Unit 7: Equilibrium

Equilibrium in chemistry is the state where the rates of the forward and reverse reactions are equal, resulting in constant concentrations of reactants and products over time. This unit examines how to describe, quantify, and manipulate equilibrium systems, with a focus on the equilibrium constant, the reaction quotient, and the application of Le Châtelier’s Principle. Students learn to calculate equilibrium concentrations from initial conditions, predict the direction of change when a system is disturbed, and analyze special cases such as solubility equilibria. Mastery of equilibrium concepts is essential for understanding acid-base chemistry, solubility, electrochemistry, and real-world chemical processes like industrial synthesis and environmental systems.

Introduction to Equilibrium

Chemical equilibrium is a dynamic process where the forward and reverse reactions occur at the same rate, so the concentrations of reactants and products remain constant. It does not mean that the amounts of reactants and products are equal, but rather that their ratio remains fixed for given conditions. The balance is maintained through continuous molecular changes, even though macroscopic properties such as color, pressure, and concentration remain unchanged.
Equilibrium can only be reached in a closed system, where no reactants or products are lost to the surroundings. In open systems, products may escape or react further, preventing the system from establishing a stable balance. The equilibrium position depends on factors like temperature, pressure, and concentration, and changes in these can shift the balance toward more products or reactants.
The concept of dynamic equilibrium applies to both physical processes, such as phase changes, and chemical reactions. For example, in a sealed container with liquid water and water vapor, molecules continuously evaporate and condense at equal rates. This illustrates that equilibrium is about equal rates, not static inactivity.
Understanding equilibrium is crucial for predicting how a reaction mixture will respond to changes in conditions. It connects directly to thermodynamics, as the equilibrium position corresponds to the minimum Gibbs free energy for the system at given conditions. This relationship allows chemists to link kinetic and thermodynamic perspectives when analyzing chemical behavior.

The Equilibrium Constant (\(K\))

The equilibrium constant (\(K\)) quantifies the ratio of product concentrations to reactant concentrations at equilibrium, each raised to the power of its stoichiometric coefficient from the balanced equation. For a general reaction \( aA + bB \rightleftharpoons cC + dD \), \( K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} \) when concentrations are in molarity. The magnitude of \(K\) indicates the extent of the reaction: large \(K\) means products are favored, small \(K\) means reactants are favored.
The value of \(K\) is constant for a given reaction at a specific temperature but changes if the temperature changes, reflecting the temperature dependence of equilibrium position. It is independent of the initial concentrations of reactants and products, meaning that different starting mixtures will yield the same \(K\) if the temperature is constant. This property makes \(K\) a powerful tool for predicting the behavior of equilibrium systems.
There are different forms of the equilibrium constant depending on the units used. \(K_c\) uses molar concentrations, while \(K_p\) uses partial pressures for gaseous equilibria, related by \( K_p = K_c (RT)^{\Delta n} \) where \( \Delta n \) is the change in moles of gas. The correct form must be chosen based on how the system is measured experimentally.
The equilibrium constant does not include pure solids or pure liquids because their concentrations do not change during the reaction. Only gaseous species and solutes in solution appear in the expression. This ensures that \(K\) reflects only the changing species that influence the position of equilibrium.

