|Chemistry 8th Edition / Chang|
|Student Study Guide
THE RELATION BETWEEN REACTANT CONCENTRATION AND TIME (13.3)
First-Order Reactions. One of the most widely encountered kinetic forms is the first-order rate equation. In this case the exponent of [A] in the rate law is 1.
For a first-order reaction the unit of the rate constant is reciprocal time, 1/t. Convenient units are s–1, h–1, etc.
The equation that relates the concentration of A remaining to the time since the reaction started is
This is a very useful equation called the integrated first-order equation. Here [A]0 is the concentration of A at time = 0, and [A] is the concentration of A at time = t. The rate constant k is the first-order rate constant. The concentration [A] decreases as the time increases. This equation allows the calculation of the rate constant k when [A]0 is known, and [A] is measured at time t. Also, once k is known, [A] can be calculated for any future time.
To determine whether a reaction is first order, we rearranged the first-order equation into the form:
ln [A] = – kt + ln [A]0
corresponding to the linear equation
y = mx + b
Here m is the slope of the line and b is the intercept on the y axis. Comparing the last two equations, we can equate y and x to experimental quantities.
y = ln [A] and x = t
Therefore, the intercept b = ln [A]0, and the slope of the line m = –k. Thus a plot of ln [A] versus t for a first-order reaction gives a straight line with a slope of –k as shown in Figure 13.1 below. If a plot of ln [A] versus t yields a curved line, rather than a straight line, the reaction is not a first-order reaction. This graphical procedure is the method used by most chemists to determine whether or not a given reaction is first order.
Figure 13.1 A plot of [A] versus time for a first-order reaction gives a curved line. A plot of ln [A] versus time gives a straight line for a first-order reaction.
Half-life. The half-life of a reaction, t1/2, is a useful concept. For a first-order reaction, the half-life is given by:
t1/2 = ln 2 k
where ln 2 (0.693) is a constant and k is the rate constant. Knowledge of the half-life allows the calculation of the rate constant k. The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of the initial value. After one half-life, the ratio [A]/[A]0 is equal to 0.5.
If the reaction continues, then [A] will drop by 1/2 again during the second half-life period as shown in Figure 13.2. After two half-life periods the fraction of the original concentration of A remaining, [A]/[A]0, will be 1/2 of the concentration remaining after the first half-life, so [A]/[A]0 = 0.5 x 1/2 = 0.25.
Figure 13.2 A plot of [A] versus time for a first-order reaction gives a curved (exponential) line. Over each half-life period, [A] drops in half.
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