SECT. 1] PHYSICAL PROPERTIES OF SEA-WATER 5 



dE = TdN-pdV + y{ixi + 0)dmi + Md0, (1) 



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where T is absolute temperature, N entropy, p pressure and fXi the specific 

 chemical potential of the i-th constituent of mass mtA As defined by (1), the 

 total energy includes both the internal and potential energies. If appreciable 

 motion of the phase occurs, the kinetic energy must also be taken into account. 

 We can convert (1) to apply to a system of unit mass by substituting the 

 intensive quantities : 



e = E/M = € + specific total energy, where e is the specific internal energy, 

 and 



7] = NjM specific entropy 



V = V jM specific volume 



Xi = mijM mass fraction of the i-th constituent of sea- 



water. 2 



The substitution yields the three relations among the intensive quantities 



de = T dr]-pdv + yixidxi + d0 (2) 



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e = Tr]-pv + 2H^i + ^ (3) 



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and the Gibbs-Duhem equation 



-qdT-v dp + ^ Xi diJLi = 0. (4) 



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Although the development of the thermodynamic relationships can be 

 carried out in terms of the general multi-constituent system, a great simplifica- 

 tion is achieved by introducing the concept oi salinity. Dittmar (1884) showed 

 that the majority of ions in sea- water are present in remarkably constant ratios 

 to one another. To a first approximation, we can consider changes of salt 

 content of sea-water to be brought about by the addition or removal of water. 

 In principle, we could define salt content as the ratio of the total mass of dis- 

 solved salts to the mass of sea-water. In practice, however, this ratio is almost 

 impossible to determine accurately, and, historically, a somewhat different 

 concept of salinity has evolved. 



1 The term M d0 represents changes of the total energy due to displacements of the 

 phase relative to the geopotential field. As pointed out by Craig (1960), this term was not 

 taken into account by Fofonoff (1959) in his application of thermodynamical considera- 

 tions to a sea-water system. His derived equations are, therefore, valid only if the phase is 

 fixed in space, i.e. if d0 is set equal to zero. 



2 Mass fractions rather than mole fractions are used throughout this section in order to 

 retain the same units as the equations of motion (cf. Eckart, Chapter 2). Although mole 

 fractions are more suitable for expressing many of the thermodynamical relationships, 

 they cannot be introduced conveniently into the dynamical equations. The mole fraction 

 for the i-th constituent in terms of the mass fraction is given by (Mxi/Mi)/'^ mi/Mi, where 

 Mi is the molecular weight of the i-th constituent. * 



