# Law of Mass Action Derivation

The law of mass action is proposed by Goldberg and Wage in 1867. The basics of this law are about the mass of the substances that react in a reaction. The law states that “The rate of a chemical reaction is proportional to the active masses of the reacting substances”.

In dilute solutions where conditions approach the ideal state, ‘active mass’ may be represented by the concentration of the reacting substances, i.e. gm-molecules or gm-ions per liter. The constant of proportionality is known as ‘the velocity constant’.

Now, let us consider a homogeneous, reversible reaction.

$A+\;B\;\rightleftharpoons\;C+\;D$

According to the law of mass action,

VF = K1 [A] [B]

Vb= K2 [C] [D]

And

Where,

VF= Velocity of the forward reaction

Vb = Velocity of the backward reaction

[A], [B], [C], [D] = Molar concentration of A, B, C, and D respectively, K1 and K2 are

constant.

At equilibrium,

VF = Vb

K1[A] [B] = K2[C] [D]

$\frac{K_1}{K_2}=\;\frac{\lbrack C\rbrack\;\lbrack D\rbrack}{\lbrack A\rbrack\;\lbrack B\rbrack}$

Since K1 and K2 are constant, the K fraction must also be a constant.

This result may be generalized as: When equilibrium is reached in a reversible reaction, at a constant temperature, the product of the molecular concentration of the resultants (RHS) divided by the product of the molecular concentrations of reactants (LHS), each concentration being raised to a power equal to the number of molecules of that substance taking part in the reaction, is constant.

Hence,

$K=\;\frac{\lbrack C\rbrack\;\lbrack D\rbrack}{\lbrack A\rbrack\;\lbrack B\rbrack}$

Where K is the equilibrium constant of the reaction.

In extension, the equilibrium constant for the general reversible reaction

$aA\;+\;bB\;+\;cC\;+\;…….\;\rightleftharpoons\;pP+\;qQ\;+\;rR…$

$K\;=\;\frac{{\lbrack P\rbrack}^p\left[Q\right]^q\left[R\right]^r}{{\lbrack A\rbrack}^a\left[B\right]^b\left[C\right]^c}$

Where a, b, c . and p, g, r are the number of molecules of reacting species.

Make sure you also check our other amazing Article on: Theories of Acids and Bases