Water Conditioning & Purification Magazine

Water Softening—Part 2 of 2: The Fundamental Theory

By Peter Meyers

Summary: In Part 1, we discussed aspects of ion selectivity as they relate to water softening, including selectivity coefficients, apparent selectivity and selectivity reversal. In Part 2, we take up the mathematical relationships of concentration, valence and capacity, which underlie the theory of water softening.


Recapping from Part 1 of this article, water softening is efficient, both in the service cycle and regeneration cycle. Chemical processes that are efficient in both the forward and reverse directions are quite unusual. A phenomenon called selectivity reversal is responsible for efficient regeneration of water softeners.

Selectivity reversal is caused by changes in total dissolved solids (TDS) for ion exchange systems that involve ions of unequal valence (usually divalent to monovalent). Low TDS favors divalent ions such as calcium (Ca), while high TDS favors monovalent ions such as sodium (Na). When a softener is in service and the TDS is low, softening resins favor calcium over sodium by a wide margin. However, during regeneration, in the presence of high TDS brine, selectivity of the resin for calcium over sodium is significantly reduced—or even reversed. This results in an ion exchange process that’s efficient during both cycles.

Selectivity coefficients
Selectivity coefficients define the equilibrium relationship between two competing ions and a specific ion exchange resin. They can be used to determine the concentration of an ion in the water (or resin) under any particular condition. The selectivity equation looks exactly like the solubility product equation. The selectivity coefficient and the selectivity equation can be manipulated the same way as other mass action relationships, such as manipulation of solubility products used in solution chemistry.

In selectivity calculations, the molar concentrations of the ions are each taken to the power of the ion’s combining ratio. Combining ratios are determined from the balanced chemical equation. When both ions have the same valence, the combining ratio is 1-to-1 (or 2-to-2, etc). The powers cancel each other out and do not affect the calculation. The chemical equations shown (see Figure 1) are typical of ion exchange reactions between ions of equal valence.

For equations involving ions of unequal valence (see Figure 2), the powers aren’t the same and don’t cancel each other out. This causes the concentration units not to cancel each other out either. That’s the mathematical explanation of why concentration is so important in divalent to monovalent ion exchange relationships.

By convention, selectivity coefficients are depicted with the ion entering the resin (ion “a”) being shown above the ion being displaced from the resin (ion “b”). In the formal equation, all solution and resin concentrations are expressed as moles per liter, although equivalent fractions are often used to simplify the math. Resin concentration is described in the form in which it’s used—that’s wet volume capacity—and is manipulated as though it were a solution concentration. Strictly speaking, activity coefficients of the ions should be included. Most selectivity coefficients published over the years were calculated without consideration of activity coefficients (see Part 1, WCP, November 2000) and they won’t be included in this discussion.

Published selectivity coefficients for familiar ions and types of resins are shown in Table 1 and Table 2. Keep in mind that these were typically calculated from concentrated solutions. Their values are to some extent dependent on the conditions of the experiment and on the accuracy of the test methods. They are accurate over a wide range of conditions, including most potable water systems. They become progressively less accurate when concentrations are very high, although even here they can be very useful tools for predicting what will happen during regeneration.

Apparent selectivity
In the case of divalent to monovalent exchanges, the relative affinity varies proportionally to the solution concentration for any given resin. “Apparent selectivity” differs from the selectivity coefficient because it varies with concentration. The mathematical explanation is the squared factor in the expression of the monovalent ions concentration. As solution concentration increases, the resin increasingly favors the monovalent ion. In order to express this concept mathematically, we have chosen to use equivalent fractions, rather than molar concentrations. This allows simplification of the rather complicated selectivity formula and effect of concentration to be plainly seen.

