Basis swapping

Secondary species can also be swapped into the basis in place of basis species. This might be desirable if the original basis species becomes extremely small in concentration, causing numerical difficulties. For example, if the aluminum basis species is Al3+, it would be desirable to swap AKOII )J for Al3+ before beginning the calculation for most natural waters, because it probably exists in concentrations orders of magnitude greater than Al3+, and convergence will be helped. 

During the calculation, minerals may dissolve completely or precipitate, and therefore need to be swapped in or out of the basis by the program as the calculation proceeds. Bethke (1996) shows clearly how this is accomplished using the techniques of linear algebra. 

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Since we could not possibly store each possible variation on the basis, it is important for us to be able at any point in the calculation to adapt the basis to match the current system. It may be necessary to change the basis (make a basis swap, in modeling vernacular) for several reasons. This chapter describes how basis swaps can be accomplished in a computer model, and Chapter 11 shows how this technique can be applied to automatically balance chemical reactions and calculate equilibrium constants. 

The modeler first encounters basis swapping in setting up a model, when it may be necessary to swap the basis to constrain the calculation. The thermodynamic dataset contains reactions written in terms of a preset basis that includes water and certain aqueous species (Na+, Ca++, K+, Cl-, HCOJ, SO4-, H+, and so on) normally encountered in a chemical analysis. Some of the members of the original basis are likely to be appropriate for a calculation. When a mineral appears at equilibrium or a gas at known fugacity appears as a constraint, however, the modeler needs to swap the mineral or gas in question into the basis in place of one of these species. 

The third step in changing the basis is to set the equilibrium constants for the revised reactions. The new equilibrium constant K j for a species reaction can be found from its value Kj before the basis swap according to... 

Conveniently, perhaps even miraculously, the equations developed in Chapter 5 to accomplish basis swaps can be used to balance chemical reactions automatically. Once the equations have been coded into a computer program, there is no need to balance reactions, compute equilibrium constants, or even determine equilibrium equations by hand. Instead, these procedures can be performed quickly and reliably on a small computer. 

To further illustrate how the basis-swapping algorithm can be used to balance reactions, we consider several ways to represent the dissolution reaction of pyrite, FeS2. Using the program RXN, we retrieve the reaction for pyrite as written in the llnl database... 

Note particularly the method for basis swapping. If for example we had wanted to have MW-12 saturated with quartz, rather than it having a silica concentration of 8.4 mg L-1, and we wanted to control the CO2 fugacity to 10-2bar, we would swap quartz for the basis species SiC>2 (aq), and CO2 (g) for the basis species HCOJ. In EQ3/6 this is done by entries in the basis switch/constraint column, as follows. It is a vital part of geochemical modeling to know how to perform this basis swapping, and to understand how and when to do it. 

A constant maturity swap, or CMS, is a basis swap in which one leg is reset periodically not to LIBOR or some other money market rate but to a long-term rate, such as the current 5-year swap rate or 5-year government bond rate. For example, the counterparties to a CMS might exchange 6-month LIBOR for the 10-year Treasury rate in eflFect on the reset date. In the U.S. market, a swap one of whose legs is reset to a government bond is referred to as a constant maturity Treasury, or CMT, swap. The other leg is usually tied to LIBOR, but may be fixed or use a different long-term rate as its reference. 

A differential swap is a basis swap in which one of the legs is calculated in a diflFerent currency. Typically, this leg is linked to a reference index rate for another currency but is denominated in the domestic currency. For example, one party might pay 6-month sterling LIBOR, in sterling, on a notional principal of 10 million and receive euro-LIBOR minus a margin, in sterling, on the same notional principal. Differential swaps are not very common and are the most difficult for a bank to hedge. 

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