Writing Equilibrium Expressions

To write an equilibrium expression, start with the balanced chemical equation and place the concentrations of the products in the numerator and the concentrations of the reactants in the denominator. Each concentration term is raised to the power of its stoichiometric coefficient from the balanced equation. For example, for \( \text{N}_2(g) + 3\text{H}_2(g) \rightleftharpoons 2\text{NH}_3(g) \), the expression is \( K_c = \frac{[\text{NH}_3]^2}{[\text{N}_2][\text{H}_2]^3} \).
Only species in the gas phase or aqueous solutions appear in the equilibrium expression, because their concentrations can change during the reaction. Pure solids and pure liquids are omitted since their molar concentrations remain constant under given conditions. For example, in \( \text{CaCO}_3(s) \rightleftharpoons \text{CaO}(s) + \text{CO}_2(g) \), the expression is \( K_p = P_{\text{CO}_2} \) since solids are excluded.
The form of the equilibrium constant depends on the type of system studied. For reactions involving gases, partial pressures are often used (\(K_p\)), while for reactions in solution, molar concentrations (\(K_c\)) are typical. The two constants are related by \( K_p = K_c (RT)^{\Delta n} \), where \(R\) is the gas constant in appropriate units, \(T\) is the absolute temperature, and \( \Delta n \) is the change in moles of gaseous species.
Reversing a chemical equation inverts the equilibrium constant (\(K_{\text{reverse}} = 1/K_{\text{forward}}\)), and multiplying the equation by a factor raises \(K\) to that power. These relationships are essential for manipulating equilibrium constants when using Hess’s Law–type logic for equilibrium systems, such as when combining multiple reactions to find the overall \(K\).

Reaction Quotient (\(Q\)) and Comparing to \(K\)

The reaction quotient (\(Q\)) has the same mathematical form as the equilibrium constant but is calculated from the concentrations or partial pressures at any point in time, not just at equilibrium. Like \(K\), \(Q\) omits pure solids and liquids and includes only the species whose amounts change during the reaction. \(Q\) is a “snapshot” of the system’s composition that allows prediction of how the reaction will proceed to reach equilibrium.
Comparing \(Q\) to \(K\) determines the direction in which the reaction must shift to reach equilibrium. If \(Q < K\), the reaction will shift toward the products (forward direction) to increase \(Q\) until it equals \(K\). If \(Q > K\), the reaction will shift toward the reactants (reverse direction) to decrease \(Q\). If \(Q = K\), the system is already at equilibrium and no net change occurs.
This concept is crucial in chemical analysis and industrial processes, where controlling reaction conditions can push the equilibrium toward desired products. For instance, in the Haber process for ammonia synthesis, monitoring \(Q\) relative to \(K\) guides adjustments in temperature, pressure, and concentrations to maximize yield while conserving resources.
Since \(Q\) can be calculated at any stage of a reaction, it is especially useful for predicting outcomes when a system is disturbed by changes in concentration, pressure, or temperature. Combining \(Q\) with Le Châtelier’s Principle provides a complete picture of how and why the system will adjust to restore equilibrium.

Le Châtelier’s Principle

Le Châtelier’s Principle states that if a system at equilibrium is disturbed by a change in concentration, pressure, volume, or temperature, the system will shift in the direction that counteracts the disturbance. This shift occurs because the rates of the forward and reverse reactions are temporarily unbalanced, prompting the system to adjust until a new equilibrium is established. The principle is a qualitative tool for predicting the effects of changes without detailed calculations.
When the concentration of a reactant is increased, the system shifts toward the products to consume the added reactant; when a product concentration is increased, the system shifts toward the reactants to consume the excess product. Conversely, decreasing the concentration of a reactant shifts the system toward the reactants, while decreasing a product shifts toward the products. These changes directly alter \(Q\) relative to \(K\), driving the shift.
Changes in pressure and volume affect equilibria involving gases. Increasing pressure (or decreasing volume) shifts equilibrium toward the side with fewer moles of gas to reduce pressure, while decreasing pressure (or increasing volume) shifts toward the side with more moles of gas. If both sides have the same number of gas moles, changes in pressure or volume have no effect on equilibrium position.
Temperature changes alter the value of \(K\) because they affect the reaction’s enthalpy. For endothermic reactions (\(\Delta H > 0\)), increasing temperature shifts equilibrium toward products, while decreasing temperature shifts toward reactants. For exothermic reactions (\(\Delta H < 0\)), increasing temperature shifts toward reactants, while decreasing temperature shifts toward products. This is because temperature changes modify the rate constants for forward and reverse reactions differently.
The addition of a catalyst speeds up the attainment of equilibrium by increasing the rates of both forward and reverse reactions equally, but it does not change the equilibrium position or the value of \(K\). Misunderstanding this point is common, but remembering that catalysts only affect reaction rates—not thermodynamic equilibria—prevents confusion.

