# Chemical Equilibrium

Chemical equilibrium occurs when the concentrations of the products and the reactants in a reaction do not change, because the forward and reverse reactions occur at the same rate. An analogy would be a boater on a river who maintains a constant position. The speed upstream of the boat equals the speed of the river itself. A satellite in space would see no apparent change; however, up close one hears the motor running the sees the current flowing. This dynamic equilibrium is achieved when there is a lack of change in a system as inputs and outputs remain in balance. In a dynamic chemical equilibrium, products and reactants interconvert rapidly at the molecular scale, while their concentrations remain constant on the macroscopic scale.

## Equilibrium Law

The equilibrium (law of mass action) law states that the concentrations of the products multiplied together and divided by the concentrations of the reactants equals the equilibrium constant ($K$ ). The equilibrium constant is a unitless number which depends on the reaction itself and the temperature of the mixture when equilibrium is attained.

The letter $K$ is reserved as the symbol for the equilibrium constant. A specific type of the equilibrium constant can be notated with a subscript:

• $K_{c}$ = concentration is in molarities
• $K_{p}$ = partial pressures of gases represent reactant and product amounts
• $K_{sp}$ = solubility product
• $K_{a}$ = acid ionization constant
• $K_{b}$ = base ionization constant
• $K_{f}$ = formation constant
• $K_{w}$ = dissociation constant for water

The specific equilibrium law depends on the equilibrium reaction under study. A general equilibrium reaction can be written as:

$\alpha A+\beta B\rightleftharpoons \sigma S+\tau T$ The general equilibrium law for the above reaction is written in its simplified form as (technically activities are used):

$K={\frac {[S]^{\sigma }[T]^{\tau }}{[A]^{\alpha }[B]^{\beta }}}$ Compounds in a liquid or solid state do not appear in the equilibrium law because they have a constant concentration (technically they have unit activity). For example for the reaction:

$CH_{4}(g)+2O_{2}(g)\rightleftharpoons CO_{2}(g)+2H_{2}O(l)$ The equilibrium law is:

$K_{c}={\frac {[CO_{2}]}{[CH_{4}][O_{2}]^{2}}}$ Aqueous solutions and gases are included in the equilibrium law. For the below reaction:

$NH_{3}(g)+H_{2}O(l)\rightleftharpoons NH_{4}^{+}(aq)+OH^{-}(aq)$ The equilibrium law is:

$K_{c}={\frac {[NH_{4}^{+}][OH^{-}]}{[NH_{3}]}}$ ## Finding the Value of the Equilibrium Constant

In the equation:

$H_{2}(g)+Cl_{2}(g)\rightleftharpoons 2HCl(g)$ The equilibrium law is:

$K_{p}={\frac {P_{HCl}^{2}}{P_{H_{2}}P_{Cl_{2}}}}$ The most direct method for finding the value of the equilibrium constant $K_{p}$ is by measuring the concentration of each of the reactants and products, and plugging in their values in the equilibrium law. For example, if the concentration at equilibrium for the above reaction are determined as $P_{H_{2}}=1.0\times 10^{-8}bar$ , $P_{Cl_{2}}=3.4\times 10^{-6}bar$ , and $P_{HCl}=0.802bar$ , they can be substituted into the equilibrium law to solve for $K_{p}$ :

$K_{p}={\frac {P_{HCl}^{2}}{P_{H_{2}}P_{Cl_{2}}}}$ becomes

$K_{p}={\frac {0.802^{2}}{(1.0\times 10^{-8})(3.4\times 10^{-6})}}$ After substituting in the concentrations, the appropriate arithmetic is performed to find the value of $K_{p}$ . In this case $K_{p}=1.89*10^{13}$ .

## Uses of the Equilibrium Law

The value of the equilibrium constant connotes the extent to which, in a chemical reaction, reactants are converted into products. Thus, from the equilibrium constant $K$ , one can infer the composition of an equilibrium mixture. If the equilibrium constant is very large (i.e. above $10^{10}$ ), the amount of products present at equilibrium is greater than the amount of reactants, which means that the reaction goes to completion. If $K$ equals 1, the amount of products present at equilibrium is the same as the amount of reactants. When $K$ is very small (i.e. below $10^{-10}$ ), the amount of products formed is extremely small; no visible reaction takes place.

### Spontaneous Reactions

A spontaneous reaction is a reaction that will proceed without any outside energy or driving force. A spontaneous reaction has an equilibrium constant greater than 1. A reaction will be nonspontaneous if the equilibrium constant is less than 1.

### The Reaction Quotient

The reaction quotient ($Q$ ) is a value that can be obtained by substituting in the values of the given concentrations into the equilibrium law. The equilibrium constant is the value $K$ when the reaction is at equilibrium. If the chemicals in the reaction are not at equilibrium, then the value obtained by the equilibrium law is called the reaction quotient. $Q$ has the same form as the equilibrium law, except $K_{c}$ is replaced by Q. Four properties may be derived from this definition of the reaction quotient, Q:

• If $Q=K_{c}$ , the reaction is at equilibrium.
• If $Q$ does not change with respect to time, the reaction is at equilibrium and thus, $Q=K_{c}$ .
• If $Q , the reaction will move to the right (the forward direction) in order to reach equilibrium.
• If $Q>K_{c}$ , the reaction will move to the left (the reverse direction) in order to reach equilibrium.