First Order Reaction

A general unimolecular reaction



where A is a reactant and P is a product is called a first-order reaction.

The rate is proportional to the concentration of a single reactant raised to the first power.

The decrease in the concentration of A over time can be written as:



Equation (2) represents the differential form of the rate law. Integration of this equation and determination of the integration constant C produces the corresponding integrated law.

Integrating equation (2) yields:

The constant of integration C can be evaluated by using boundary conditions. When t = 0, [A] = [A]₀. [A]₀ is the original concentration of A.

Substituting into equation (3) gives:
Therefore the value of the constant of integration is:
Substituting (5) into (4) leads to:

Plotting   ln[A]   or   ln[A] / [A]₀   against time creates a straight line with slope   -k. The plot should be linear up to a conversion of about 90%.

Equation (6) can also be written as: This means that the concentration of A decreases exponentially as a function of time.

The rate constant k can also be determined from the half-life t_1/2. Half-life is the time it takes for the concentration to fall from [A]₀ to [A]₀ / 2.

According to equation (6) is obtained:



Pseudo First Order Reaction

A and B react to produce P:



If the initial concentration of the reactant A is much larger than the concentration of B, the concentration of A will not change appreciably during the course of the reaction The concentration of the reactant in excess will remain almost constant. Thus the rate's dependence on B can be isolated and the rate law can be written


Equation (1) represents the differential form of the rate law. Integration of this equation and evaluation of the integration constant C produces the corresponding integrated law.

Substituting [B] = c into equation (1) yields:

Integrating equation (2) gives: The constant of integration C can be evaluated by using boundary conditions. At t = 0 the concentration of B is c₀.

Therefore
Accordingly is obtained:
If the decrease in concentration of B is followed by photometric measurement the Beer' Law must be taken into account.

Combining equation (4) and Beer' Law

A = absorbance, e = molar absorbtivity with units of L · mol ^-1 cm ^-1
c = concentration of the compound in solution, expressed in mol · L ^-1
P₀= radiant power for radiation entering; P= radiant power for radiation leaving

gives the relationship between k' and lnA:



One needs only monitor the relative concentration of B as a function of time to obtain the pseudo-first order rate constant k'. The value of k' can then be divided by the known, constant concentration of the excess compound to obtain the true constant second order k:


The pseudo-first order rate constant k' can be also determined from the half-life t_1/2.

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