Calculating Equilibrium Concentrations

Equilibrium concentrations can be calculated using the ICE table method (Initial, Change, Equilibrium), which organizes data and applies the equilibrium constant expression. Start by listing the initial concentrations, then define the changes in terms of a variable (commonly \(x\)), and finally express equilibrium concentrations as the sum of the initial amounts and the changes. This method ensures a systematic approach to solving equilibrium problems.
Once the equilibrium concentrations are expressed in terms of \(x\), substitute them into the equilibrium constant expression and solve for \(x\). The resulting equation can be linear, quadratic, or occasionally require approximation methods when \(K\) is very large or very small. Approximations, such as neglecting \(x\) in addition or subtraction, must be justified by showing that the change is insignificant compared to the initial concentration.
When \(K\) is very small (\(< 10^{-3}\)), the reaction lies far to the reactant side, and the change in reactant concentration is often negligible. Conversely, when \(K\) is very large (\(> 10^3\)), the reaction lies far to the product side, and the change in product concentration is often negligible. Recognizing these extreme cases can simplify calculations and save time on exams.
For gaseous equilibria, partial pressures can be used instead of concentrations, with ICE tables organized in units of atm or torr. The same algebraic approach applies, but the equilibrium expression uses partial pressures in \(K_p\) form. When switching between \(K_c\) and \(K_p\), remember to apply \(K_p = K_c (RT)^{\Delta n}\) with consistent units for \(R\) and \(T\).
Checking the final answer is essential: all equilibrium concentrations must be positive, and substituting them back into the equilibrium expression should reproduce the given value of \(K\). If results contradict these checks, re-examine sign conventions, algebra, and any approximations made during the calculation.

Equilibrium with Pressure and Gases (\(K_p\) vs. \(K_c\))

For gaseous reactions, equilibrium can be expressed in terms of partial pressures (\(K_p\)) rather than concentrations (\(K_c\)). The relationship between the two is \( K_p = K_c (RT)^{\Delta n} \), where \(R\) is the gas constant in units of \( \text{L·atm·mol}^{-1}\text{·K}^{-1} \), \(T\) is the absolute temperature in kelvins, and \( \Delta n \) is the change in moles of gaseous products minus moles of gaseous reactants. This formula ensures proper conversion between the two equilibrium constants depending on measurement units.
\(K_p\) is especially useful in reactions where pressures are measured more easily or more accurately than concentrations, such as industrial gas-phase processes. For example, in the Haber process \( \text{N}_2(g) + 3\text{H}_2(g) \rightleftharpoons 2\text{NH}_3(g) \), partial pressures are often monitored directly to control yield. The use of \(K_p\) allows engineers to make real-time adjustments to maintain desired equilibrium conditions.
When \( \Delta n = 0 \) for a gaseous reaction, \(K_p\) and \(K_c\) are numerically equal because the \( (RT)^{\Delta n} \) term becomes 1. Recognizing this special case simplifies calculations and avoids unnecessary conversions. This often occurs in balanced reactions where the total number of moles of gas on both sides is the same.
The same mathematical rules for manipulating equilibrium constants apply to both \(K_p\) and \(K_c\): reversing a reaction inverts the constant, and multiplying the reaction by a factor raises the constant to that power. This consistency ensures that whether using concentration or pressure, the relative positions of equilibrium remain predictable and mathematically linked.

