21:2/ Thermodynamics and Biology 387 



of mechanical energy dissipated, a fixed number of heat energy units 

 were generated. Moreover, if heat energy were used to operate a 

 machine, the same ratio of energies was valid. By extending the con- 

 cept of energy to include electric and magnetic energy, chemical energy, 

 and finally mass energy, it has been possible to retain the conservation 

 of energy as a fundamental law. 



Another name for this fundamental law is the first law of thermodynamics. 

 Symbolically, it may be written 



dE = 8Q - 8W (1) 



In this expression, E is the internal energy, Q the heat put into the 

 system, and W the work done by the system. The symbol 8 is used 

 instead of d for differences, because neither 8Q nor 8W is an exact 

 differential. A differential is the difference in a thermodynamic 

 quantity when the system is changed from one equilibrium state to a 

 neighboring equilibrium state. (The states are defined in thermo- 

 dynamics by the pressure p, volume V, temperature T, and concentra- 

 tions c t .) In the above expression, dE is an exact differential because it 

 depends only on the initial and final states, whereas 8Q and 8W will 

 vary with the path between these two states. In fact, if one considers a 

 heat engine going around a cycle, dE, dT, dp, dV, and dc will all be 

 zero for a complete cycle or any integral number of cycles. In contrast, 

 8W and 8Q will increase with each complete cycle. 



It is always preferable to work with exact differentials, if this is 

 possible. Those who have studied differential equations will know 

 that it is often possible to multiply by a suitable function, known as an 

 integrating factor, to make a differential exact. This can be done for 

 both 8Q and 8W. It is then possible to rewrite the first law using exact 

 differentials only. 



The differential of added heat 8Q may be made exact by dividing 

 by the absolute temperature. The resultant differential dS, where 



dS = ^ (2) 



is called the differential of entropy. (For reasons not germane to our 

 present discussion, dS can be calculated only for reversible changes 

 between equilibrium states.) The entropy S is interpreted in statistical 

 mechanics as a measure of the disorder of the components of the system. 

 In information theory, the entropy is a measure of the information to be 

 gained by determining the locations, and so on, of all the parts of the 

 system. From the point of view of thermodynamics, the importance of 

 entropy is that it returns to its original value after a complete cycle; 

 that is, dS is an exact differential. 



