Chemistry 8th Edition / Chang | ||||||
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Student Study Guide |
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Index | 13.1 | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 |

THE RELATION BETWEEN REACTANT CONCENTRATION AND TIME (13.3)

STUDY OBJECTIVES

- Determine for first- and second-order reactions the concentration of a reactant at any time after the reaction has started, or the time required for a given fraction of the sample to react.
- Determine the half-life of a first-order reaction.
- Apply rate law equations and graphs to determine whether a reaction is first order or second order.

** 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.

A products

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, t

t _{1/2}=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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