An alternative term to that of half-life is the lifetime of the reaction; it is denoted by t. Its value is the reciprocal of the first order rate constant, i.e., t = 1/k. We gain further insight into the meaning of t from Eqn [1.10]:

In words, t is the time taken for [A] to fall to 1/e, or 0.368, of its initial value.

Q: What are the values of the half-life and lifetime of the reaction shown in the previous question? Draw a solid vertical line at the half-life, and a dashed line at the lifetime, on the decay curve.

A: Since k = 0.5 s-1,then from Eqn [1.13], t1/2 = 0.693/0.5 = 1.39 s, and from Section 1.4.3, t = 1/0.5 = 2 s. Notice that the lifetime is longer than the half-life by a factor of 1/Log, or 1.44.

The requisite graph is generated using the following functions.

First we define the lines to be drawn. These are generated using the Graphics function of which more details can be found in the Mathematica help browser.

halfLifeLine = Graphics[

{AbsoluteThickness, Line[{{1.39, 0}, {1.39, 10}}]}]; lifeTimeLine = Graphics[{Dashing[{0.05, 0.02, 0.05, 0.02}], Line[{{2.0, 0}, {2.0, 10}}]}];

The lines are displayed together with the plot of Eqn [1.10] by using the Show function. Recall that we assigned the plot of Eqn [1.10] to the name gph1 in the previous example.

Show[{gph1, halfLifeLine,lifeTimeLine}, AxesLabel -> {"Time ", "Concentration" }] ;

Concentration

Concentration Figure 1.3. Exponential decay with [A]0 = 10 mmol L-1 and k = 0.5 s-1, showing the value of the half-life of the reaction (vertical solid line) and the lifetime of the reaction (vertical dashed line), respectively.
 Graphics [ primatives, represents a two-dimensional graphical image. See option —> value] help browser for a list of primatives and options Showg, g2, shows several plots and graphics objects combined ... , option —>value]

Dealing with graphics.

Dealing with graphics.