Reaction Rates

Reaction rates are defined as the change in the concentration of either reactants or products with respect to a change in time.  Reactant concentrations will decrease, and so have a negative sign, while product concentrations will increase and carry a positive sign.  For a simple reaction like, A  B,  the reaction rate can be expressed as :   where the triangle or "delta" means "change" and the squared brackets mean concentration in mol/L or molarity(M).  The script "t" is time.  Reaction rates can refer to : the initial reaction rate where time, t = 0 and no product has accumulated,  instantaneous rates that occur at a particular point in time, or average rates of reaction that represent the average of a series of instantaneous rates.

Let's look at an example reaction --- the reversible gas-phase reaction between ethylene and ozone; one of many reactions that are responsible for the formation of photochemical smog.  The reaction arrow in this equation shows that this is a reversible reaction. The balanced equation for the reaction looks like this: In this balanced equation all the coefficients for reactants and products are equal to 1 so the product concentration increases at the same rate as the reactant concentration declines.  The rate for this reaction can be expressed as: When both concentration and time data is available calculations for instaneous, initial, and average rates of reaction can be calculated directly. However, reaction rates cannot be determined from looking at a balanced equation.  All reaction rates must be determined from experimental data.

It is not uncommon to have very sparse data from which to calculate a reaction rate for a particular chemical change.  Following is a summary  of other methods that can be used when only standard tabularized constants or partial data are available.
 

When time data is not available, it is possible to calculate the rate of a reaction from standard tables of  reaction rate constants. Reaction rate constants are specific to each reaction and remain constant even as the reaction progresses (as long as reaction conditions such as temperature are held constant).  When a k is available for the reaction of interest, the rate can be calculated using the  rate law or rate equation.  For a chemical equation of the form
 

                                                         aA + bB + . . .   cC + dD . . .

where the lowercase letters are coefficients, the rate law is equal to k times the product of the reactants and has the form :

                                                                    Rate = k[A]^a[B]^b

Reaction rates are determined from experimental data.

When k values are not available, the reaction rate can be calculated using a graphical interpretation of the rate law equation.  The order of the rate law equation for a particular reaction is defined as the sum of the exponents.  So, for the equation above, the order would be (a + b).  When concentration is plotted versus time first order rates form a straight line only when concentration data is trasformed to the natural log of concentration.  Second order reaction rates plotted in this way form a straight line only when the inverse of of concentration is plotted versus time.  The following table summarizes the information that can be gleaned from plotting transformed concentration data versus time.
 

Reaction Order	Units of k	Rate Law Equation
0th Order change in reaction rate independant of concentration	mol/L* s	Rate=k
	1 /  s	Rate=k[A]
	L/mol * s	Rate=k[A]2 or k[A][B]

When insufficient data is available to plot and determine the best straight line fit,  another method may be used to determine the reaction rate. Consider the experimental data shown in the following table for the reaction A + B   C.  The concentration of A is held constant during experiments 1 and 2 (yellow) while the concentration of B is doubled.  Any change seen in the rate of formation of C in experiments 1 and 2 must be due to the reaction order of B.
 

When B is doubled, no change is seen in the initial rate of C.  Therefore, the rate order with respect to B must be 0th.  What about A?  Between experiments 1 and 3 (peach) the concentration of B remains constant but the concentration of A is tripled.  What happened to C during the tripling of A? it increased by a factor of 9( or 3 * 3) so a is 2nd order.  The rate equation then is  Rate = k[A]²[B]⁰ with an overall 2nd order rate.  Solving for k=4.0 x 10^-5/(.100)² = 4.03 x 10^-3 1/mol * s.
 

In some cases, no direct experimental data is available but values for temperature, constants like , E_a(activation energy), and R, the gas constant are available.  Then the Arhenius Equation can be used to calculate a k from which the rate can be determined.  The Arrhenius Equation is:

                                                                                    
where A is a constant related to the geometry needed for a collision to produce a chemical reaction, e is a constant approximately 2.7218, R is the gas constant = 8.314 J/ mol * K, and T is the temperature in K (Kelvin).  The Arrhenius equation can also be used to evaluate the effect of temperature, geometry and activation energy on the rate constant.  For example, when the activation energy is very large, k decreases.  Look for the Arrhenius' Equation Calculator to explore the relationship between temperature and other variables in kinetics calculations.