# 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: 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: 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: 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: Fraction of sites occupied by antivenom: Fraction of free sites: Concentration of venom in the blood: Concentration of antivenom in the blood: Concentration of venom-antivenom product in the blood:
 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