Derive the integrated equation for first order kinetics and give its characteristics

First order kinetics is a type of reaction rate that depends on the concentration of a single reactant. The rate law for first order kinetics is expressed as:

Rate = k[A]

where:

• Rate is the speed at which the reactants turn into products.
• k is the rate constant.
• [A] is the concentration of the reactant.

To derive the integrated rate law for first order kinetics, we start with the differential rate law:

d[A]/dt = -k[A]

This equation says that the rate of disappearance of A is proportional to the concentration of A.

We can separate the variables and integrate:

∫d[A]/[A] = -k ∫dt

This gives us the natural logarithm:

ln[A] = -kt + C

where C is the integration constant.

To solve for C, we can use the initial condition that at time t=0, [A]=[A]₀ (the initial concentration of A). This gives us:

ln[A]₀ = C

Substituting this back into our equation gives us the integrated rate law for first order kinetics:

ln[A] = -kt + ln[A]₀

or

ln([A]₀/[A]) = kt

This equation tells us that the natural logarithm of the ratio of the initial concentration to the current concentration is directly proportional to time.

Characteristics of first order kinetics:

1. The rate of the reaction depends on the concentration of a single reactant.
2. The half-life of the reaction (the time it takes for half of the reactant to be used up) is independent of the initial concentration.
3. The plot of ln[A] versus time gives a straight line with a slope of -k.
4. The units of the rate constant k are s⁻¹ (per second), as the reaction rate is concentration per unit time.