8 FOFONOFF [chap. 1 



The third derivatives give us the additional useful relationships 



d^g 1 8cp d'^v 



dp dT^ T dp dT^ 



d^g 1 dcp d^fjL 



ds 8T^ ^ ~T~ds ^ W^' 



(19) 

 (20) 



To determine the Gibbs function, g, completely — except for an arbitrary 

 constant — we would have to know absolute values of the three first derivatives ; 

 i.e. entropy, specific volume and chemical potential difference as functions of 

 temperature, pressure and salinity. However, this cannot be done as neither 

 entropy nor chemical potential difference is completely specified in terms of 

 equilibrium-state properties. As will be seen later, entropy, rj, can be determined 

 except for a linear function of salinity, i or, from (16), /x except for a linear 

 function of temperature. This implies that the Gibbs function is arbitrary to 

 the extent of a function of the form aT + bTs + cs + d, where a, b, c and d are 

 constants. Conversely, it is clear that we do not require a knowledge of this 

 arbitrary function to describe the equilibrium state. 



Our knowledge of the three first derivatives — specific volume, entropy and 

 chemical potential difference — is now considered in more detail. 



2. Equation of State for Sea-Water 



Our present knowledge of the specific volume of sea-water as a function of 

 temperature, pressure and salinity is based on three sets of measurements. The 

 first of these was the determination of specific gravity as a function of chlorinity 

 at 0°C and atmospheric pressure. These determinations were made with 

 pycnometers by Forsch et al. (1902) and the results were expressed by the 

 empirical formula 



cTo = - 0.069 +1.4708(7/- 1.570 X 10-3(7/2 + 3.98 X 10-6OP, (21) 



where o-q, the specific gravity anomaly, is defined in terms of the specific gravity 

 So by CTo=103(so- 1). 



A total of 24 samples of sea-water, all from surface waters, were used in the 

 determinations. Most of the samples were collected in the Baltic, North Sea 

 and the North Atlantic Ocean, and are now recognized not to be adequately 

 representative of all ocean waters. Thompson and Wirth (1931) carried out 

 determinations of ao on 36 samples, including sea-water from the Pacific and 

 Indian Oceans, and from depths to 1000 m. A comparison with ctq, computed 

 from (21), showed that their measured values were higher than those given 



1 An alternative interpretation of the arbitrary linear function of salinity in the 

 definition of -q is obtained by introducing a combined partial entropy of the salts, 

 r]s= —8ixs/8t- Then, the specific entropy, rj, is given by stjs + ( 1 + 'S)>7w The partial 

 entropies, rjs, tjw, contain arbitrary constants giving rise to an arbitrary linear function 

 of salinity in rj. 



