If I said how is my reaction going for 5 minutes? I initially had a certain amount of concentration of A, and I had my reaction going for 5 minutes. How much A do I have left? Well, integrated rate law can tell me that. Differential Rate Law cannot tell me that. Differential Rate Law can say how long it's going to take for the whole reaction to occur, but the Integrated Rate Law talks about the reaction at a specific moment in time.
So that's the big difference between differential and integrated. If you're asking for a specific moment in time, you cannot use Differential Rate Law. So hopefully that helps you in determining when do you use which equation, because I know that it can sometimes get confusing. Hope that helped. Previous Unit Chemical Reactions. Next Unit Chemical Solutions.
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Explanation Transcript. Chemistry Kinetic-Molecular Theory. Science Biology Chemistry Physics. However, the integrated first-order rate law is usually written in the form of the exponential decay equation. For a reaction that is second-order overall, and first-order in two reactants, A and B, our rate law is given by:. There are two possible scenarios here.
The first is that the initial concentrations of A and B are equal, which simplifies things greatly. This is the standard form for second-order rate law, and the integrated rate law will be the same as above. Note here that a plot of [A] versus t will yield a straight line with the slope -k. The y-intercept of this plot will be the initial concentration of A, [A] 0.
The important thing is not necessarily to be able to derive each integrated rate law from calculus, but to know the forms, and which plots will yield straight lines for each reaction order. A summary of the various integrated rate laws, including the different plots that will yield straight lines, can be used as a resource. The integrated rate law for a zero-order reaction also has the form of the equation of a straight line:. Figure 3 shows a plot of [NH 3 ] versus t for the decomposition of ammonia on a hot tungsten wire and for the decomposition of ammonia on hot quartz SiO 2.
The decomposition of NH 3 on hot tungsten is zero order; the plot is a straight line. The decomposition of NH 3 on hot quartz is not zero order it is first order. From the slope of the line for the zero-order decomposition, we can determine the rate constant:.
In each succeeding half-life, half of the remaining concentration of the reactant is consumed. Using the decomposition of hydrogen peroxide Figure 1 in Chapter During the second half-life from 6. The concentration of H 2 O 2 decreases by half during each successive period of 6. The decomposition of hydrogen peroxide is a first-order reaction, and, as can be shown, the half-life of a first-order reaction is independent of the concentration of the reactant.
However, half-lives of reactions with other orders depend on the concentrations of the reactants. We can derive an equation for determining the half-life of a first-order reaction from the alternate form of the integrated rate law as follows:. We can see that the half-life of a first-order reaction is inversely proportional to the rate constant k. A fast reaction shorter half-life will have a larger k ; a slow reaction longer half-life will have a smaller k.
Solution The half-life for the decomposition of H 2 O 2 is 2. Check Your Learning The first-order radioactive decay of iodine exhibits a rate constant of 0. What is the half-life for this decay? We can derive the equation for calculating the half-life of a second order as follows:. Consequently, we find the use of the half-life concept to be more complex for second-order reactions than for first-order reactions. Unlike with first-order reactions, the rate constant of a second-order reaction cannot be calculated directly from the half-life unless the initial concentration is known.
We can derive an equation for calculating the half-life of a zero order reaction as follows:. The half-life of a zero-order reaction increases as the initial concentration increases. Equations for both differential and integrated rate laws and the corresponding half-lives for zero-, first-, and second-order reactions are summarized in Table Differential rate laws can be determined by the method of initial rates or other methods. We measure values for the initial rates of a reaction at different concentrations of the reactants.
From these measurements, we determine the order of the reaction in each reactant. Integrated rate laws are determined by integration of the corresponding differential rate laws. Rate constants for those rate laws are determined from measurements of concentration at various times during a reaction. The half-life of a reaction is the time required to decrease the amount of a given reactant by one-half.
The half-life of a zero-order reaction decreases as the initial concentration of the reactant in the reaction decreases. The half-life of a first-order reaction is independent of concentration, and the half-life of a second-order reaction decreases as the concentration increases.
What is the half-life of this reaction? What fraction of the cyclopropane remains after 0. Fluorine is a radioactive isotope that decays by positron emission to form oxygen with a half-life of Physicians use 18 F to study the brain by injecting a quantity of fluoro-substituted glucose into the blood of a patient.
The glucose accumulates in the regions where the brain is active and needs nourishment. The isomerization of cyclobutene to butadiene is first-order and the rate constant has been measured as 2. Determine the partial pressure of cyclobutene and its concentration after The plot is nicely linear, so the reaction is second order. The reaction is first order. Skip to content Chapter Learning Objectives By the end of this section, you will be able to:.
Answer: Figure 1.
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