# What is the Law of Mass Action in Chemistry?

Benjamin Sirulnik, Saranya Chatterjee
• Author
Benjamin Sirulnik

Ben has taught and tutored students in grades 4-12 in math and science as well as intro chemistry and physics for over 4 years. He has a BA in Chemistry from Carleton College where he worked as an Organic Chemistry TA and Intro Chemistry tutor. He contributed to research on biological nanosensors in the Bioengineering department at Northeastern University.

• Instructor
Saranya Chatterjee

Saranya has a masters degree in Chemistry and in Secondary Education. She has taught high school, AP chemistry for 2 years and is teaching undergraduate college chemistry for 3 years.

Understand the definitions of the Law of Mass Action. Learn more about equilibrium constants when applying this law and see its real-life applications here. Updated: 03/19/2022

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## Law of Mass Action

The law of mass action states that the rate of a chemical reaction is proportional to the active masses of the reacting substances at a constant temperature. Active masses are substances that are actively reacting within a chemical reaction. For example, in a closed bottle of soda, there is constant interconversion between carbonic acid (H2CO3), bicarbonate (HCO3-), and carbon dioxide (CO2) gas. Since all three substances are reacting, they are active masses in the reaction. When the cap of the soda bottle is removed, carbon dioxide gas continuously leaves the liquid. Once a carbon dioxide molecule has left the liquid, it is no longer an active mass, and therefore no longer affects the reaction rate by the law of mass action.

Some reactions occur only in the gaseous phase. For example, dinitrogen tetraoxide gas decomposes into nitrogen dioxide gas. In a closed system, these gases are both active masses. When applying the law of mass action, active masses in the gas phase are usually expressed in terms of partial pressure instead of molarity. Neither partial pressure nor molarity indicate the total quantity of a substance; however, they do indicate the relative amount of a substance within a liquid or gaseous system. Since the law of mass action describes proportionality of active masses, only the relative quantities are required, making concentration and partial pressure ideal for calculations.

### History of the Law of Mass Action

The discovery of the law of mass action began with the understanding that many reactions not only proceed in the forward direction, but proceed in the reverse direction as well. French chemist Claude Louis Berthollet was one of the early chemists to demonstrate chemical equilibrium, the ability of reactions to react forward and backward until there is no net change in the quantity of reactants or products and equilibrium concentrations are reached.

Building on Berthollet's work, Norwegian chemists Cato Maximilian Guldberg and Peter Waage discovered that reactants and products of these equilibrium reactions have stoichiometric coefficients. They realized that these stoichiometric coefficients, combined with the concentrations of the reactants and products, allowed them to predict the reaction rate. This discovery was formalized as the law of mass action and yielded the mass action equation.

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## Law of Mass Action Equation

The law of mass action relates stoichiometric reaction coefficients and reactant and product concentrations. The generic chemical equation used for describing the law of mass action in the law of mass action equation is:

$$aA + bB \to cC + dD$$

In this equation, A and B are reactants, C and D are products, and a, b, c, and d are their respective stoichiometric coefficients.

### The Equilibrium Constants in Law of Mass Action

The formula for the equilibrium constant (Keq) is expressed as follows:

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

This equation is derived from the law of mass action, and the understanding that at equilibrium, the forward reaction rate is equivalent to the reverse reaction rate. For the reaction described above, the forward and reverse reaction rates are expressed below. The variables kf and kr are rate constants for the forward and reverse reaction.

$$rate_{forward} = k_f[A]^a[B]^b$$

$$rate_{reverse} = k_r[C]^c[D]^d$$

At equilibrium, the forward and reverse rates are equal:

$$k_f[A]^a[B]^b = k_r[C]^c[D]^d$$

By dividing each side of the equation by {eq}k_r[A]^a[B]^b {/eq}, the rate constants are isolated to one side. The fraction kf/kr at equilibrium is named Keq. This yields the law of mass action equation:

$$K_{eq} = \frac{k_f}{k_r} = \frac{[A]^a[B]^b}{[C]^c[D]^d}$$

### Relationship of Kc, Kp and Kx

#### Equilibrium Constant for Gaseous Reactions

Kc, Kp, and Kx represent the equilibrium constant calculated from active mass concentration, pressure, and molar concentration respectively. In gaseous phase reactions, the law of mass action equation uses partial pressures of each active mass. The law of mass action equation for gaseous phase reactions is shown below. In this equation, Kp is the equilibrium constant for pressure and P indicates partial pressure.

$$K_{p} = \frac{(P_A)^a(P_B)^b}{(P_C)^c(P_D)^d}$$

#### Relating Kp and Kc

Through mathematical derivation, Kp can be related to Kc. For any ideal gas, {eq}P = nRT {/eq}, where P is pressure, n is moles, R is the ideal gas constant, and T is temperature. Replacing moles with concentration of a gas, we relate partial pressure to concentration in the equation {eq}P_x = C_xRT {/eq}, where x is a single reactant or product in the gaseous phase.

This equation can be plugged into the law of mass action equation for pressure above.

$$K_p = \frac{(C_CRT)^c*(C_DRT)^d}{(C_ART)^a*(C_BRT)^b}$$

Next, the RT term is factored out along with the exponents. The exponents that are in the numerator are added together and those in the denominator are subtracted to yield the final exponent.

$$K_p = \frac{(C_C)(C_D)}{(C_A)(C_B)}*(RT)^{(a+b-c-d)}$$

This equation can be further simplified. Since {eq}\frac{(C_C)(C_D)}{(C_A)(C_B)} {/eq} is equivalent to to Kc, Kc replaces the fraction:

$$K_p = K_cRT^{(a+b-c-d)}$$

The factored-out exponents (a+b-c-d) represent the difference between moles of products and moles of reactants in the original chemical equation. This is named {eq}\Delta n {/eq}, and is added to yield the final equation relating Kc to Kp:

$$K_p = K_c(RT)^{\Delta n}$$

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#### What is meant by Law of Mass Action?

The law of mass action states that the rate of a reaction is proportional to its active masses. Chemical species in a reaction are considered 'active' if they are able actively participate in the reaction.

#### Why is the Law of Mass Action important?

The law of mass action is important because it allows chemists to determine an equilibrium constant for reversible reactions. This constant allows the prediction of the quantities of each active mass at chemical equilibrium.

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