Chemical diffusion derivation from thermodynamic factor and self-diffusion

[DiffusConfuse]

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= X_AD_B+X_BD_A

meaning the chemical diffusion of B is:

DC_B=X_AD_B=X_ADT_B(1+∂lnγ/dlnX_B)

and the chemical potential of B is:

μ_A=μ⁰_0.5B2+RTlnγ_Bx_B=μ⁰_0.5B2+RTlnp^0.5_B2

finally the mole fractions of X_B and X_A vary accordingly:

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

X_A = 1/(3+x)

and then DC_B is:

DC_B=DT_B(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 X_A in the overall diffusion equation combined with a derivation including the mole fraction X_B from the chemical potential, but can't work it out exactly.

Thank you in advance for any clarification on this topic!

 

[eljose79]

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!