# Kinetics problem solving strategy

Problems that deal with Kinetics usually involve determining reaction order, finding rate constant and quantities of reactants or products after a given time.

## Determining reaction order

There are few ways to determine reaction order. Probably the most universal way is to calculate it mathemathically, that is to find how many times the rate of the reaction increases (decreases) when higher (lower) concentrations of reagents are used. For example, if rate of the reaction doubles when the concentration of certain reagent is increased two-fold, reaction is of first order in relation to that reagent (more of this method in Worked Example 1).

Second way is to determine it from a graph where a mathemathical function of concentration of reagent A is plotted upon time . Specifically, if dependence of concentration of reagent A on time is linear then the reaction is of zeroth order in relation to A, if dependence of ln [A] on time is linear, then reaction is of first order in relation to A, if dependence of 1/[A] on time is linear then the reaction is of second order. If neither of these mathemathical functions of [A] is linear, then other usage of other method is advised.

Third way is to determine it from the half-life of a reagent. If each half-life is two times shorter then it's predecessor, then the reaction is of zeroth order. If half-lifes remain constant throughout the reaction, then it is of first order. If each half-life is two times longer than it's predecessor, then it is of second order. If it impossible to determine, other methods must be used.

## Determining rate constant

Rate constant can be ontained by several ways, such as calculated from known date (as in worked example 1), determined from slope of the reaction graph or from half-life (this way is common only in first order reactions such as radioactive decay).

For rate constant to be determined reaction rate law must be known. If law is known, rate is determined by substituting rate and concentrations with known data. If rate is not known, equation system must be used and concentrations and relative rates must be known or determined.

Another way that is commonly used experimentically is to plot linear graph and calculate it's slope. Once again rate law must be known or determined in plotting process to use proper methemathical function to get linear graph. If, for example, reaction is of first order, dependance of natural logarithm of [A] on time must be plotted. Then assuming that the graph is correct any two points may be chosen and rate determined by dividing differences of logarithmed consentration by differences in time (k = Δln[A]/Δt or k = (ln[A]1 - ln[A]2) / (t1 - t2). Note that different formulas are used for different rate laws (k = Δ[A]/Δt for zeroth order reactions and k = (1/[A])/t for second, while equations for other rate laws may be got by integrating differential kinetic equations).

Third way used for first order reactions is to determine rate constant from equation k = ln 2 / t1/2 , where t1/2 is the half-life of a reaction. In radioactive reactions λ symbol is used instead of k in noting rate constants.This method will be explored in worked example 2.

## Determining concentration after time

Various concentrations after certain time has passed are determined by using integrated law equations, so reaction law must be known. Purely mathemathical method is used - if we have a first order equation, we substitute known time, rate constant and initial concentration with known numbers into integrated first order kinetic equation ln [A] = ln [A]0 - kt. This equation may be used in different ways - rate conctant may be found if we know any two concentrations and difference of time (essentially it is the same method as fnding slope of a plotted kinetic graph), elapsed time may be found if we known concenteation at initial and final moment as well as rate constant, initial concentration can be found if we know initial concentration and rate conctant as well as elapsed time.

## Worked example 1

Problem: following reaction is taking place: A + B → C. Find reaction orders with respect to A and B and the rate constant by using the following information about reaction rates:

 [A] [B] Rate 0.1 0.1 0.003 0.2 0.1 0.006 0.1 0.2 0.012

Solution: by examining second and third rows it can be seen that by doubling the concentration of A the rate of reaction increases two-fold. It is safe to conclude that the reaction is of first order with respect to A since 21 = 2. Then by examining second and fourth rows we notice that by doubling the concentration of B the rate increases four times. By following the same logic we conclude that the reaction is of second order with respect  to B since 22 = 4. Rate equation can be written as rate = k [A]1[B]2. Rate consant can be calculated from any row, we will use second one in this example. By substituting letters with kwown values we get equation 0.003 = k × 0.1 × 0.12 ; k = 0.003 / 0.1 / 0.12 = 3

## Worked example 2

Problem: find quantity of radioactive element [A] left after 1000 years, if we know that half of it decays in 1500 years and initially we have 1000 grams of it.

Solution: it is known that radioactive decay obeys laws of first order kinetics. Half life is given so we may find rate constant  - λ = ln 2 / t1/2,  λ = 0.693 / 1500 years = 4.6 × 10-4 years-1. Since mass of radiactive substance is proportional to quantity, it may be used in first order integrated equation instead of concentration (this is the case in most of situations). So ln m(A) = ln (1000 g) - 4.6 × 10-4 years-1 × 1000 years = 6.91 - 0.46 = 6.45, [A] = e6.45 = 633 (grams)