Developing the Equations

Our development of the equations you will need, in order to model the behavior of venom and/or antivenom in the human body, will be quick and dirty, but also relatively straight-forward. You may want to read through this information more than once to get a feel for the procedure.



Batch Reactor Design Equation:

 

The design equation for a batch reactor is:
Design 
      Equation

 

If we assume the reactor is well-mixed, then the reaction rate will be constant over every differential portion of volume, and we'll get:
Design 
      Equation

 

Finally, combining the number of moles of venom in the blood, NV, with the volume will give us the concentration of venom in the blood:
Design 
      Equation

 

The development is similar for the other relevant concentrations for antivenom and venom-antivenom product in the blood, CA and CP, respectively.


 

Analogous to Catalytic Reactions?

Believe it or not, but the way in which venom and antivenom interact with receptor sites on the diaphragm muscle may be modeled as a catalytic reaction. You may want to read Chapter Ten of Elements of Chemical Reaction Engineering, 3rd edition (or Chapter Six of the 2nd edtion), for coverage of this topic in greater detail. The most convenient way to discuss site coverage by either venom or antivenom is to refer to the fraction of sites that are unoccupied (i.e., free sites), occupied by venom, or occupied by antivenom, which are represented by fS, fSV, and fSA, respectively. (Oddly enough, the fraction of sites occupied by the venom-antivenom product, fSP, is never used, since the rate of product leaving a site is considered to be instantaneous.)



 

Combined Rate Laws and Material Balances:

Here we combine the rate laws and the material balances for each of the important variables in our problem.

 

Fraction of sites occupied by venom:
Description1
Reaction1

 

Fraction of sites occupied by antivenom:
Description2
Reaction2

 

Fraction of free sites:
Description3
Reaction3

 

Concentration of venom in the blood:
Description4
Reaction4

 

Concentration of antivenom in the blood:
Description5
Reaction5

 

Concentration of venom-antivenom product in the blood:
Description6
Reaction6

 

where:
V = venom
A = antivenom
P = venom-antivenom product
S = unoccupied site
S0 = initial number of sites
SV = site occupied by venom
SA = site occupied by antivenom