Integrated Rate Laws
Method of Initial Rates 
The variation in concentration with time provides a highly detailed description of how fast the reaction is occurring. In many circumstances, though, it is desirable to have a simple, approximate measure of the reaction rate, and the half-life provides such a measure.
The half-life, t1/2, is the time it takes for one-half of the original amount of material to react. If the initial concentration of a reactant A is 0.100 mole L-1, the half-life is the time at which [A] = 0.0500 mole L-1.
Intuitively, the faster the reaction, the shorter the half-life. The rate of the reaction is proportional to the rate constant; thus the larger the rate constant, the shorter the half-life.
The following exercises illustrate how the half-life depends upon the concentration of reactant for zero-, first-, and second-order reactions.
The graph at the left shows how the concentration of reactant A varies with time for a zero-order reaction. To create the plot, enter an initial concentration for A in the box and click the "Create Plot" button.
Use the cursor to determine the half-life of the reaction. Do this by locating the point on the concentration-time curve that corresponds with one-half the original amount of reactant. While the cursor is over this point, press the left mouse button to display the time, which is the half-life.
Perform this measurement for several different initial concentrations. For each initial concentration, enter the half-life in the appropriate box and add this point to the graph at the right. The dependent variable in the graph at the right is the half-life. Select an option for the independent variable: [A]0, ln [A]0, or 1/[A]0.
Which plot gives a straight line?
Write the integrated rate law for a zero-order reaction, and derive an expression for the half-life as a function of the initial concentration.
Is your relationship consistent with the plot you found to yield a straight line?
After at least five points are plotted, click on the "Plot Line" button, which will plot the line-of-best-fit on the graph at the right. Use the slope and/or intercept to determine the rate constant for the reaction.
Perform the same measurements as described for the zero-order reaction.
Which plot gives a straight line?
Write the integrated rate law for a first-order reaction, and derive an expression for the half-life as a function of the initial concentration.
Is your relationship consistent with the plot you found to yield a straight line?
After at least five points are plotted, click on the "Plot Line" button, which will plot the line-of-best-fit on the graph at the right. Use the slope and/or intercept to determine the rate constant for the reaction.
Perform the same measurements as described for the zero-order reaction.
Which plot gives a straight line?
Write the integrated rate law for a second-order reaction, and derive an expression for the half-life as a function of the initial concentration.
Is your relationship consistent with the plot you found to yield a straight line?
After at least five points are plotted, click on the "Plot Line" button, which will plot the line-of-best-fit on the graph at the right. Use the slope and/or intercept to determine the rate constant for the reaction.
Integrated Rate Laws
Method of Initial Rates
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