.. _example-phreeqc-manual-22:

22 - Gas Solubilities
=====================

CO₂ dissolving in water at pressures up to hundreds of atmospheres, which is
where the usual assumptions about gases stop working.

At ordinary pressure a gas is near enough ideal and its solubility is
proportional to its partial pressure -- Henry's law. Neither holds at the
pressures found in a deep saline formation, and this example is about the
corrections that take their place.

The practical context is CO₂ storage: how much carbon dioxide a formation can
hold is a solubility question at exactly these pressures.


Fugacity, not pressure
----------------------

.. raw:: html
   :file: study_22 - Gas Solubilities.html

A real gas under compression is not ideal: the molecules occupy space and
attract each other, so its effective pressure -- its **fugacity** -- departs
from the measured one, and it is the fugacity that drives dissolution.

PHREEQC uses a Peng-Robinson equation of state to compute it, which is why
these calculations need a database carrying the relevant parameters.


The results
-----------

.. figure:: Pressure.svg
   :alt: Dissolved CO2 against total pressure at four temperatures from 25 to 100 C
   :align: center

   CO₂ in solution against pressure, at 25, 50, 75 and 100 °C.

   Each curve starts straight -- that is Henry's law -- and then bends over.
   The departure is the non-ideality: doubling the pressure well above a
   hundred atmospheres does not double the amount dissolved, so a Henry's-law
   estimate overpredicts storage capacity, and increasingly so with depth.

   Solubility also **falls with temperature** across the whole range, as it
   does for most gases. Depth therefore pushes two ways: more pressure helps,
   more heat does not.

.. figure:: P_minus_Vm.svg
   :alt: Dissolved CO2 against the molar volume of the gas at four temperatures
   :align: center

   The same results against the gas's molar volume. This is the
   non-ideality shown directly -- for an ideal gas the molar volume would be
   fixed by pressure and temperature alone, and the spread between these
   curves is the equation of state doing its work.


What it shows
-------------

That a model is only valid over the range its thermodynamics were built for,
and that the way to know is to take it to the edge of that range and look.

:ref:`example-phreeqc-manual-02` makes the same point by comparing against
measured solubilities; this one makes it by changing the physics the
calculation rests on.


Source
------

* Parkhurst, D. L. and Appelo, C. A. J. (2013). *Description of input and
  examples for PHREEQC version 3.* U.S. Geological Survey Techniques and
  Methods, book 6, chapter A43. This is Example 22 of that manual.
* Peng, D.-Y. and Robinson, D. B. (1976). *A new two-constant equation of
  state.* Industrial & Engineering Chemistry Fundamentals 15, 59-64.