# 14.5 Integrated Rate Laws ## Integrated Rate Laws

The primary purpose of the integrated rate laws is that they allow us to calculate concentration changes over time.  Each equation is specific to its order so the order of a reactant must be known before one can calculate its change in concentration over time.  There are four variables in the equation: [A], [A]0, k, and t.  As long as three are know the fourth can be calculated.

Zero Order  1st Order  2nd Order  ### Why are They Called 'Integrated' Rate Laws

The 'integrated' rate laws are so called as they are each the solution of a differential equation which is solved by separation of variables followed by integration.  The derivation of the integrated rate laws is generally not required of students of General Chemistry (whether high school or college).

### Half Life Formulas

The half life of a reactant is the time it takes until only half of the reactant remains.  From the above table it can be seen that the half life for a Zero order or 2nd order reaction depends upon the initial concentration of the reactant.  For a Zero order reactant, the half life becomes shorter and shorter over time.  For a 2nd order reactant the half life becomes longer and longer over time.  It is only for a 1st order reactant that the half life is independent of the initial reactant concentration which means that it will remain constant over time.

Radioactive decay proceeds according to 1st order kinetics and each radioactive nuclide has a characteristic half life.  Had radioactive decay been Zero or 2nd order this would not have been the case as the half life of a nuclide would have been dependent upon how much one initially had.

Take for example francium-223 which has a half life of only 22 minutes.  If you started with a 100g sample then only 50g of francium-223 would remain after 22 minutes.  After an additional 22 minutes only 25g would remain, and after an additional 22 minutes only 12.5g would remain, etc. ### Plots Involving the Integrated Rate Laws

The final place the integrated rate laws are useful is in graphical relationships between the reactant concentration or a function of the reactant concentration vs time.  Each of the integrated rate laws can be matched up to the slope-intercept equation of a line to determine what should be plotted and the x-and y-axes to obtain a straight line.

For a Zero order reactant a plot of concentration vs time is linear.

For a 1st order reactant a plot of natural log of concentration vs time is linear.

For a 2nd order reactant a plot of 1 over concentration vs time is linear.

### Plots for Zero Order Integrated Rate Laws

The following table shows that for a Zero order reactant plotting [A] vs time will yield a straight line having a negative slope (the slope of the line will equal -k and can be used to determine the value of the rate constant). The following shows the three relevant plots for a zero order reactant and how that only a plot of concentration vs time is linear. ### Plots for 1st Order Integrated Rate Laws

The following table shows that for a 1st order reactant plotting ln[A] vs time will yield a straight line having a negative slope (the slope of the line will equal -k and can be used to determine the value of the rate constant). The following shows the three relevant plots for a 1st order reactant and how that only a plot of natural log concentration vs time is linear. ### Plots for 2nd Order Integrated Rate Laws

The following table shows that for a 2nd order reactant plotting 1/[A] vs time will yield a straight line having a positive slope (the slope of the line will equal +k and can be used to determine the value of the rate constant). The following shows the three relevant plots for a 2nd order reactant and how that only a plot of 1 over concentration vs time is linear.  