Derivation of rate equations: conservation principles

Conservation principles, in conjunction with the thermodynamic concept that potential differences drive fluxes, can be used to develop rate equations that describe the behavior of physical systems.  Recall, from physics, some of the conservation principles:

conservation of mass

conservation of momentum  (linear and angular)

conservation of energy

conservation of charge

A more mathematical way to state a conservation principle is

You can fill in the blank with the conserved quantity of your choice.  If we consider a finite amount of time, say Dt, then we can write the conservation principle as

for any conserved quantity.  A more specific mathematical notation for the accumulation is

That is, the accumulation in time period Dt is equal to the amount present at the end of the time period, i.e., at time (t+Dt), less the amount present at the start of the time period, i.e., at time t.  Likewise, we can write for the gain and loss terms

and

Thus, a conservation balance can be written as

This, in turn, can be rearranged to the form

Note, that if we take the limit as (Dt ® 0) of both sides of the above relation, the left hand side looks suspiciously like the derivative of the conserved quantity, i.e.,

while the right hand side, which does not depend on Dt, remains unchanged.  Thus, the conservation balance produces the rate equation

Mathematical application of a conservation principle, i.e., finding appropriate mathematical expressions for bookkeeping to keep track of conserved quantities, leads directly to formulation of rate equations through the definition of the derivative.  However, simple bookkeeping is not sufficient.  Understanding and elucidating the basic physical and chemical phenomena of a situation are required before one can develop a reasonable mathematical model to describe the situation.  As a result, mathematical models of physical systems are only as good as the physics and chemistry which go into developing the description.  When developing mathematical models one needs to keep in mind the actual situation versus model approximations and simplifications.  Understanding and applying the thermodynamic concepts about potential differences driving fluxes will aid in developing physically meaningful rate equations.