Resin capacity
Did anyone notice that the resin concentration has an effect on apparent selectivity? Low solution concentrations favor divalent ions and high resin concentrations also favor divalent ions. Consequently, low resin concentrations are more efficient during regeneration. If a softening resin could be produced with a reduced capacity around half of a typical cation resin, the apparent selectivity could be made favorable both in the service exchange, and during regeneration. This would improve salt usage efficiency to almost 100 percent (vs. perhaps 50-to-60 percent in a typical water softener), albeit at the expense of making the softener twice as large.

* Note, do not confuse low cross-linked cation resins with resins that are only surface sulfonated. Here, the concentration of the portion of the resin that’s functionalized is similar to regular resins and the inert core does nothing to reduce selectivity.
It turns out that making a polystyrene resin with very low cross-linking is probably not practical to do. Many attempts have been made to reduce crosslinkage and/or sulfonation in order to reduce capacity (and cost). The result has been mixed. The finished product is physically and chemically unstable and prone to release of undesirable leachables. However, we can take advantage of our understanding to optimize the softener systems we’re currently using. If we make sure we control brine concentration well above 8 percent (preferably around 12-to-14 percent), this will go a long way toward making the softener operate as efficiently as the other design parameters allow.

Selectivity reversal
Before leaving the subject of selectivity reversal, it’s important we briefly consider anionic water softeners. Anionic softening is used for nitrate removal as well as for specific contaminants such as arsenic.

Believe it or not, the selectivity coefficient for sulfate vs. nitrate of a regular strong base anion resin is around 0.1 (meaning that nitrate is greatly preferred over sulfate). However, because sulfate is divalent, concentration is important. At 500 parts per million (ppm) TDS, the apparent selectivity for sulfate vs. nitrate is the selectivity coefficient (0.1) times the ratio of the resin capacity (around 1.4 Molar), divided by the solution concentration (around 0.01 Molar). This means the apparent selectivity for sulfate vs. nitrate is around 1.4! Here’s a case where selectivity reversal has indeed occurred. In fact, this is why regular anion resins will dump nitrate in favor of sulfate, when the TDS is low.

Due to selectivity reversal, the dumping phenomenon doesn’t occur if the TDS is above about 1,500 ppm. The so-called nitrate selective resins have advantages over ordinary anion exchange resins, not only because they prefer nitrate to all other ions but because they don’t prefer sulfate ions at low solution concentrations. The most common version (tri-ethylamine functionality) has a sulfate-to-nitrate selectivity coefficient of around 0.01; and thus, won’t dump nitrates over sulfates unless the TDS is very low (below approximately 150 ppm, this resin will still dump nitrates in favor of sulfates).

The selectivity coefficient for arsenate is similar to that of sulfate and both ions are divalent. Both sulfate and arsenate will begin to leak once all the chloride exchange sites on the resin have been used up. This means the operating capacity of an anion resin for arsenate must be based on the combined concentration of arsenate plus sulfate.

Arsenate dumping can also occur whenever the apparent selectivity favors chloride (at TDS greater than about 1,500 ppm). The subject of arsenic removal by ion exchange is a fairly hot topic. Whether or not you believe arsenic removal by ion exchange is practical, it should be understood that arsenic will dump off of anion exchange resins if the chemistry of the water favors dumping and is not closely monitored.

Conclusion
There’s never enough room in a short article to cover a subject thoroughly. As closing thoughts, here are a few practical suggestions based on selectivity reversal principles. Remember to keep the brine draw wide open so the eductor sucks properly and the softener will regenerate just fine. Yes, contact time is also important but if you can’t have both, concentration is far more important. The higher the TDS, the worse a softener works (high TDS causes lower capacity and higher leakage because the resin has less preference for hardness). Oh yes, remember to close the bypass when the softener is in service.

About the author
Peter Meyers is technical manager for ResinTech Inc., a resin manufacturer and distributor in Cherry Hill, N.J. He has nearly 30 years industry experience covering a wide range of applications from demineralizers, polishers and softeners to industrial process design. He also is a member of the WC&P Technical Review Committee. Meyers can be reached at (856) 354-1152, (856) 354-6165 (fax) or email: pmeyers@resintech.com

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