# How does increasing the ionic strength of an aqueous solution decrease the activity coefficient?

Activity is a new concept for me, and I am having a little bit of trouble understanding it.

For the equation:

$$\ce{aA + bB<=>cC + dD}$$

$$K_c= \frac{[\ce{A}]^a[\ce{B}]^b}{[\ce{C}]^c[\ce{D}]^d}$$

As I understand it, this is not the true equilibrium constant expression for the chemical equation above. My textbook calls $K_c$ in this instance the "concentration quotient." The technically correct equilibrium constant expression is:

$$K=\frac{(a_\ce{A})^a(a_\ce{B})^b}{(a_\ce{C})^c(a_\ce{D})^d}$$

Where $K$, in this equation, is the "true" equilibrium constant determined by the activity, $a$, of each species in equilibrium with each other. The activity is represented by the equation: $a_\ce{A}=\gamma_\ce{A} [\ce{A}]$, where $\gamma$ is the activity coefficient.

My textbook Quantitative Chemical Analysis (Ninth Edition, by Daniel C. Harris), states that "the equilibrium 'constant' is not really constant." And then it goes into how increasing ionic strength increases the solubility of ionic compounds in aqueous solution. It then later states "the activity coefficients must decrease with increasing ionic strength."

This would seem to contradict the idea that increasing ionic strength increases the solubility of an ionic compound, would it not? If the ionic strength increases the solubility of an ionic compound, then wouldn't that mean that the concentrations of the ions that make up that ionic compound would increase, and thus the activity coefficient, therefore, must increase to show an increase in activity, otherwise known as the "effective concentration?"

Or is the activity independent of the ionic strength such that in order to keep the activity the same under higher concentrations, the activity coefficient must be smaller? This would then bring up the question: is the equilibrium constant, $K$, (determined by the activities of the components in the equilibrium expression) constant even under different ionic strengths, while the "concentration coefficient" (determined merely by the concentrations of the components in equilibrium with one another) does vary under different ionic strengths? I'm not sure I'm getting this very well.

Thermodynamic activity is essentially the same the chemical potential. At the solubility limit of a compound, the dissolved species is in equilibrium with the solid species. For example, a saturated solution of sodium chloride is in equilibrium with solid sodium chloride crystals. In such equilibria, the chemical potential of the dissolved sodium chloride is equal to the chemical potential of the solid sodium chloride crystals.

If the ionic strength increases the solubility of an ionic compound, then wouldn't that mean that the concentrations of the ions that make up that ionic compound would increase, and thus the activity coefficient, therefore, must increase to show an increase in activity, otherwise known as the "effective concentration?"

Let's assume that increasing the ionic strength doesn't alter the crystals of solid sodium chloride that dissolved sodium chloride is in equilibrium with. That means the chemical potential of the crystal is unchanged. At the solubility limit, the dissolved sodium chloride is still in equilibrium with the crystals, and as before, this equilibrium requires that the crystals have the same chemical potential (aka thermodynamic activity) of the dissolved species.

If there is more of the dissolved species (a higher concentation), the only way the activity can stay the same is if the activity coefficient goes down.

So you were right when you wrote:

...the activity [is] independent of the ionic strength such that in order to keep the activity the same under higher concentrations, the activity coefficient must be smaller...

This would then bring up the question: is the equilibrium constant, $K$, (determined by the activities of the components in the equilibrium expression) constant even under different ionic strengths, while the "concentration coefficient" (determined merely by the concentrations of the components in equilibrium with one another) does vary under different ionic strengths?

The equilibrium constant is a ratio of thermodynamic activities. The "concentration coefficient" can vary and will not be constant any time that activity coefficients are not 1.0. Ionic strength is one of many variables that can affect activity coefficients. So to conclude:

• Yes, the equilibrium constant $K$ is determined by the activities of the components in the equilibrium expression.
• The "concentration coefficient" $K_c$ can vary as a function of ionic strength (or other variables).
• So, the activity must remain the same even under different ionic strengths? – kmcmillan Feb 14 '16 at 20:47
• At the solubility limit, yes. So if the solubility of $\cf{MgSO4}$ under ionic strength A is 27 g/L, and the solubility under ionic strength B is 40 g/L, then (a) the activity of $\cf{MgSO4}$ at 27 g/L under condition A is the same as the activity of $\cf{MgSO4}$ at 40 g/L under condition B, and (b) the activity coefficients are obviously different. This is because the activity of crystalline magnesium sulfate does not change as in equilibrium with dissolved $\cf{MgSO4}$ under both conditions. – Curt F. Feb 14 '16 at 20:55
• Okay, so because the magnesium sulfate's activity would remain the same, the activities of the magnesium cation and the sulfate anion would also remain the same, meaning the equilibrium constant/solubility product constant (as determined by those activities) would also remain the same under different ionic strengths, right? – kmcmillan Feb 14 '16 at 21:25