388 Thermodynamics and Biology /2I :2 



The differential of work done 8 W may be represented as a sum of 

 inexact differentials, each of which can then be made exact by suitable 

 integrating factors. If the system is a gas, then 8W is particularly 

 simple; dividing it by the pressure/? gives the differential of volume dV. 

 In other words, 



8W = pdV 



For more complex systems, it is convenient to discuss the difference 

 8W, defined by 



W = 8W - pdV (3) 



This work, other than expansion, may be: elastic or mechanical, 8W' M \ 

 electromagnetic, 8W E ; or chemical, 8W C . In each case, it is possible 

 to find an expression similar to pdV. 



For any elastic or mechanical type of work, one may always write 



8W' M = Fd£ 



where F is a suitably defined force and d£ the displacement in the direc- 

 tion of the force. If many forces are present, 8 W' M is the sum of terms 

 such as the preceding equation, that is, 



8W' M = f F& (4) 



;= i 



Likewise, the differential of the electromagnetic work, represented as 

 the sum of products of potentials e, times differentials of charge g, is 

 given by 



M 



8W' E = J e t dq t (5) 



For biomolecular studies, the most important term of this type is 

 often the differential of the chemical work 8W' C . It may be represented 

 as 



8W' C = 2 «<IWM) 

 1 = 1 



where n t is the number of moles of the i th substance, U t its osmotic 

 pressure, and \/c { the volume per mole of this substance. (Note that 

 U t is analogous to P and n t d(l/c t ) to dV.) In the case of ideal solutions, 

 it is possible to further simplify the preceding equation. Just as for one 

 mole of an ideal gas 



pV= RT 

 so for ideal solutions 



II(l/c) = RT that is, n = cRT 

 Substituting for Ifj, one finds 



N 



8W' C = - 2 nftTdQnct) (6) 