Effects of Temperature, Pressure, and Volume on \(K\)

The equilibrium constant \(K\) is temperature-dependent but unaffected by changes in concentration, pressure, or volume alone. This is because \(K\) is derived from the ratio of rate constants for the forward and reverse reactions, and only temperature changes can alter these rate constants by affecting activation energies. Therefore, only temperature shifts can change the numerical value of \(K\).
For endothermic reactions (\(\Delta H > 0\)), increasing temperature increases \(K\) because the added thermal energy favors the forward reaction, producing more products at equilibrium. For exothermic reactions (\(\Delta H < 0\)), increasing temperature decreases \(K\) because the extra heat acts like a product, favoring the reverse reaction. These relationships follow directly from Le Châtelier’s Principle and can be predicted from the sign of \(\Delta H\).
Changes in pressure or volume do not alter \(K\) but can shift the equilibrium position toward reactants or products. For gaseous systems, increasing pressure (by decreasing volume) shifts the equilibrium toward the side with fewer moles of gas, while decreasing pressure shifts it toward the side with more moles. If the number of moles of gas is the same on both sides, pressure and volume changes have no effect on the equilibrium position.
Understanding the effect of temperature on \(K\) is critical for optimizing industrial reactions. For instance, the synthesis of ammonia is exothermic, so lowering the temperature increases \(K\), but too low a temperature slows the rate excessively. In practice, a compromise temperature is chosen to balance a favorable equilibrium constant with a reasonable reaction rate, often with the aid of a catalyst.
These principles connect equilibrium thermodynamics with kinetics: temperature changes shift \(K\) by altering rate constants, while pressure and volume changes temporarily upset the reaction quotient \(Q\) but leave \(K\) unchanged. Recognizing this distinction helps avoid common mistakes when predicting system responses to various stresses.

Solubility Equilibria (\(K_{sp}\))

Solubility equilibria describe the equilibrium established when an ionic solid dissolves in water to produce ions, and some of those ions recombine to re-form the solid. The equilibrium constant for this process is the solubility product constant, \(K_{sp}\), which is specific to each compound at a given temperature. For example, for \( \text{PbCl}_2(s) \rightleftharpoons \text{Pb}^{2+}(aq) + 2\text{Cl}^-(aq) \), \(K_{sp} = [\text{Pb}^{2+}][\text{Cl}^-]^2\).
The magnitude of \(K_{sp}\) reflects how soluble a substance is: larger \(K_{sp}\) values indicate greater solubility, while smaller values indicate limited solubility. \(K_{sp}\) depends only on temperature and is unaffected by the presence of pure solids or liquids. However, it can be used to calculate molar solubility, which is the number of moles of solute that dissolve per liter of solution.
Solubility product expressions must be written from the balanced dissolution equation, ensuring that the exponents match the stoichiometric coefficients of the ions. This ensures correct relationships between ion concentrations when solving for molar solubility. For salts with different ion ratios, such as \( \text{CaF}_2 \) or \( \text{Fe(OH)}_3 \), the relationship between molar solubility and ion concentrations will vary accordingly.
Comparing the ion product (\(Q_{sp}\)) to \(K_{sp}\) predicts precipitation: if \(Q_{sp} > K_{sp}\), the solution is supersaturated and precipitation occurs; if \(Q_{sp} < K_{sp}\), the solution is unsaturated and no precipitate forms; if \(Q_{sp} = K_{sp}\), the system is at equilibrium and the solution is saturated. This concept is vital for understanding selective precipitation techniques in qualitative analysis.

Common Ion Effect

The common ion effect occurs when a solution contains an ion that is also a product of the dissolution of a salt, reducing the solubility of that salt. This effect shifts the dissolution equilibrium toward the solid form, as predicted by Le Châtelier’s Principle, because adding more of a product ion drives the reaction in the reverse direction. For instance, adding NaCl to a saturated AgCl solution decreases AgCl’s solubility by increasing \([\text{Cl}^-]\).
The common ion effect is important in controlling precipitation in chemical processes. In water treatment, for example, adding a compound with a common ion can precipitate undesirable ions from solution, aiding in purification. Conversely, in some situations such as medical treatments, the common ion effect can be used to reduce absorption of certain ions in the body.
Calculations involving the common ion effect often require setting up an ICE table that accounts for the initial concentration of the common ion and the change due to dissolution. The presence of a significant initial concentration of the common ion reduces the change in concentration (\(x\)) associated with dissolution, leading to a lower calculated molar solubility. This demonstrates the quantitative impact of the common ion effect on solubility.
This effect connects solubility equilibria with acid-base equilibria because many salts contain ions that can hydrolyze or react with acids or bases. For example, adding HCl to a CaF₂ solution increases solubility by removing \(\text{F}^-\) via HF formation, showing that common ion effects can be overridden by additional equilibria in the system.

