PNG 550
Reactive Transport in the Subsurface

1.2 Primary and secondary species

PrintPrint

Here we will go through a few examples on how we choose primary and secondary species using the tableau method. Details are referred to in chapter 2, Morel and Hering, 1993. 

How do we categorize the primary and secondary species of the system? Here we go through several steps for the choice of primary and secondary species. 

  1. What are the species?

    List of species (5): H+, OH-, H2CO30, HCO3-, CO32-, (H2O is typically included implicitly).

     
  2. How many algebraic relationships do we have that define the dependence between activities of different species?
     

    We have 3 fast aqueous reactions, which means that we have 3 laws of mass action, as shown in three expressions defining the equilibrium constant for each reaction. 

  3. How many primary species do we have and what are they?

    Here we have 5 species in total and 3 dependencies (equations). So we should have 5-3 = 2 primary species. The primary species should be defined so that all other secondary species can be written in terms of primary species.
  • Can we choose H+ and OH- as primary species? No, because we cannot write the carbonate species as combinations of H+ and OH-,
  • Can we choose H+ and CO32-? Yes. See the following table. The species in the top row are primary species. The first left column includes all species.

    Species H+ CO32- (H2O)
    H+ 1< 0 0
    OH- -1 0 1
    H2CO30 2 1 >0
    HCO3- 1 1 0
    CO32- 0 1 0

We can write all species in terms of H+ and CO32-:

$$\begin{array}{l}
\mathrm{H}^{+}=\mathrm{H}^{+} \\
\mathrm{CO}_{3}^{2-}=\mathrm{CO}_{3}^{2-} \\
\mathrm{OH}^{-}=\left(\mathrm{H}_{2} \mathrm{O}\right)-\mathrm{H}^{+} \\
\mathrm{H}_{2} \mathrm{CO}_{3}=2 \mathrm{H}^{+}+\mathrm{CO}_{3}^{2-} \\
\mathrm{HCO}_{3}^{-}=\mathrm{H}^{+}+\mathrm{CO}_{3}^{2-}
\end{array}$$

Here ${OH}^{-}$, $\mathrm{H}_{2} \mathrm{CO}_{3}{}^{0}$, and ${HCO}_{3}^{-}$ are secondary species.

  1. Is the choice of primary species unique? No. As long as the secondary species can be written as combinations of primary species, the list is legitimate. For example, we can also use OH- and $\mathrm{HCO}_{3}$ as primary species in this example.

Species OH- HCO3- (H2O)
H+ -1 0 1
OH- 1 0 0
H2CO3 -1 1 1
HCO3- 0 1 0
CO32- 1 1 -1

We can then write all species in terms of primary species:

$$\begin{array}{l} \mathrm{H}^{+}=\left(\mathrm{H}_{2} \mathrm{O}\right)-\mathrm{OH}^{-} \\ \mathrm{OH}^{-}=\mathrm{OH}^{-} \\ \mathrm{H}_{2} \mathrm{CO}_{3}=\mathrm{HCO}_{3}^{-}+\mathrm{H}_{2} \mathrm{O}-\mathrm{OH}^{-} \\ \mathrm{HCO}_{3}^{-}=\mathrm{HCO}_{3}^{-} \\ \mathrm{CO}_{3}{ }^{2-}=\mathrm{HCO}_{3}^{-}-\mathrm{H}_{2} \mathrm{O}+\mathrm{OH}^{-} \end{array}$$

Similarly, (H+, HCO3-), (H+, H2CO3), (OH-, CO32-) are also legitimate choices for primary species. However (H2CO3, CO32-), (HCO3-, CO32-), are not. You can practice using these species to write the expression of secondary species.

Take home practice 1.1:

If we impose the charge balance condition in example 1, we will then an additional algebraic relationship: $$\mathrm{H}^{+}=\mathrm{OH}^{-}+\mathrm{HCO}_{3}^{-}+2 \mathrm{CO}_{3}^{2-}$$ In this case how many primary species will we have? How many secondary species? What are they?

Click here for the answer In example 1, we have 5 species and 3 reactions. Now we have 1 more relationship through this charge balance condition. This means that we have 4 algebraic relationships in total. In this case we will then have 1 = 5-4 degree of freedom so that as long as we specify 1 condition for the system, the system is defined.

If we add an additional species Ca2+ in the closed carbonate system in example 1, we then have the following reactions in addition to those in Example 1:

$$\begin{array}{l}
\mathrm{CaCO}_{3}^{0} \Leftrightarrow \mathrm{Ca}^{2+}+\mathrm{CO}_{3}^{2-} \\
\mathrm{CaHCO}_{3}^{+} \Leftrightarrow \mathrm{Ca}^{2+}+\mathrm{HCO}_{3}^{-} \\
\mathrm{CaOH}^{+} \Leftrightarrow \mathrm{Ca}^{2+}+\mathrm{OH}^{-}
\end{array}$$

  1. How many species do we have in total? 
  2. How many dependencies if we know the equilibrium constants of all these reactions above?
  3. What is the number of primary species?
  4. What are the primary and secondary species? How many different sets of primary species can you come up with?

Click here for the answer
  1. Now we have 9 species, with the additional 4 species of Ca2+, CaCO30, CaOH+, and CaHCO3+.
  2. Originally we have 3 dependencies. Now we have added 3 more dependencies through the above 3 reactions so we have 6 dependencies.
  3. The total number of primary species is 9 - 6 = 3
  4. An example is (Ca2+, H+, HCO3-). All other species can be written in terms of these three primary species. They are however not the only set of primary species.