## Chemical Equilibrium: Chemical Activities of Substances

James Richard Fromm

The active mass or chemical activity of any chemical system, including any chemical reaction, is simply the product of the chemical activities of each of its component substances. The component substances are are the reactants or the products of a chemical reaction.

Example. The rate of the forward chemical reaction CaCO3 CaO + CO2 is then proportional to the chemical activity of CaCO3, while the rate of the reverse reaction, CaO + CO2 CaCO3 is proportional both to the activity of CaO and to the activity of CO2. These two rates must be equal if the reaction is at equilibrium.

Likewise, the driving force of the reaction of CaCO3 is directly proportional to the chemical activity of CaCO3 while the driving force of the reverse reaction is directly proportional both to the activity of CaO and to the activity of CO2. The two driving forces must be equal if the reaction is at equilibrium. In either case the condition at equilibrium can be written as ka(CaCO3) = k'a(CaO)a(CO2).

In the above equation the constants k and k' are the proportionality constants for the reaction in the forward and reverse directions. Since the quotient of two constants is a constant, grouping the activity terms on one side gives K = a(CaO)a(CO2)/a(CaCO3)

The same expression is obtained, in this example and in general, regardless of whether a rate approach or a driving force approach is used. This resulting constant K is called the equilibrium constant for the equilibrium, and is always written with the activities of the products (right side) in the numerator and the activities of the reactants (left side) in the denominator. If the equilibrium reaction were written in the reverse direction, as it could be, the value of the equilibrium constant would be the reciprocal of its value for the equilibrium written for its original direction. The form of any equilibrium constant is always completely specified by the stoichiometry of the equilibrium to which it refers.

Example. If the equilibrium is that of hydrogen and iodine with hydrogen iodide, written as

H2(g) + I2(g) HI(g) + HI(g), then K = a2(HI)/a(H2)a(I2) If the form

1/2 H2(g) + 1/2 I2(g) HI(g) is preferred then K = a(HI)/(the square root of)a(H2)(the square root of)a(I2)

The numeric value of the equilibrium constant always depends on the form of the equilibrium to which it refers. In the example above, the value of the first constant is the square of the value of the second. Also, the activity of a product or of a reactant appears in the equilibrium constant as many times as it appears in the equilibrium which that constant describes.

Example. The equilibrium 2NO2 N2O4 is described by the equilibrium constant

K = a(N2O4)/a2(NO2)

Any equilibrium however complex can be described by an equilibrium constant if its stoichiometry is known. For a reaction as complex as 6A + 7B + 8C 2D + 3E + 4F the constant would be

K = a2(D)a3(E)a4(F) /a6(A)a7(B)a8(C)

#### Approximations to Activity

Chemical activity can in some cases be measured directly, but it is often more convenient to take advantage of the fact that for many substances the active mass per unit volume is directly proportional to the actual amount of substance per unit volume. Actual amount of substance per unit volume is an easily measurable quantity. For solutions, the amount of substance per unit volume is the mol/m3 or more often the mol/dm3. The mol/dm3, or mol/liter, is simply the molar concentration M.

For gases, the ideal gas law pV = nRT is rearranged to p = nRT/V. This form of the ideal gas law shows that pressure p is directly proportional to amount of substance per unit volume n/V. As a consequence, pressure in convenient units (kPa, atm, or torr) is often used in place of the activity of gases.

For pure substances which are solids or liquids, the amount of substance is a constant number of moles/unit volume which is the quotient of the density of the substance and its molar mass (m/V divided by m/n is n/V). This is really just the reciprocal, or inverse, of the molar volume. The constant value for pure solids and liquids, and the values of RT> for gases, are usually incorporated into the numeric value of the equilibrium constant. The effect of this incorporation is that the activity of a pure solid or liquid appears as unity and the activity of a gas appears as its pressure.

#### Breakdown of Approximations to Activity

The approximations to activity of concentration, of pressure, and of constancy for pure substances will not accurately reflect the actual behavior of matter under all conditions. The approximation of pressure holds well under the same conditions as does the ideal gas law from which it was derived, when the pressure is low and the temperature high. Under other conditions more elaborate equations of state such as the van der Waals equation must be used. The approximation of activities by concentrations is no longer reasonably accurate for aqueous solutions whose concentration is greater than about one mol/L, and for many solutes significant differences are found at even lower concentrations. The assumption that the activity of pure substances is constant is very good unless their density is changed, as might happen at very high pressures, but this is uncommon.

When the chemical activity is no longer accurately described by these approximations but the approximations are still used, it appears that the constant has become a variable - its numeric value does not remain the same as the concentration or pressure are varied. Actually, the numeric value of the constant is still what is was before but the concentration or pressure are now so high that the approximation of active mass being proportional to actual moles/unit volume no longer is realistic. The molecules of gas or solute are now to some extent interfering with each other in reaction and some of the actual mass is not active mass. A variety of ways to deal with this problem have been used.

The simplest method, and the only one we will discuss, is to introduce a "finagle factor" to make the value of the constant under conditions of low pressure or concentration hold at higher values. This "finagle factor" is called the activity coefficient. The activity coefficient is that number which, when multiplied by the actual pressure or concentration, gives the activity of the substance as it is under conditions of low pressure or low concentration. The value of the activity coefficient of a gas or solute usually ranges downward from unity as the pressure or concentration is increased. For example, the mean activity coefficient of aqueous KCl decreases from 0.97 at 0.001 molar to 0.61 at 1.0 molar. A salt which has ions of greater charge, like aqueous ZnSO4, has a mean activity coefficient of 0.70 at 0.001 molar which decreases to 0.05 at 1.0 molar. In extremely concentrated aqueous solutions, activity coefficients sometimes even have values which are greater than one.