666 
R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1 
Ocean Sci., 6, 633-677, 2010 
www.ocean-sci.net/6/633/2010/ 
The Raoult constant is a——0.57 (Feistel et al., 2008). 
The density of ideal-gas vapour as a function of temper 
ature and pressure is estimated from the ideal-gas equation, 
P 
RwT 
The specific gas constant of water is 
R w =461.518 05 Jkg -1 K -1 . 
(A35) 
Eqs. (5.44)-(5.47) with respect to small changes in the in 
dependent variables Sa, T, P, p v and p w to obtain: 
S A g s ss AS A - (/7 -fV + gf- S A g s ST )AT (A39) 
“ (^w +g S P -SAg s SP ) AP 
= f w -f w + g s -S A gt-^ + -^ 
A8 Conditions for seawater in equilibrium with liquid 
water (Sect. 5.6) 
To determine equilibrium conditions for two samples of wa 
ter and seawater that are separated by a semi-permeable 
membrane and have different pressures, P w and P s , respec 
tively, we first linearize the three Eqs. (5.41)-(5.43) with re 
spect to small changes of the six unknowns S A , T, P s , P w , 
p s and p w to obtain: 
S A g s ss AS A + (/7 -f?-g s T + S A g S ST )AT (A36) 
/1 s s \ s A P w 
“ (p® +8p ~ Sa8sp ) AP 
pS pW 
— fW , f S I S r- s , r f_ 
— — J + / + £ — ¿Ags H s W 
p* p w 
A P w / \ 
„w,w AT ^ r FfW „WfW\ A „w /aq-7\ 
- p frp AT + —pW - [ 2 fp + P fpp) A P (A37) 
pW 
_ w fW _ £ 
P Jp p w 
- p s /r P AT + ^ - (2/p + P s /pp) Ap S (A38) 
Flere, p s is the density of pure water under the pressure P s 
of seawater, and / s is the related Flelmholtz function of liq 
uid water. To iteratively solve the system (Eqs. 5.41-5.43) 
for S A , T, P s , P w , p s and p w using Eqs. (A36)-(A38), 
three further equations must be added which specify addi 
tional conditions such as AP =0, AP w =0 and AS a —0 cor 
responding to the temperature, the pressure of the pure-water 
sample and the salinity of the seawater being specified. 
Trivial estimates such as P S =P W or Sa=0 suffice as initial 
values to start the iteration of Eqs. (5.41-5.43). 
A9 Equilibrium conditions for seawater, ice and water 
vapour (Sect. 5.7) 
To determine conditions under which seawater, ice and wa 
ter vapour exist in equilibrium, we first linearize the four 
(fr - g : r h ) A T + - g^j A P (A40) 
= P Ih - fV _ I_ 
8 1 p v 
- p V / r V p AP + ^ - (2/ p v + p v /^) Ap v (A41) 
- p W / r W p AP + ^ - (2/ p w + p w /7) Ap w (A42) 
_ w f w 
P Jp p W- 
To obtain the Eqs. (A39) and (A40), we first expanded 
Eqs. (5.44) and (5.45) and then simplified them by using 
Eqs. (A41) and (A42). To compute the triple point, an ar 
bitrary independent fifth condition is required. If e.g. the 
salinity is given, this additional equation is ASa=0; if the 
pressure or the temperature is known, one uses AP=0 or 
A P=0, respectively. With this condition specified, the four 
relations (Eqs. A39-A42) can be used to iteratively deter 
mine the other four seawater triple-point properties. 
Suitable initial values can be obtained from approxi 
mate equations which link salinity, freezing temperature and 
vapour pressure, roughly estimated from Fig. 7, as, 
P P t - S A x 3000 Pa (A43) 
P « P t — x 60 K. (A44) 
The starting value for the vapour density p v is taken from the 
ideal-gas law, p v ~/£ (R^T), and the liquid-water density is 
initialized with its pure-water triple-point value, p w =p7- 
A10 Equilibrium conditions for liquid water and water 
vapour in air (Sect. 5.8) 
To determine conditions for equilibrium between water 
vapour in air and pure liquid water, we express the chemical 
potentials in Eq. (5.48) by means of Eqs. (5.49) and (5.51),