Differential Rate Law of Complex Reactions

[dissolver]

If I have an equation consisting of n reactants, then the rate would be:

Rate $$= k[A_1]^{p_1}[A_2]^{p_2}[A_3]^{p_3}...[A_n]^{p_n}$$

My question is how do I know what the rate is? Like it gives me a rate, but which reactant has this rate? Zumdahl's chemistry is so ambiguous on this. I don't understand how you can tell which reactant the calculated rate belongs to. Are all of the rates the same or does this simply give you the overall reaction rate disregarding the individual rates of the various reactants (since when determining the proportionality constant k, the book uses the overall rates of change in concentration of all reactants).

 

[GCT]

The rate from the stoichiometric equation for a simple case of a rate determining step with respect to a particular reactant or product should be
the rate in general divided its coefficient when dealing with the rate equation only in terms of one of the agents (the actual situation or derived).

When dealing with a general rate equation as you've posted here, it is with respect to the order of the dynamics, the A1, A2,A3,A4 (rarely termolecular and even trimolecular) describes the specific reactants/products.

 

[dissolver]

So in the above equation, if I divide the rate by the coefficient, I get the individual rate for that reactant?

Also, why is the rate equal to the rate divided by the coefficient?

 

[dissolver]

I read some online material about this.

The rate law computes the rate equal to the rate of any species divided by it's coefficient. And the rates of the individual species are equal to the products of their respective coefficients and the computed rate. Is this correct?

So if [tex]R =-\frac{\delta [A]}{a \delta t}=-\frac{\delta }{b \delta t}=\frac{\delta [C]}{c \delta t}= \frac{\delta [D]}{d \delta t}[/tex] is proportional to the rate equation, then that means you can compute the rate of any species in the reaction given sufficient concentration data and rates based on the species whose rate is under study? Then if I divide the computed rate by the specie's coefficient I arrive at R once again, which I can use to find the rate of any other species.

Please disabuse my reasoning if it is incorrect.

 

[sdekivit]

sounds OK to me :)

this formula is kind of logic, because when you have the reaction say 2A + B --> 3 C with R = x (mol/L*s)

then when we look at the reaction equation we see the when the overall reaction rate is x then A will react twice as fast. that's why $$R = - \frac {1} {2} \frac {d[A]} {dt}$$

When we look at B at a reaction rate x, the rate at which B reacts will be x too.

When looking at C, C is formed three times as fast as the overall reaction rate, that's why $$R = \frac {1} {3} \frac {d[C]} {dt}$$

 

[dissolver]

$$\frac{\delta A}{a \delta t}$$ is the overall reaction rate? I thought the slowest elementary process was the overall reaction rate.

 

[GCT]

the overall reaction rate can be discussed in the context of the actual rate equation, which may be with respect to its order. You can rearrange the equations or mathematically derive them if you wish to understand the disappearance or appearance rate for one particular agent.

Rate equations and their applications are models, they aren't fundamental as in some of the very important equations in quantum mechanics. That is they aren't exact. If such a mathematical model or derivations, which can get really complex depending on the nature of the rate dynamics in question, adequately describes the experimental rate, then it is frequently employed for that specific situation.

In essence, all of it is a lot of math, and is almost always imperfect in describing everything about a reaction, that's why you'll see that (if you go on to study physical chemistry) in a lot of cases, the rate equations pertain to one aspect of the reaction stage; that is at the initiation stages, or with assumptions of steady state as in the Michaelis-Menten principles.

So it's kind of useless to understand the rate subtopic in its essence, rather you should try to acquire a mathematical perspective which means deriving some equations for yourself.