature, concentration gradient, molecular weight, and the presence of other solutes, as well as solvent and solute-solvent interactions
  • #1
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I am working through a paper in which the chemical diffusion coefficient is related to the self-diffusion (or tracer) through the following derivation, on which I am a little hung up.

AB2 metal.

Overall diffusion D= XADB+XBDA

meaning the chemical diffusion of B is:

DCB=XADB=XADTB(1+∂lnγ/dlnXB)

and the chemical potential of B is:

μA00.5B2+RTlnγBxB00.5B2+RTlnp0.5B2

finally the mole fractions of XB and XA vary accordingly:

XB = (2+x)/(3+x)

XA = 1/(3+x)

and then DCB is:

DCB=DTB(2+x/2) ∂ln(pB2)/dx

I am confused as to where the (2+x)/2 term comes from.

I believe it is from the mole fraction XA in the overall diffusion equation combined with a derivation including the mole fraction XB from the chemical potential, but can't work it out exactly.

Thank you in advance for any clarification on this topic!
 
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  • #2


Thank you for your question. I am happy to provide some clarification on the derivation you are working through.

The (2+x)/2 term comes from the mole fractions of XB and XA in the chemical diffusion equation. Let's break it down step by step:

1. The overall diffusion equation for AB2 metal is D = XADB + XBDA, where D is the overall diffusion coefficient, XA and XB are the mole fractions of A and B, and DA and DB are the self-diffusion coefficients of A and B, respectively.

2. We can rewrite the equation as D = DTB(2+X)/2, where DTB is the tracer diffusion coefficient of B.

3. To understand where the (2+X)/2 term comes from, we need to look at the mole fractions of A and B. From the given information, we know that XB = (2+X)/(3+X) and XA = 1/(3+X).

4. The mole fraction of B, XB, is multiplied by the self-diffusion coefficient of A, DA, in the overall diffusion equation. This gives us (2+X)/(3+X) * DA.

5. Similarly, the mole fraction of A, XA, is multiplied by the self-diffusion coefficient of B, DB, in the overall diffusion equation. This gives us 1/(3+X) * DB.

6. Combining these two terms, we get (2+X)/(3+X) * DA + 1/(3+X) * DB. Simplifying this expression gives us (2+X+1)/(3+X) * DA = (3+X)/(3+X) * DA = (2+X)/2 * DA.

7. Therefore, the (2+X)/2 term comes from the combination of the mole fractions of A and B in the overall diffusion equation.

I hope this helps clarify the derivation for you. Please let me know if you have any further questions or need any additional clarification. Best of luck with your research!
 

1. What is chemical diffusion?

Chemical diffusion is the process by which molecules and particles move from an area of high concentration to an area of low concentration, driven by the random motion of particles. It is a fundamental process in chemistry that plays a crucial role in many natural and industrial processes.

2. How is chemical diffusion related to thermodynamic factor?

The thermodynamic factor is a measure of the deviation of a solution's behavior from that of an ideal solution. In the context of chemical diffusion, it is used to describe the effect of non-ideal interactions between molecules on the diffusion process. A lower thermodynamic factor indicates stronger interactions and slower diffusion.

3. What is self-diffusion?

Self-diffusion is a type of chemical diffusion in which molecules of the same substance move from one location to another. It is also known as homogenous diffusion and is important in understanding the behavior of pure substances.

4. How is chemical diffusion derived from thermodynamic factor and self-diffusion?

The derivation of chemical diffusion from thermodynamic factor and self-diffusion involves applying Fick's laws of diffusion to a non-ideal solution. This results in an expression for the diffusion coefficient that takes into account the effects of both the thermodynamic factor and self-diffusion. This equation can then be used to calculate the rate of diffusion in a non-ideal solution.

5. What factors can affect chemical diffusion?

Several factors can affect chemical diffusion, including temperature, concentration gradient, molecular weight, and the presence of other solutes. Additionally, the type of solvent and the nature of the solute-solvent interactions can also influence the rate of diffusion. These factors should be considered when studying and predicting diffusion behavior in a given system.

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