Coupled Equilibria

Coupled equilibria occur when two or more equilibrium processes share common species, meaning a shift in one equilibrium can influence the others. These systems are interconnected, and the overall position depends on the combined effects of all equilibria present. This concept is often seen in solutions where solubility equilibria interact with acid-base equilibria.
A common example is the dissolution of a salt containing a basic anion, such as \(\text{CaCO}_3\), in acidic solution. The solubility equilibrium \( \text{CaCO}_3(s) \rightleftharpoons \text{Ca}^{2+}(aq) + \text{CO}_3^{2-}(aq) \) is coupled with the acid-base reaction \(\text{CO}_3^{2-} + \text{H}^+ \rightleftharpoons \text{HCO}_3^-\), which removes \(\text{CO}_3^{2-}\) from the system and shifts dissolution toward more products, increasing solubility.
Another example involves metal hydroxides in acidic environments. For instance, \(\text{Al(OH)}_3\) dissolves more readily in acidic solutions because \(\text{OH}^-\) ions are consumed by reaction with \(\text{H}^+\) to form water. This removal of a product ion drives the dissolution equilibrium forward, demonstrating how controlling pH can manipulate solubility.
Coupled equilibria are important in environmental and biological systems, where multiple equilibria often operate simultaneously. In natural waters, for example, carbonate equilibria, gas-phase CO₂ equilibrium, and mineral solubility equilibria all interact, meaning that changes in one (such as CO₂ levels) can cascade into shifts in others, affecting pH and aquatic life sustainability.

Common Misconceptions

One common misconception is thinking that equilibrium means the forward and reverse reactions have stopped. In reality, equilibrium is dynamic — both reactions continue to occur, but at equal rates, so there is no net change in concentrations. This misunderstanding often leads students to incorrectly describe equilibrium as a static state rather than a continuous molecular process.
Students sometimes believe that the concentrations of reactants and products are equal at equilibrium, but this is not necessarily true. The actual equilibrium concentrations depend on the value of \(K\) and the stoichiometry of the reaction. For example, in a reaction with a very large \(K\), the product concentration will be much higher than the reactant concentration, even though the system is at equilibrium.
Another error is assuming that catalysts change the equilibrium position or the value of \(K\). Catalysts only speed up the rates of both forward and reverse reactions equally, allowing the system to reach equilibrium faster. They do not alter the final concentrations or the thermodynamic favorability of the reaction.
When applying Le Châtelier’s Principle, some students incorrectly think that changes in concentration, pressure, or volume can change \(K\). These variables only cause a temporary shift in the reaction quotient (\(Q\)); \(K\) changes only with temperature because it is tied to the thermodynamic properties of the reaction. Recognizing this distinction prevents misinterpretation of equilibrium data after disturbances.
In solubility equilibria, learners may wrongly assume that adding a common ion always decreases solubility. While this is usually true, certain coupled equilibria — such as acid-base reactions consuming the common ion — can reverse the effect and increase solubility. This shows that multiple equilibria must be considered together to accurately predict changes in a system.
Another misconception is confusing \(Q\) and \(K\) by treating them as interchangeable. While both have the same form, \(K\) applies only at equilibrium and is constant for a given temperature, whereas \(Q\) can be calculated at any point and changes until equilibrium is reached. Misusing \(Q\) can lead to incorrect predictions about the direction of shift in a reaction.