Derive law of mass action thermodynamically?

 

Question: Derive law of mass action thermodynamically?

The law of mass action describes the relationship between the concentrations of reactants and products in a reversible chemical reaction at equilibrium. It states that the product of the concentrations of the products raised to their stoichiometric coefficients divided by the product of the concentrations of the reactants raised to their stoichiometric coefficients is equal to a constant at a given temperature and pressure. Mathematically, this can be expressed as:

Kc = ([C]^c [D]^d) / ([A]^a [B]^b)

where Kc is the equilibrium constant, [A], [B], [C], and [D] are the concentrations of the reactants and products, and a, b, c, and d are their stoichiometric coefficients.

The law of mass action can be derived thermodynamically using the principles of free energy and entropy. At equilibrium, the Gibbs free energy change (ΔG) of the reaction is zero, which means that the system is in a state of minimum free energy. Mathematically, this can be expressed as:

ΔG = ΔH - TΔS = 0

where ΔH is the enthalpy change of the reaction, T is the temperature, and ΔS is the entropy change of the system.

Rearranging this equation, we can express the equilibrium constant (Kc) in terms of the enthalpy and entropy changes of the reaction and the temperature:

Kc = e^(-ΔG/RT) = e^(-ΔH/RT) * e^(ΔS/R)

where R is the gas constant.

The first exponential term on the right-hand side of this equation represents the Boltzmann factor, which is a measure of the probability that a reaction will occur. The second exponential term represents the entropy of the system, which is a measure of the number of ways in which the particles in the system can be arranged.

At equilibrium, the forward and reverse reaction rates are equal, which means that the rate constants for the forward and reverse reactions are equal. This can be expressed mathematically as:

k1[A]^a[B]^b = k2[C]^c[D]^d

where k1 and k2 are the rate constants for the forward and reverse reactions, respectively.

Substituting this equation into the expression for the equilibrium constant (Kc), we obtain:

Kc = k2/k1 = ([C]^c [D]^d) / ([A]^a [B]^b)

This is the law of mass action, which relates the equilibrium constant to the concentrations of the reactants and products at equilibrium.

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