>Consider the reaction A + B --> P which takes place in a solution of
>varying volume. What is the expression for dP/dt as a function of A, B,
>and d(vol)/dt?
It depends on why the volume is changing or, more specifically, whether
or not the volume change is accompanied by flow of P. For example, volume
could be decreasing because there is a flow out of the "system", with this
flow carrying P. I will presume that the volume under consideration is
surrounded by a membrane across which the solute may flow, but not P.
In this case, the amount of P in the volume is [P]*V, the concentration of
P times the volume. Presuming that A+B -> P is the only reaction involving
P (i.e., it's not consumed), the net rate of production of P by chemical
reaction is defined by some rate functions r([A],[B],[P]) **where r is on a
per unit volume basis**. What this is depends on the type of kinetics you're
considering. (I'm allowing that P may have a feed-back inhibition effect on
the rate.) For mass-action kinetics, this would just be r([A],[B],[P]) =
k*[A]*[B]. Then by conservation of mass, the rate of change in the amount of
P, d([P]*V)/dt, must equal the **total rate of production** V*r([A],[B],[P]):
d([P]*V)/dt = V*r([A],[B],[P]).
(Total rate of production = volume*(rate/volume).) Applying the chain-rule
of differentiation,
d([P]*V)/dt = V*d[P]/dt + [P]*dV/dt = V*r([A],[B],[P]).
Rearranging, one obtains:
d[P]/dt = r([A],[B],[P]) - [P]*(1/V)*dV/dt = r([A],[B],[P]) - [P]*(d ln(V)/dt).
The first term is the net synthesis term, in the absence of volume change,
the second term is dilution effect--if the amount of P stays the same and
the volume is getting bigger (dV/dt > 0), then the concentration of P will
drop.
Paul
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Paul M. Schlosser, Ph.D.
Chemical Industry Institute of Toxicology (CIIT)
P.O. Box 12137
6 Davis Drive (not needed for regular mail)
Research Triangle Park, NC 27709
Ph:(919) 558-1243; FAX:(919) 558-1300; schlosser at ciit.org
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