Chemical Kinetics�Lecture 3

CML101

July 31, 2024

First Order Reactions

Consider a simple 1st order rxn: A → P

How much A is left after time t? Integrate:

Differential form:

First order reactions

The integrated form of first order rate law:

Can be rearranged to give:

[A]₀ is the initial concentration of A (t=0).

[A]_t is the concentration of A at some time, t, during the course of the reaction.

For a first-order reaction, the concentration at any time, labeled [A]_t, decreases in a characteristic negative exponential way. The rate at which it approaches a zero concentration is dictated by the value of the rate constant

If a reaction is first-order, a plot of ln [A]_t vs. t will yield a straight line with a slope of –k.

Plots of [A] as a function of time for various rate constants k. The rate constant of a given curve is provided in the figure.

The natural log of reactant concentration as a function of time for a first-order chemical

reaction

The length of time required for half of the reactant to disappear is called the half-life of the reaction and is written as t_1/2

Notice that the half-life for a first-order reaction is independent of the initial concentration, and only the rate constant of the reaction influences the half-life

First order Kinetics – Radioactive Decay

The radioactive decay of an unstable nucleus is an important example of a process that follows a first-order rate law. Choosing Cu⁶⁴ as an example, we have the transformation

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The emission of a β-particle occurs with the formation of a stable isotope of zinc. The probability of this occurrence in the time interval dt is simply proportional to dt.

-dN is the number of copper nuclei that disintegrate in the interval dt.

N₀ being the number of Cu⁶⁴ nuclei present at t = 0, N the number at any time t

λ Is the decay constant and is related to he half life by eqn 3 above.

First order Kinetics – Bacterial Growth

A bacterial colony grows most commonly by cell division. Thus

where dN is the number of cells that divide in the time interval dt, and λ_g is a constant. This growth law is very similar to the law of radioactive decay, except that the negative sign is missing. Upon integration we obtain

The generation time, t_g , is the time required for the population to double ; that is, when t = t_g, N = 2N₀.

Express N/N₀ in terms of t_g

The growth law, as given in the previous page, is not applicable during the entire history of a bacterial colony. A typical population curve, N versus t, is shown in the adjacent figure. There is an initial induction period, followed by a during which the exponential growth occurs, as described by the equation on the previous page. The population growth slows, then stops ; in the final phase the population drops as the bacteria die off more rapidly than they are produced.

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Equation describes the growth only during the exponential phase in the interval. The leveling off occurs as the supply of nutrients is exhausted. Finally, if the environment becomes sufficiently hostile (due to lack of nutrients or increased concentrations of toxic substances), the colony dies.

Bacterial Growth in the Lab

www.youtube.com/watch?v=jhbEQclX9mk

Hydrolysis of Cisplatin (anticancer drug)

Cisplatin, the first “inorganic” anticancer drug to be discovered, is unique in its ability to cause complete remission of the relatively rare, but deadly cancers of the reproductive organs in young adults. The structures of cisplatin and its hydrolysis product have been given above. Both platinum compounds have four groups arranged in a square plane around a Pt(II) ion. The reaction shown is important because cisplatin, the form in which the drug is administered, is not the form in which the drug is active. Instead, at least one chloride ion must be replaced by water to produce a species that reacts with deoxyribonucleic acid (DNA) to prevent cell division and tumor growth. Consequently, the kinetics of the reaction in the figure have been studied extensively to find ways of maximizing the concentration of the active species. -- Example of first order reaction

Excited State Decay

Second-Order Reactions

NOBr(g) → NO(g) + Br₂ (g)

Slope = k_r

k_r = 2.01 dm³mol^-1s^-1

Another form of Second Order rate law

A + B → P

Derive the expression below for the rate law provided on the previous page!

Derive the integrated rate law if the reaction is written as below, with the reaction

being first order with respect to A and first order with respect to B.

aA + bB → P

A plot of the expression on the left against t should be a straight line from which kr can be obtained.

The problem below shows how the reaction rate law can be found by combining the method of isolation and the predictions of integrated rate laws

Draw graphs using any plotting software and convince yourself!

Reversible Reactions