3.2 Law of Mass Action

3.2 Law of Mass Action#

Reaction quotient#

To measure chemical equilibrium, we need to define the concept of the reaction quotient (\(Q_c\)). For any balanced chemical reaction, this term signifies the ratio of the concentrations of products in the numerator to the concentrations of reactants in the denominator, with each concentration elevated to the power corresponding to its stoichiometric coefficient.

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

(22)#\[Q_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]

In equation (22), square brackets “\([ \, ]\)” are used to indicate molar concentrations (\(\pu{mol L-1}\) or \(\pu{M}\)) of a chemical constituent.

The Law of Mass Action and \(K\)#

Extensive studies and observations have determined that \(Q_c\) had the same value at equilibrium at constant temperature, regardless of the initial concentrations. Therefore, \(Q_c\) becomes the equilibrium constant for a system at equilibrium, \(K_c\).

(23)#\[Q_c = \frac{[C]^c[D]^d}{[A]^a[B]^b} = K_c\]

Equation (23) is the law of mass action. This expression can be simplified as follows:

(24)#\[K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]

This expression is also called the equilibrium expression. If the equilibrium concentrations of all reactants and products are known in a given chemical reaction, we can calculate \(K_c\).

Gas phase concentrations#

In reality, gas concentrations are often expressed in partial pressures or as percent of the composition of Earth’s atmosphere. Therefore, concentrations of gases are commonly expressed in units of atmosphere or \(\pu{atm}\). Gas concentrations can be expressed in equations as \(P\) (partial pressure).

Using the ideal gas law equation, we can also convert gas concentrations from \(\pu{mol L-1}\) to \(\pu{atm}\).

(25)#\[PV = n R T\]

where, \(P\) is pressure in \(\pu{atm}\), \(V\) is volume in \(\pu{L}\), \(n\) is number of moles, \(T\) is temperature in \(\pu{K}\), and \(R\) is gas law constant and has the value of \(\pu{0.0821 L atm mol-1 K-1}\) or \(\pu{8.314 J mol-1 K}\) at STP. Note that the ideal gas law is derived from a combination of Boyle’s, Charles’, and Avogadro’s laws. To convert gas concentrations from \(\pu{mol L-1}\) to \(\pu{atm}\), rearrange Eq. (25) as follows:

(26)#\[\frac{n\ (\pu{mol})}{V\ (\pu{L})} = \frac{P\ (\pu{atm}) }{R\ (\pu{atm L mol-1 K-1}) \times T\ (\pu{K})}\]

The ABC’s of gas: Avogadro, Boyle, Charles - TED-Ed

The equilibrium expression can be rewritten for reactions that involve gases only as

(27)#\[K_p = \frac{{P_C}^c {P_D}^d}{{P_A}^a {P_B}^b}\]

The subscript, \(p\), in \(K_p\) refers to gas concentrations expressed in \(\pu{atm}\), while the subscript, \(c\), in \(K_c\) refers to gas concentrations expressed in \(\pu{mol L-1}\). Because gas concentrations are interchangeable in units, it follows that \(K_p\) and \(K_c\) are correlated as follows:

(28)#\[K_p = K_c (RT)^{\Delta n}\]

where \(n\) is the number of moles of a chemical species in a reaction and \(\Delta n = \sum \text{Total}\, n\, \text{of products} - \sum \text{Total}\, n\, \text{of reactants}\).

\(K\) and \(\Delta_{rxn}G\)#

As seen in Eq. (24), \(K\) indicates a ratio of product concentrations over reactant concentrations at equilibrium. The magnitude of this ratio (\(K\)) can be interpreted mathematically to deduce relative concentrations of reactants or the products at equilibrium in a chemical reaction. When \(K\) is very large, we can expect the numerator (product concentrations) to be very large relative to the denominator (reactant concentrations) and vice versa. In other words, we can interpret a high \(K\) value to indicate the products in a reaction to being more favored and that the reaction proceeds spontaneously as written (to the right). Likewise, a low \(K\) can mean a spontaneous reaction to the left. As a rule of thumb, a \(K\) value of \(\ge 10^2\) is considered high, and a value of \(\le 10^{−2}\) is considered low.

Table 9 Magnitude of \(K\) and significance in reactions.#
\(K\) value	Reaction Direction	Spontaneity
\(K \gg 1\)	Products are favored	Spontaneous to right
\(K \ll 1\)	Reactants are favored	Spontaneous to left

Influence of \(K\) on the direction of reactions.#

Since the magnitude of \(K\) is related to the reaction’s spontaneity, \(K\) is directly related to \(\Delta_{rxn}G^\circ\). When \(K \gg 1\), \(\Delta_{rxn}G^\circ\) is negative and when \(K \ll 1\), \(\Delta_{rxn}G^\circ\) is positive. Therefore, we can also use \(\Delta_{rxn}G^\circ\) to determine \(K\). In the hypothetical reaction,

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

\(Q_C\) can be written as shown in Eq. (22). Using thermodynamic data, \(\Delta_{rxn}G^\circ\) can be determined. In nonstandard conditions, it can shown from thermodynamic principles that

\[\Delta_{rxn}G^\circ = \Delta_{rxn}G^\circ{} + \mathrm{R}T\ln Q_c\]

Under equilibrium conditions, \(\Delta_{rxn}G^\circ{}=0\) and \(Q_c=K_c\), therefore the above equation can be simplified as follows:

