ENTROPY 55 



of entropy, to make it a definite property of the state of a substance, is one 

 of the offices of the Second Law. 



It is therefore a part and a consequence of the Second Law to affirm that 

 when an ideal gas undergoes a free expansion, it experiences a gain of entropy 

 despite the fact that it receives no heat from the world without! Nor is the 

 affirmation confined to ideal gases; it would be true of any substance, though 

 in general a free expansion would be attended with a change in temperature. 

 Nothing, therefore, could be more wrong than to repeat pur first definition 

 of change-of-entropy with the words "in a reversible way" left out. 



One now begins to see why the concept of entropy is so much harder to 

 receive than that of energy. Every scientist is accustomed by now to the 

 "conservation" of energy: whereby it is meant that if the energy of a system 

 rises or falls by any amount, it is because there has been an inflow or an 

 efflux of just that amount from or into the outer world. Nothing of the 

 sort can be said of entropy, of which we have just seen that it may vary even 

 within a system which is having no transactions at all with the outer world. 

 One may not speak of the conservation of entropy excepting under the sharp 

 and severe restriction that all of the processes in the system and in the out- 

 side world are reversible: and "reversible" must be used in the full sense 

 adumbrated in a previous footnote, whereby no transfer of heat is reversible 

 unless the body whence it comes and the body to which it goes are of identi- 

 cal temperature. Yet nothing of all this contradicts the assertion that 

 entropy and energy are, both of them, uniquely determined functions of the 

 state of a system — functions, therefore, of any two of the three variables 

 P and V and T, for substances of a single kind in a single phase. 



Since the equating of entropy-change to inflow of heat divided by tem- 

 perature is something that positively must not be done for an irreversible 

 process, we must seek other ways to assess it. 



One such way has already been stated. If it is possible to start from a 

 particular state, and thence to arrive by reversible ways at both the begin- 

 ning and the end of the irreversible process in question, all the necessary 

 knowledge is at hand; for now by integrating dQ/T along the two ways we 

 find two quantities, of which the difference is the entropy-change desired. 



Applied in the special case which we have been considering, this method 

 has given 



AS = (Cp - C„) In (Ti/To) (3) 



not, however, a useful expression as yet, since it contains a quantity (To) 

 which does not figure at all in the irreversible process in question— a process 

 which, I recall, is the free expansion of one mole of ideal gas, at a constant 

 temperature Ti, from a higher pressure and lower volume which we will 

 denote by P' and V, to a lower pressure and a higher volume which we will 



