Degree of dissociation for a substance undergoing decay with first order kinetics

The degree of dissociation can be defined as the fraction of the molecules that have dissociated into their constituent atoms or ions.

Formula for first order kinetics

The rate of decay for a substance undergoing first-order kinetics is given by the formula:

rate = k[A]

where k is the rate constant and [A] is the concentration of the substance.

Formula for degree of dissociation

The degree of dissociation for a substance undergoing decay with first order kinetics is given by the formula:

α = 1 – e–kt

where α is the degree of dissociation, k is the rate constant, and t is the time.

Explanation

The formula for the degree of dissociation can be derived by considering the rate law for a first-order reaction. In a first-order reaction, the rate of decay is proportional to the concentration of the substance. This can be expressed mathematically as:

rate = k[A]

where k is the rate constant and [A] is the concentration of the substance.

The degree of dissociation can be defined as the fraction of the molecules that have dissociated into their constituent atoms or ions. Therefore, the degree of dissociation can be expressed as:

α = n / n0

where n is the number of molecules that have dissociated and n0 is the initial number of molecules.

The number of molecules that have dissociated can be calculated from the concentration of the substance using the formula:

n = [A](1 - α)N

where N is the Avogadro's number.

Substituting this expression for n into the equation for the degree of dissociation, we get:

α = 1 - [A](1 - α)N / n0

Simplifying this expression, we get:

α = 1 - e^-kt

where k = ln2 / t, and t is the half-life of the substance.

Therefore, the degree of dissociation for a substance undergoing decay with first order kinetics is given by the formula:

α = 1 – e–kt