The Change of Concentration with Time

Rate laws tell us what the rate of a reaction is in terms of the concentration(s) of reactant(s)

• Rate laws can be converted into equations that tell us what the concentrations of the reactants (or products) are at any time during the course of the reaction.

A reaction with a single reactant, where the reaction rate is linearly proportional to the concentration of the reactant (i.e. the 1^st power of the reactant) is a first order reaction:

• What this says is that the rate at which the concentration of reactant A decreases over time (i.e. the rate at which it is used up in the reaction) is proportional to the concentration of reactant A. This makes sense, in that we expect the rate to slow down as A is used up (and its concentration decreases)
• The proportionality constant, k, will have units of inverse time, like sec^-1 or min^-1, so that k[A] will have units of M/sec or M/min (which are appropriate units for rates of change of concentration)

This can be rearranged to relate the effects of a change in time to a change in the concentration of reactant A:

Using calculus, this equation can be transformed (i.e. integrated) to yield an equation that relates the concentration of A at the start of the reaction [A]₀, to its concentration at any other time t, [A]_t:

Rearranging to solve for [A]_t:

This equation relates the concentration of reactant A after some time t, if given the initial concentration ([A]₀) and rate constant k. This equation actually has the form of a linear equation, y = mx + b

• y = ln[A]_t
• slope (m) of the line = -k
• x = t
• y-intercept of the line = ln[A]₀

The conversion of methyl isonitrile (CH₃NC) to acetonitrile (CH₃CN) is a first order reaction:

CH₃NC -> CH₃CN

• CH₃NC is a gas and so the relative amount can be monitored by pressure
• Here is data collected that follows the above reaction as a function of time

• A plot of pressure vs time looks like this:

• A plot the ln(Pressure) vs time looks like this:

• The slope of the line is approximately -5.25 x 10^-5 s^-1. This is the negative value of the rate constant, k.
• The y-intercept of the line is approximately 5.00 (dimensionless units). This value is equal to the natural log of the initial concentration, or ln[A]₀
• To solve for [A]₀ calculate e^ln[A]0. It is approximately 148 (units will now be torr) - close to the value of 150 in the above table.

For a first order reaction the equation:

can be used to determine:

1. The concentration of a reactant remaining at any time after reaction has started (i.e. [A]_t), if you know k, A₀, and t
2. The time required for a given fraction of a sample to react (i.e. solving for t if you know the ratio of [A]_t/[A]₀) and k
3. The time required for a reactant concentration to reach a certain level - as in "half-life" calculations (see below)

The half-life of a reaction, also known as t_1/2, is the amount of time it takes for the concentration to drop to one-half of it's initial level

• In other words, the point in time where [A]_t = 1/2[A]₀
• We can determine t_1/2 in a first order reaction by substituting in 1/2[A]₀ for [A]_t:

Note that the half-life is independent of the concentration. This means that if you randomly choose some time to calculate the concentration of reactant, exactly 0.693/k seconds later, the concentration will be 1/2 of what it was

• Radioactive decay behaves this way also

A second order reaction, by definition, can be the result of:

• A reaction involving a single reactant whose rate depends upon the reactant concentration raised to the second power:

Rate of reaction = k[A]²

• A reaction involving two reactants, whose rate depends on the first power of each reactant:

Rate of reaction = k[A][B]

• For a reaction that is second order in just one reactant (A), the rate law is given by:

Rate = -D[A]/Dt = k [A]²

• This is another linear function (y = mx +b) with
• Slope (m) = rate constant, k
• x values are time
• y values are corresponding inverse concentrations of reactant
• the y-intercept is the inverse of the value of the starting concentration

• What is the half-life (t_1/2) of a second order reaction?

• For second-order reactions, the half-life (t_1/2) is dependent upon the initial concentration