Integrated Rate Laws and Activation Energy

There's a new bit of OWL assignment posted, make sure you take a look.

Rate laws can tell us a lot about a reaction, but a simple rate law doesn't do a great job of telling us how fast or slow a reaction really is.  In order to incorporate a time component, we can integrate the rate law expressions.  The integrated rate laws (IRLs) give us a way to monitor the way concentrations change over time.

0^th Order IRL → [A]_t = -kt + [A]₀

1^st Order IRL → ln[A]_t = -kt + ln[A]₀

2^nd Order IRL → {1/[A]_t} = -kt + {1/[A]₀}

IRLs can also be used to determine the order of a reaction with respect to a given reactant.  All of the IRLs listed above are equations of lines.  If we plot [A]_t vs. t and the result is a straight line, then the process must be 0^th order w.r.t. [A].  Likewise, a linear plot of ln[A]_t vs. t implies a 1^st order process, and a linear plot of {1/[A]_t} vs. t implies a 2^nd order process.

Why do reactions have the rates they do?  This is a function of the amount of energy required to make a reaction occur.  Recall from Collision Theory that collisions must occur and those collisions must be oriented and energetic.  The energy required to get a reaction started is the activation energy, E_a.  Activation energy is described by the Arrhenius equation:

k = A exp(-E_a/RT)

where:

k = rate law constant

A = frequency factor

E_a = activation energy

R = universal gas constant, 8.314 J/mol.K

T = temperature in units of Kelvin

Although this is an elegant little bit of mathematics, it's not the most useful form of the Arrhenius equation.  If 2 sets of conditions are known, we can set up a ratio of the Arrhenius equation for each run and ultimately find that:

ln(k₁/k₂) = (E_a/R)({1/T₂} - {1/T₁})

The comparative form works well, but it has a notable flaw.  We have to assume that both of the points of data that we have a quite good and accurate.  Hopefully this is the case, but it might not be, leading to error.  If we want to average out some of that error, we can do another transformation of the Arrhenius equation to form a line:

ln(k) = (-E_a/R)(1/T) + ln(A)

Friday we'll look more closely at activation energy and what it means in terms of reaction mechanisms.