UNION OF ANTIBODY WITH ANTIGEN 125 



of getting dE is by measuring ciQ and dlJ^ and taking the difference. 

 It looks as if equation (3) is a trivial tautology. 



This is not the case, however, because there is an essential dif- 

 ference between dE on one hand and dQ and dW on the other 

 (Klotz, 1950). For dE is an exact differential, and dQ and dlV are 

 not. The meaning of the mathematical term exact differential is dis- 

 cussed in textbooks of the calculus. Here we need only recall that, 

 if a differential dX is exact, the values of A' at two different points, 

 .Yi and Xo, depend solely on the initial and final values of the in- 

 dependent variables of which X is a function, whereas, if dX is inexact, 

 the values of X depend upon the particular route we take from Xi 

 to Xo. In physics, if the pressure P and volume V of steam in an en- 

 gine are fixed, the values of the other variables such as the tem- 

 perature T are thereby determined. Since the values of P and V 

 determine the state of the system, P and V are called the independent 

 variables. We could have chosen other sets of two, such as P and 

 T or V and T, but in the study of heat engines, where thermodynamics 

 originated, the set P, V is particularly useful. 



We find that specifying P and V does not uniquely determine 

 Q or W , for the amount of heat a system may take up can vary in 

 spite of this, and it is well known that the portion of the heat a 

 machine converts into work depends on the efficiency of the machine. 

 Consequently, dQ and dW are inexact differentials and final values 

 of Q and VV depend not merely on the final values of P and V, but 

 on the route we choose in getting from the state Pi, V\ to Fo. f^2- 

 Two possible routes are shown schematically in Fig. 9-5. 



On the other hand, the value of E is completely determined by P 

 and V, and no matter what route we take from Pi, ]\ to P^. V2, 

 the final value of E, E^, will he the same. Consequently, if we go from 

 point 1 to point 2, then back to point 1 (this we call a reversible 

 cyclic process), AP must equal zero, while AQ and AfF will in 

 general be different from zero. All this is a mere restatement of the 

 first law of thermodynamics, but it is of the greatest importance. 



A thermodynamic quantity which depends only upon the values 

 of the independent variables is called a thermodynamic junction. 

 Thus, the total energy E is such a function. Knowing that P is a 

 thermodynamic function, we can write other expressions which are 

 also thermodynamic functions. For example, if we write 



