Integrated Rate Laws 1 1

How are “integrated rate laws” different from “rate laws?” A rate law gives the speed of reaction at a given point in time, and how it is affected by the concentration of the reactants. An integrated rate law gives the change in concentration from time zero up to a given point in time. It describes how the rate changes over time. 2 2

Integrated Rate Law The following formulae are given on the AP data sheet: They can be derived using calculus: First order reactions Second order reactions Chem FAQ: How do I calculate reaction rate using a rate law? 3

Use of the Integrated Rate of 1st Order Rxns Example Dinitrogen pentoxide is not very stable. In the gas phase or dissolved in a nonaqueous solvent, like CCl4, it decomposes by a 1st order rxn into dinitrogen tetroxide and molecular oxygen. 2 N2O5  2 N2O4 + O2 The rate law is Rate = k[N2O5] At 45oC, the rate constant for the rxn is 6.22 x 10-4s-1. If the initial concentration of N2O5 at 45oC is 0.500 M, what will its concentration be after exactly one hour? Ans. 0.053 M 4

Half-life (t½) of a Reactant "Half-life" of a reactant is the time it takes for ½ of the reactant to react. It is NOT half the time it takes for all of the reactant to react. For example, t½ of the radioisotope I-131 is 8.0 days. Starting with 20 g of I-131, after 8.0 days, there would be 10 g left. After a total of 16.0 days (two half-lives), there would be…… 5 g left

Application of Half-life I-131 has a half-life of 8 days. After 4 half-lives (32 days), it is down to 1/16 of its original concentration.

If we start with 20.0 g of I-131, after 40 days, how much of I-131 is still there? (Half-life of I-131 = 8 days.) What percent of I-131 remains after 24 days? What fraction of I-131 remains after 48 days?

Example: A patient is given a certain amt of I-131 as part of a diagnostic procedure for a thyroid disorder. What fraction of the initial I-131 would be present in a patient after 25 days if none of it were eliminated through natural body processes? t1/2 = 8.02 days What to do if # days is not an exact multiple of the half-life? We solve this problem using the integrated rate law for 1st order rxn?

Half-life is a measure of the speed of reaction: The shorter the half-life, the faster is the reaction. By definition, after one half-life, [A]t = ½ [A]o For a 1st order rxn, Note that half-life is not affected by concentration. How do you know that? There is no concentration term in the equation.

Half-life = 8.02 days, after 25 days?

The half-life of I-132 is 2.295h. What percentage remains after 24 hours? (Previous problem was for I-131.) % is based on 100 Ans. 0.071 % I-132 remains.

Integrated Rate Law of 2nd Order Reactions They are of several types: Rate=k[A]2, Rate=k[A]1[B]1 and Rate=k[A]2[B]0, etc… 2nd order means the powers must add up to 2. The integrated equation is of the form ChemFAQ: How does concentration vary with time in a second order reaction?

Example:Nitrosyl chloride, NOCl, decomposes slowly to NO and Cl2. 2NOCl 2NO + Cl2 The rate law shows that the rate is second order in NOCl. Rate = k[NOCl]2 The rate constant k equals 0.020 L mol-1 s-1 at a certain temp. If the initial conc of NOCl in a closed reaction vessel is 0.050 M, what will the concentration be after 35 minutes? Ans. 0.016 M

Half-life of 2nd Order Reactions Derived from Note that unlike 1st order reactions, half-life of 2nd order rxns depend on the initial conc of the reactant. 2HI(g)  H2(g) + I2 (g) has the rate law, Rate = k[HI]2 with k = 0.079 L mol-1s-1 at 508oC. What is the half-life of this rxn at this temp when the initial HI concentration is 0.10 M? Ans. 1.3x102s

Using a graph to find K for 2nd Order Reactions How can we determine rate constant graphically? Which are the variables? Rearrange ChemFAQ: How does concentration vary with time in a second order reaction? y = mx + b What do we plot on the x-axis? y-axis? What does the slope represent?

Using a graph to find K for 1st Order Reactions Slope is calculated to be - 6.0x10-4 s-1. So, k = 6.0x10-4 s-1 Rate = k[N2O5] Units are correct! What is the slope equal to? slope = - k So, k = - slope Note: (-) does not mean k is negative but it has the opposite sign of the slope.

Determining the order of a reaction by seeing which plot gives a striaght line. Second order: Zero order:[A] vs time First order ln [A] vs time:

Summary of Integrated Rate Laws 1st Order Rxn 2nd Order Rxn