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### Topic: Gibbs Free Energy - Two Phases  (Read 446 times)

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#### vegard85

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##### Gibbs Free Energy - Two Phases
« on: May 16, 2020, 05:42:24 PM »
Hi

What are the Gibbs free Energy of a system containing multiple species in both gas and liquid phase?

The specific system I'm trying to solve by minimizing the Gibbs free Energy are:
SO2(g)  SO2(aq)
SO2(aq) + H2O(l)  HSO-3(aq) + H+(aq)

CO2(g)  CO2(aq)
CO2(aq) + H2O(l)  HCO-3(aq) + H+(aq)

I don't want to solve it using Henry's Law & Equilibrium constants because I want to add constraints on some of the variables later on.

My best guess are the below function but I'm not sure if it's correct. (The pressure are assumed constant at 1atm)
$$\Delta T = T - 298.15$$
$$n_{gas} = n_{SO_{2(g)}} + n_{CO_{2(g)}}$$ $$n_{liq} = n_{SO_{2(aq)}} + n_{HSO^-_{3(aq)}} + n_{CO_{2(aq)}} + n_{HCO^-_{3(aq)}} + n_{H_2O_{(l)}} + n_{H^+_{(aq)}}$$
$$\mu_{SO_{2(g)}} = G_{f,SO_{2(g)}}^0 - S^0_{SO_{2(g)}} \Delta T + RT \ln \left\{ \frac{n_{SO_{2(g)}}}{n_{gas}} \right\}$$ $$\mu_{CO_{2(g)}} = G_{f,CO_{2(g)}}^0 - S^0_{CO_{2(g)}} \Delta T + RT \ln \left\{ \frac{n_{CO_{2(g)}}}{n_{gas}} \right\}$$ $$\mu_{HSO^-_{3(aq)}} = G_{f,HSO^-_{3(aq)}}^0 - S^0_{HSO^-_{3(aq)}} \Delta T + RT \ln \left\{ \frac{n_{HSO^-_{3(aq)}}}{n_{liq}} \right\}$$ etc...
$$G = \mu_{SO_{2(g)}} n_{SO_{2(g)}} + \mu_{CO_{2(g)}} n_{CO_{2(g)}} + \mu_{HSO^-_{3(aq)}} n_{HSO^-_{3(aq)}} + \mu_{HCO^-_{3(aq)}} n_{HCO^-_{3(aq)}} + \mu_{H^+_{(aq)}} n_{H^+_{(aq)}} + \mu_{H_2O{(l)}} n_{H_2O{(l)}}$$

#### vegard85

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##### Re: Gibbs Free Energy - Two Phases
« Reply #1 on: May 25, 2020, 01:49:25 PM »
After some digging I found out - made an error by using molar fractions everywhere

The standard potentials given in tables (Hf°,S°,Gf°) for all the species are not necessarily in molar fractions.
For solutes dissolved in water they are in molality
For solvent (H2O) it's in molar fraction
For gas it's in partial pressure

Ref: https://www.chem.wisc.edu/deptfiles/genchem/netorial/modules/thermodynamics/table.htm

This seems to work...
$$\mu_{SO_{2(g)}} = \mu^0_{SO_2(g)} - \bar{S}^0_{SO_{2(g)}} \Delta T + RT \ln p_{SO_{2(g)}}$$ $$\mu_{H^+_{(aq)}} = \mu^*_{H^+_{(aq)}} - \bar{S}^*_{H^+_{(aq)}} \Delta T + RT \ln m_{H^+_{(aq)}}$$ $$\mu_{H_2O_{(l)}} = \mu^0_{H_2O_{(l)}} - \bar{S}^0_{H_2O{(l)}} \Delta T + RT \ln x_{H_2O_{(l)}}$$