24 



FOFONOFF 



[chap. 1 



Taylor series about the final equilibrium values. By substituting the expansions 

 into (56) and neglecting terms of third order and higher, we obtain 



Aw = 



1m? 



l^r(A.-Ao^.2 



Z|_„, _,,)(..-.,) + 1^(^.-0^' 



(67) 



= —ra\m%\1m^ h^<p, 



where the symbol 8^ is introduced for the second-order differential operator in 

 (57). The differential form of the change Acp can be transformed so that the 

 derivatives are given in terms of temperature rather than enthalpy. The 

 expression for Acp becomes 



d(p St^A^ 



Acp = - 



mim2 

 2w2 



OT"' 



T"<P 



(58) 



where 





7^2 



{S2-Sl)-. 



ds dT \T) 



We can conclude from (58) that the final temperature at equilibrium is given by 



mi&i + m2&2 niim^ ^T^h 



& = 



(59) 



m 2m2 Cp 



Thus, the final temperature depends on the variation of specific heat with 

 temperature and salinity and on 



"2" ds dT \Tl 



(52-Sl)2 



which may be interpreted as the heat of mixing of sea- water ; i.e. the heat that 

 must be added or removed to maintain constant temperature when one gram 

 of sea-water of salinity S2 is mixed with a large mass of sea-water of salinity si. 

 By substituting r] for cp in (57), we can calculate the increase of entropy of 

 the system on reaching equilibrium. The expression for the entropy increase 

 can be reduced to 



Amr] 



mim2 I[{ h2-hi)-hs{s2-si)] 2 , i_ ^ /, _o \2 



2w2 



mim2 

 2m2 



rp2 +T 8S ^ ' '^ 



(60) 



where 



hs = h- 



8h\ 

 ds/T.p 



^ dT dT\T 



From (60) we can see that both the specific heat, Cp, and dixjds must be 

 positive for the entropy to increase for all initial combinations of temperatures 



