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How is pH calculated in MSE when there is no H+ ion?

Table of Contents

Overview

Definitions of symbols, superscripts, and subscripts

The calculation of pH using the OLI software

The AQ thermodynamic framework (using OLI Studio: Stream Analyzer Version 12.0)

The MSE thermodynamic framework (using OLI Studio: Stream Analyzer Version 12.0)

Conclusion

Overview

pH is an important reported value in the OLI software. Traditionally the definition of pH is:

pH = -log [H+]

Where [H+] is the concentration of the hydrogen ion (in mole/kg H2O).

A more strict definition of pH leverages the activity of the hydrogen ion:

pH = -log (aH+) = -log (mH+γH+m,∞)

In OLI's Mixed-Solvent Electrolyte (MSE) thermodynamic framework, the hydrogen ion does not exist. Instead, the MSE framework uses the hydronium ion (H3O+). 

This raises the question: how does OLI calculate pH when there is no hydrogen ion in the MSE model?

(For further technical details, please see the following publication: Modeling acid–base equilibria and phase behavior in mixed-solvent electrolyte systems. Fluid Phase Equilibria, 256, 34-41)

Definitions of symbols, superscripts, and subscripts

To start, we will define the relevant symbols:

γjm,∞ activity coefficient of species j, on the basis of molality (mole/kg H2O) and infinite dilution reference state (asymmetrical), i.e., γjm,∞→ 1 as mj → 0.0

γjX,∞ activity coefficient of species j, on the basis of mole fraction and infinite dilution reference state (symmetrical), i.e., γjX,∞→ 1 as Xj → 1

γjX activity coefficient of species j, on the basis of mole fraction and the fused salt reference state (symmetrical), i.e., γjX→ 1 as Xj → 1

Xw The mole fraction of water

XH+ The mole fraction of the hydrogen ion

aH+ The activity of the hydrogen ion

Mw The molecular weight of water, 18.0154 g/mole

 

- Infinite dilution in water reference state

m - Molality basis

X - Mole fraction basis

The calculation of pH using the OLI software

The AQ thermodynamic framework (using OLI Studio: Stream Analyzer Version 12.0)

This simulation features a sample at 25⁰C, 1.0 atmospheres, 55.5082 moles of H2O and 0.0001 moles of HCl. The AQ thermodynamic framework was selected.

After running the simulation, the calculated pH with the AQ framework was 4.00503.

Let's understand how OLI calculated this using the mole fraction basis, which is the default internal unit basis.

In the calculation Output tab, we find the following output species and thermodynamic values (these values can be displayed by following the steps in this article).

First, we convert the mole fraction (x-based) activity coefficient for the hydrogen ion into a molal (m-based) activity coefficient.

aH+m = aH+x * (1000/Mw)

Since aH+X = γH+x,∞ * XH+

Using the values from the Output tab, we can determine the activity of the hydrogen ion on the
m-based scale.

aH+X = 0.988482 * 1.80153E-06 = 1.78078E-06

aH+m = 1.78078E-06 * (1000/18.0154) = 9.88477E-05

Finally, pH = -log(aH+m ) = 4.00503

The MSE thermodynamic framework (using OLI Studio: Stream Analyzer Version 12.0)

Following the same procedure as the above example, we ran the simulation using the MSE thermodynamic framework. The Output results are as follows:

The calculation steps we followed above required an apparent concentration of the hydrogen ion, which does not exist in the MSE framework.  However, the following relationship must be true:

H3O+ = H+ + H2O

At equilibrium, the activities on both sides of the chemical equation must be equal. Therefore, we can write the x-based relationship:

aH3O+x = aH+x * aH2Ox

We can rearrange this equation to solve for the activity of the hydrogen ion:

aH+x= aH3O+x / aH2Ox

We already know that aH+m = aH+x * (1000/Mw)

We can use this expression to arrive at:

aH+m = aH3O+x / aH2Ox * (1000/Mw)

We can now take the values from the MSE calculation Output tab and enter into the equation:

pH = -log (aH+m) = -log (aH3O+x / aH2Ox) - log(1000/Mw)

Where:

 aH3O+x = γH3O+x * XH3O+= (0.988515)(1.80153E-06) = 1.78083E-06

 aH2Ox = γH2Ox * XH2O = (1.000)(0.999996) = 0.999996

pH = -log (aH+m) = - log(1.78083E-06 / 0.999996) - log (55.509) = 4.00501

 

Conclusion

You can determine the pH of a solution when there is no hydrogen ion but you also have hydronium ion.

 

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