## Law of Mass Action in Chemical Equilibrium

**Law Mass Action** formula in chemical equilibrium was first developed by two Norweigan chemists, Guldberg and Waage in 1867 to derive a quantitative equation to calculate the equilibrium constant of a reaction in terms of molar concentration or active masses of reactants and products in chemistry. The law of mass action states, at a constant temperature, the rate of a chemical reaction is proportional to the active masses of reacting substances. Therefore, the definition and expression of active masses in mass action law is an important consideration to help equilibrium chemistry.

For ordinary systems of dilute solution and gas molecule, the active mass may be molar concentration and partial pressure but for pure crystalline solid or liquid, the active mass is taken as unity according to law mass action, since the rate of reaction does not affect equilibrium concentration formula.

### Definition of Active Mass in Equilibrium Reaction

Concentration or active masses formula effect by thermodynamics mass action law. Therefore, the active mass of gas, solid, and liquid defines by the law of mass action as, When the solution is dilute and the system behaves ideally the concentration is represented as the molar concentration. The unit of molar concentration moles/lit. For example, esterification of acetic acid in the alcohol solution.

Partial gas pressure is expressed in the atmosphere for the gas molecules concentration when the pressure of the system is very low. For example, the formation of water reaction and formation of carbon dioxide gas, active mass express in molar concentration, and partial pressure by the law of mass action. But for pure solid and pure liquid, active mass or concentration is assumed to be unity since their mass does not affect the rate of reaction.

### Definition of Concentration from Equilibrium Formula

Let us state the equilibrium formula and mass of action law to a general chemical reaction, A + B ⇆ C + D. As the reaction proceeds in the forward direction, the concentration of reactants decreases, and the forward kinetics reaction rate also decreases. But when the products are getting accumulated in the system, the reverse reaction also starts.

Rate of forward reaction (R_{f}) ∝ C_{A} × C_{B}

∴ R_{f} = k_{f} × C_{A} × C_{B }Rate of backward reaction (R_{b}) ∝ C_{C} × C_{D}

∴ R_{b} = k_{b} × C_{C} × C_{D}

Where C_{A}, C_{B,} and C_{C}, C_{D} are the concentration of forwarding and reverse reactions respectively in molar units. K_{f} and K_{b} are the rate constants at a given temperature.

### Mass Action Law Formula in Chemistry

As the reactions proceed in the forward reaction rate decreasing but the reverse reaction rate increases and attained a state when they are equal concentration. Therefore the point where forward and reverse reaction rates equal is called the equilibrium point of the chemical reaction. There will be no further change in the concentration of the system. Therefore, at equilibrium, R_{f} = R_{b }or, k_{f} × C_{A} × C_{B} = k_{b} × C_{C} × C_{D}, where C_{A}, C_{B}, C_{C,} and C_{D} are the equilibrium point concentration of reactants and products. But k_{f}/k_{b} = k_{c} = concentration equilibrium constant derive by mass action formula.

### Mass Action Law Equilibrium Concentration

Let us consider a general balance chemical reaction to define mass action law in terms of concentration for learning chemistry

Ɣ_{1}A_{1} + Ɣ_{2}A_{2} ⇆ Ɣ_{3}A_{3} + Ɣ_{4}A_{4}

where Ɣ_{1}, Ɣ_{2}, Ɣ_{3}, and Ɣ_{4} are stoichiometric coefficients.

∴ k_{c} = (C_{3}^{γ3} × C_{4}^{γ4})/(C_{1}^{γ1} × C_{2}^{γ2})

where k_{C} = concentration equilibrium constant of the reaction and C_{1}, C_{2}, C_{3}, and C_{4} are the equilibrium concentration of the reactants and product. Therefore, the values of the k_{C} of a chemical reaction from the mass action law affect the mode of writing or balance of the stoichiometric chemical equation. If a chemical equation multiplied by n then the general rule for writing equilibrium concentration n will be raised to the power or, k_{C} = (k′_{C})^{n}, where k_{C}, and k′_{c} are not equal in magnitude. Therefore the concentration equilibrium constant depends on the modes of balancing chemical equations.

### Pressure Equilibrium Constant Formula in Chemistry

When all the reactants and products are the gas-phase obey the ideal gas law or real gas laws, the concentration formula is expressed in partial pressure by mass action law. Let the chemical equation for the above expression

Ɣ_{1}A_{1} + Ɣ_{2}A_{2} ⇆ Ɣ_{3}A_{3} + Ɣ_{4}A_{4}

where Ɣ_{1}, Ɣ_{2}, Ɣ_{3}, and Ɣ_{4} are stoichiometric coefficients.

∴ k_{P} = (P_{3}^{γ3} × P_{4}^{γ4})/(P_{1}^{γ1} × P_{2}^{γ2})

Where k_{P} = pressure equilibrium constant of the reaction. P_{1}, P_{2}, P_{3}, and P_{4} are the equilibrium concentration in terms of the partial pressure of reacting components.

### Mole Fraction Equilibrium Constant Formula

Ɣ_{1}A_{1} + Ɣ_{2}A_{2} ⇆ Ɣ_{3}A_{3} + Ɣ_{4}A_{4}

where Ɣ_{1}, Ɣ_{2}, Ɣ_{3}, and Ɣ_{4} are stoichiometric coefficients.

∴ k_{x} = (x_{3}^{γ3} × x_{4}^{γ4})/(x_{1}^{γ1} × x_{2}^{γ2})

Where k_{x} = mole fraction equilibrium constant and x_{1}, x_{2}, x_{3}, and x_{4} are the concentration in terms of mole fraction of reacting components derive by mass action formula.

### k_{P} and k_{C} Relation from Mass Action Law

Let us consider the general equilibrium concentration reaction which expresses the relation between k_{P} and k_{C }gas and liquid phase by mass action law.

Formation of hydrochloric acid solution from hydrogen and chlorine molecule. In this reaction, the total number of reactant molecules, and of resultant molecules are the same.

H_{2} + Cl_{2} ⇆ 2HCl

k_{P} = k_{C} (RT)^{2-(1+1)} = K_{C}

For the reactions in which the number of molecules of reactants differs from the resultant molecules of the chemical reaction.

PCl_{5} ⇆ PCl_{3} + Cl_{2}

k_{P} = k_{C} (RT)^{(1+1) -1} = K_{C} RT

Therefore, the expression of k_{P} and k_{C} of the above chemical reaction in terms of concentration derive by the application of the mass of action law formula.

### Relation of k_{P} and k_{x} from Mass Action Law

k_{P} = k_{x} (P)^{Δγ}

The mass of action law in equilibrium chemistry define before the thermodynamics approach developed and definition, formula, and the relation of the different equilibrium constant of balanced chemical reaction expressed in terms of active masses or molar concentration, the partial gas pressure, and mole fraction. Therefore, the law of mass action recognized that the chemical equilibrium is a dynamic, not static equilibrium. How the system behaves if any parameter like temp, pressure, chemical catalyst, etc of the system at equilibrium changed describe by Le Chatelier Principle and Van’t Hoff equation but equilibrium concentration define by mass action law formula.