(29)#\[\begin{split}\begin{align} 0 &= \Delta_{rxn}G^\circ{} + \mathrm{R}T\ln K_c \\ \Delta_{rxn}G^\circ{} &= -\mathrm{R}T\ln K_c \end{align}\end{split}\]

The above equation can be simplified into a non-logarithmic form as follows:

(30)#\[K_c = e^{-\left({\dfrac{\Delta_{rxn}G^\circ{}}{\mathrm{R}T}}\right)}\]

Chemical Activities#

In chemical thermodynamics, concentrations are dimensionless (unitless). These dimensionless or idealized concentrations are called activities. The activity of a chemical constituent is an important concept and is used to address the influence of physical and chemical conditions in the bulk solution. Chemical species in solution interact with nearby chemical species, including chemical bonding, van der Waals (dipole), and long-range electrostatic interactions. Sometimes, when concentrations of solutes are very high, they crowd each other. The intensity of all these interactions depends on the distance between the molecules and their concentrations. The activities inherently account for all these factors and are called “thermodynamic concentrations.” Because these values are dimensionless, \(K_c\) of any aqueous phase chemical reaction is dimensionless.

Activities are derived from the same fundamental thermodynamic principles covered thus far. We use “\(\{\, \}\)” (curly brackets or braces) around chemical species names to indicate activity. An activity coefficient (\(\gamma_i\)) is used to convert concentrations to activities as follows:

(31)#\[\{A\} = \gamma_A [A]\]

where \(\{A\}\) is activity, \([A]\) is concentration, and \(\gamma_A\) is the activity coefficient. By convention, \(\gamma_A = 1\) for ideal solutes and dilute solutions (e.g., rainwater and freshwater systems) and is \(< 1\) in all other cases. Also, by convention, \(\{A\} = 1\) for pure solids (e.g., mineral phases, precipitates, etc.) and pure liquids (e.g., water).

Activity coefficients are usually determined using a solution’s ionic strength (\(I\)). Ionic strength is a measure of the combined concentration of all ions present in an aqueous solution and is calculated as follows:

(32)#\[I = \frac{1}{2} \sum_{i=1}^{n} c_i z_i^2\]

where, \(c_i\) is concentration of each ion in \(\pu{mol L-1 }\) and \(z_i\) is charge of each ion in \(\pu{eq mol-1}\). The activity coefficient, \(\gamma_i\), for each ion using one of the following equations:

(33)#\[\begin{split}\begin{aligned} -\log \gamma_i &= A z_i^2 \sqrt{I}\, && \text{for}\, I<0.005 \\ -\log \gamma_i &= \dfrac{A z_i^2 \sqrt{I}}{1+Ba_i\sqrt{I}}\, && \text{for}\, 0.005<I<0.1\\ -\log \gamma_i &= A z_i^2 \sqrt{I} \left( \dfrac{\sqrt{I}}{1+\sqrt{I}} -0.2I\right)\, && \text{for}\, 0.1<I<0.5 \end{aligned}\end{split}\]

where \(A\) and \(B\) are called Debye-Hückel constants and vary with temperature. At \(\pu{25 ^\circ C}\), \(A=\pu{0.511}\) and \(B=\pu{0.329e8}\). \(a_i\) is ion size parameter and is provided in Table 10.

Table 10 Ion size parameters (\(a_i\)) of major ions#
\(a_i, \, \pu{cm}\)	Ion
\(\pu{3e-8}\)	\(\text{All ions (exceptions below)}\)
\(\pu{4e-8}\)	\(\ce{Na+}\), \(\ce{SO4^2-}\), \(\ce{PO4^3-}\)
\(\pu{6e-8}\)	\(\ce{Ca^2+}\)
\(\pu{8e-8}\)	\(\ce{Mg^2+}\)

Units of \(K\)#

As seen in the previous section, the activity, or the effective concentration, of ions is less than that indicated by the actual concentration. Ions become more widely separated as solutions become more dilute, and the residual interionic interactions become less significant. Thus, in extremely dilute solutions, the effective concentrations of the ions (their activities) are equal to the actual concentrations. In freshwater environmental systems, aqueous ions are present in very small concentrations; therefore, activities and concentrations are similar.

Below is a summary of the main conventions of activities and effective concentrations:

Activities are dimensionless (unitless) quantities and are essentially “adjusted” concentrations.
A substance’s activity and molar concentration are roughly equal for relatively dilute solutions.
Activities for pure condensed phases (solids and liquids) are equal to 1.

Due to this last consideration, \(Q_c\) and \(K_c\) expressions do not contain terms for solids or liquids (being numerically equal to 1, these terms do not affect the expression’s value).

For homogeneous equilibrium (the reactants and products are present in a single solution), evaluation of reaction quotients (\(Q_c\)), and equilibrium constants (\(K_c\)), it is a common convention to omit units of concentrations (as they are equivalent to activities) of reactants and products, resulting in \(K\) values are also dimensionless.

A heterogeneous equilibrium is a system in which reactants and products are found in two or more phases. The phases may combine solid, liquid, or gas phases and solutions. When dealing with these equilibria, remember that solids and pure liquids do not appear in equilibrium constant expressions (the activities of pure solids, pure liquids, and solvents are equal to unity or \(=1\)). If a chemical reaction involves heterogeneous phases of reactants and products (usually a mixture of gases and liquids), it is not uncommon to see \(K\) values with units.