Reaction Kinetics Part I: The Truth behind Rate Equations
 focuschemistry
 February 06, 2018
 Reaction Kinetics
Many A Level students profess that they have not much problems with this chapter, as they can handle standard questions reasonably well.
However, when one probes deeper into details, gaps in understanding show up. One of them is the rate equation (or rate law)
The Rate Equation: what is it?
For a hypothetical reaction,
2A (aq) + B(aq) → 3C (aq) …………(1)
We notice that for every 2 moles of A that has reacted, 1 mole of B would have reacted, forming 3 moles of C.
Thus,
Rate of A disappearing = 2 x Rate of B disappearing = 1.5 x Rate of C appearing
Or
– ½Rate A = Rate B = ⅓Rate C 
Note:
Rate of reactants (disappearing) are opposite in sign compared to rate of products changing (appearing).
What does the rate of a reaction depend on? From secondary school, we were taught about certain factors that can increase the reaction rate, namely:
a. Concentration of reactants (pressure if the reactant is gaseous)
b. Temperature
c. Absence/ Presence of catalyst
Combining these factors, we obtain for the above reaction (1):
Rate = k[A]^{m}[B]^{n}
The temperature and catalyst effect is embodied in the “k” term in the rate equation. The “k” term is known as rate coefficient. However, it is more commonly known as rate constant.
This is a misnomer as the value of this term is not truly constant. It depends on temperature and activation energy (which is affected by the absence or presence of catalyst.
The famous Arrhenius equation relates the value of k with temperature and concentration.
How do we define rate?
Rate is defined as the change in the concentration of reactant or product with time.
Mathematically,
Rate  = 

A plot of concentration vs time is shown below.
Two types of rate can be calculated from the graph: Average rate (R1) and instantaneous rate (R2)
Difference between average and instantaneous rate:
Average rate  ( 

)  measures the average rate within a time frame (between t1 and t2) 
Instantaneous rate  ( 

)  measures the rate at a particular time(t1) 
Notice that average rate ≠instantaneous rate!
Mathematically, 

≠ 

However, if ∆time is very small, ∆time → d(time)
That means, average rate can be used to approximate instantaneous rate if the time duration is very small compared to the entire duration of the reaction.
We see this in a question from RI (2000):
The results shown were obtained at 25 degree Celsius to study the kinetics of the following reaction:
A(g) + B(g) → C(g)
Experiment Number  1  2  3 
Initial total pressure of A and B, P_{initial}/kPa  10.00  13.00  17.00 
Initial partial pressure of A, P_{Ai}/kPa  3.00  6.00  3.00 
Total pressure at 25 s, P_{total}/kPa  9.80  12.60  16.60 
(i) If Pc is the partial pressure of C at 25 s, show that P_{c} = P_{initial} – P_{total}
(ii) Hence determine the order of reaction with respect to A and that with respect to B, stating any assumptions made.
Discussion
(i) The relationship can be easily shown by considering a table showing the pressures of all species at t = 0 sec and t = 25 sec
A(g) +  B(g) →  C(g)  
At t=0 sec (kPa)  P_{Ai}  P_{Bi}  0 
Change (kPa)  Pc  Pc  +Pc 
At 25 sec (kPa)  P_{Ai}Pc  P_{Bi}Pc  Pc 
P_{total} = (P_{Ai}P_{c})+(P_{Bi}P_{c})+P_{c}  = P_{Ai}+P_{Bi} –P_{c} 
= P_{initial} – P_{c} 
Thus P_{c} = P_{initial} – P_{total} (shown)
(ii) To find the order of reaction with respect to A and B, we often use the initial rate method :
Experiment  Initial P_{A}(kPa)  Initial P_{B}(kPa)  Initial rate 
1  3  7  ? 
2  6  7  ? 
3  3  14  ? 
Notice that we do not have the initial rate data for all 3 experiments?
Typically, the initial rate is determined from a plot of concentration (of reactant) vs time plot, and taking the instantaneous rate at time = 0 sec by taking the tangent to the curve at time = 0 sec.
However, this plot is not available to us for any of the experiments.
What shall we do?
We could use the data of Pc at t =25 sec!
As C is a product, we know that the partial pressure of C at t = 0 sec must be 0.
This gives us 2 points on a partial pressure of C vs time plot:
For experiment 1:
The average rate of C formation in the 1st 25 seconds  = Gradient of red line  

From this example, we can see that
In the absence of initial rate data, and in the absence of a concentration vs time graph, we can approximate initial rate by using average rate of the product. All we need is just one data point of the concentration of product curve. The other point on this curve to determine average is conveniently provided by initial conditions (concentration of product = 0 at time = 0 sec)
Thus the “rate” in the rate equation is actually instantaneous rate, and not average rate as what many literature claims. But in the absence of most data, this instantaneous rate can be approximated by average rate.
The shorter this time frame in the determination of average rate, the better the approximation.
To find out more about rate equations, check out at our A Level chemistry classes!