orders+from+graphing

Reactant orders are determined using experimental data. Other than using reactant concentrations, plots of concentration versus time can also be used.


 * First-Order Rate Constants**

As an example, look at the reaction of CH 3 NC (g) --> CH 3 CN (g):

Plot (a) is pressure versus time, and plot (b) is the natural log of pressure versus time.

//A plot of the natural log of concentration (or pressure) versus time for a **first order** reactant will yield a l**inear equation with a slope = - k**.//

The slope of the line in plot (b) is calculated using rise over run:

slope = m = change in P/change in t

m = (5.0 - 4.0)/(0 - 20,000) = -0.000050 torr/sec

therefore, k = - m = 0.000050 torr/sec.


 * Second-Order Rate Constants:**


 * Second-order kinetics** is similar in that a plot of the //inverse concentration// or pressure versus time will yield a linear equation where the **slope = k**.



Similar to first-order kinetics, the slope will provide the rate constant.

slope = m = change in 1/[A]/change in t

m = (2.0 - 0)/(1 - 0) = 2.0 uM/sec

therefore, k = m = 2.0 uM/sec.


 * First-order Half-Life**

The half-life equation for first-order reactions (such as radioactive decay) is:


 * t 1/2 = 0.693 / k**

Knowing half-life or the rate constant allows for determining the other.

Look at plot (a) for the reaction CH 3 NC (g) --> CH 3 CN (g). The initial pressure was 150 torr, so at its half-life, it would be 75 torr. This corresponds to approximately 13,000 seconds. But since the rate constant is known (0.000050 torr/sec), it can be calculated:

t 1/2 = 0.693 / 0.000050 =13,860 sec = 14,000 sec taking significant figures into account.

And of course, the half-life can be used to calculate the rate constant in this situation, as well.


 * Second-order Half-Life**

The half-life equation for second-order kinetics is slightly different:


 * t 1/2 = 1/(k[A]**** 0 ****)**

where [A]** 0 ** is the initial concentration of the reactant.

The left-hand plot of [A] vs. t for the reaction A + A --> P is difficult to use in terms of estimating the half-life, so the equation can do this for us:

t 1/2 = 1/(2.0 x 10) = 0.05 sec

In conclusion, the rate constant, order, and half-life are all mathematically related to one another, and make dealing with kinetics a bit easier.