Hi, I've spent days trying to solve some equations in a paper (referenced below) that describes it as a "straightforward, albeit lengthy integration," but I can't work out the "straightforward" bit. The notation is also odd, which doesn't seem to help my problem. Perhaps someone could help?(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to solve for the function [tex]\beta^{(0)}[/tex], which is a function of T and P (though P is held constant). Ultimately, I need a value of [tex]\beta^{(0)}[/tex] at a specific temperature T.

I am given in the paper:

[tex]\beta^{(0)L}=(\frac{\partial\beta^{(0)}}{\partial T})_{P}[/tex]

[tex]\beta^{(0)J}=(\frac{\partial\beta^{(0)L}}{\partial T})_{P}+(2/T)\beta^{(0)L}[/tex]

Also given,

[tex]\beta^{(0)J}=6U_{5}+\frac{2U_{6}}{T}+\frac{U_{7}}{T^{2}}+\frac{526U_{8}}{T(T-263)^{3}}[/tex]

where the U values are empirical constants.

I (think) I managed to solve the following expression for [tex]\beta^{(0)}[/tex]:

[tex]\beta^{(0)}=\int\beta^{(0)J}\frac{T}{1+2ln(T)}\partial T[/tex]

but I cannot figure out how to integrate that expression.

Thus, I also tried to express the functions as a PDE:

[tex]\beta^{(0)J}=(\frac{\partial^{2}\beta^{(0)}}{\partial T^{2}})_{P}+\frac{2}{T}(\frac{\partial\beta^{(0)}}{\partial T})_{P}[/tex]

and then substitute the U-series empirical expression of [tex]\beta^{(0)J}[/tex] and subtract it from each side. However, I am as at much of a loss to solve that expression for [tex]\beta^{(0)}[/tex] as the integral expression above.

Can anyone help me out at all? These equations are important to work out some geochemical thermodynamics I need to set up an experiment.

Reference:

Rogers, P. S. Z. and Pitzer, K. S., High-Temperature Thermodynamic Properties of Aqueous Sodium-Sulfate Solutions.Journal of Physical Chemistry85(20), 2886 (1981).

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# 2nd order PDE (I think?)

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