S12E3 - Method of Initial Rates

The Form of the Rate Law is Determined Experimentally

To calculate the form of the rate law, we use the Method of Initial Rates.

Method of Initial Rates

ex:  Consider the reaction below,

     NH₄⁺(aq)  +  NO₂^-(aq)  →  N₂(g)  +  2H₂O(l)

     So,  rate  =  - Δ [NH₄⁺] / Δt  =  k [NH₄⁺]ⁿ [NO₂^-]^m

We need to solve for n, m, and k.  Here's the above information again, in an easier-to-follow handwritten format:

To solve for n, m, and then k, we will use the data from 3 experiments in Table 12-4 below:

-----

➞ FIRST - Calculating n and m, in:  
rate  =  k [NH₄⁺]ⁿ [NO₂^-]^m

- ratio of rates 1 and 2:  the image below shows how we can use experiments 1 and 2 data from Table 12-4.  This allows us to isolate and solve for "m"

- ratio of rates 2 and 3:  the image below shows how we can use experiments 2 and 3 data from Table 12-4.  This allows us to isolate and solve for "n"

As you can see above, we've chosen ratios such that only n or m remains unknown, due to one of them "cancelling out."

So, n = 1 and m = 1, and the reaction is 2nd order overall:

rate  =  k [NH₄⁺]¹ [NO₂^-]¹

Calculating the Rate Constant, k

➞ NEXT - Calculating the rate constant k:  we can use any of the 3 experiments to find k.

Using data from Experiment 1 in Table 12-4, we get:

                              rate  =  k [NH₄⁺]¹ [NO₂^-]¹

  1.35 x 10^-7 mol / L^.s  =  k (0.100M)¹ (0.0050M)¹

  1.35 x 10^-7 mol / L^.s  =  k (0.00050M²)

      0.00027 L / mol^.s  =  k

When solving for k, the units for k are the most difficult part.  So be careful.

=====

Determining a Differential Rate Law

Remember, a "differential rate law" expresses a reaction rate as a function of reactant concentrations.

ex:  Using the data listed below in Table 12-5, calculate the order for each reactant, the overall reaction order, and the value of the rate constant, for the following reaction:

BrO₃^-  +  5Br^-  +  6H⁺  ⟶  3Br₂  +  3H₂O

We cannot use the coefficients of the balanced reaction as our "orders" for each reactant, so we have:

rate  =  k [BrO₃^-]ⁿ [Br^-]^m [H⁺]^p

To solve for n, choose experiments 1 and 2 in Table 12-5 because this enables "m" and "p" unknowns to drop out.  See here...

Similarly, we can solve for m and p by choosing the appropriate experiments:

➞ to solve for m, divide:  rate 3 / rate 2
➞ to solve for p, divide:  rate 4 / rate 1

 =====

Our next topic in SECTION 12 - Chemical Kinetics,

We'll discuss all there is to know about the First Order, Second Order, and Zero Order Integrated Rate Laws...