## 1. Introduction

The International Thermodynamic Equation Of Seawater—2010 (TEOS-10) has been adopted as the international standard for the thermophysical properties of (i) seawater, (ii) ice Ih, and (iii) humid air. The TEOS-10 manual (IOC et al. 2010) summarizes the thermodynamic definitions of seawater, ice Ih, and humid air. The way that the thermodynamic potentials of these three substances were made consistent with each other is described in Feistel et al. (2008), and the scientific background to the announcement of this international standard is summarized in Pawlowicz et al. (2012). The terminology “ice Ih” stands for the ordinary hexagonal form of ice that is the naturally abundant form of ice, relevant for the pressure and temperature ranges found in the ocean and atmosphere [see Fig. 1b of Feistel et al. (2010)].

The temperature at which seawater begins freezing is determined from examining the thermodynamic equilibrium between the seawater and ice phases, with the relevant equilibrium condition being that the chemical potential of water in the seawater phase is equal to the chemical potential of water in the ice phase (Feistel and Hagen 1998; Feistel and Wagner 2005). Here we cast the freezing temperature in terms of the Conservative Temperature

The adiabatic lapse rate of ice is shown to be much greater than that of seawater (often 10 times as large), implying that under isentropic vertical motion, the variation of the in situ temperature of ice with pressure is much larger than for seawater.

In this paper, we consider the quantities that are conserved when ice melts into seawater. Writing equations for these conserved quantities (including enthalpy) leads to closed expressions for the Absolute Salinity and Conservative Temperature of the seawater after the melting or freezing has occurred [see Eqs. (8) and (9) below]. These equations apply at finite amplitude and do not assume the ice and seawater to be near to a state of thermodynamic equilibrium. This approach can be linearized to give an expression for the ratio of the changes in Conservative Temperature and Absolute Salinity when a vanishingly small amount of ice melts into a large mass of seawater. This result of this linearization [see Eqs. (16) and (18) below] is comparable to that of Gade (1979), although our approach is more general because it is based on the rigorous conservation of three basic thermodynamic properties (mass, salt, and enthalpy), so that it applies without approximation at finite amplitude, and we also include the dependence of seawater enthalpy on salinity.

This analysis is extended to the melting of sea ice, which is treated as a coarse-grained mixture of pure ice in which pockets of brine are trapped and the salinity of the pockets of brine is determined by thermodynamic equilibrium between the brine and the surrounding ice. This brine salinity has the same value in all pockets with equal temperatures and pressures, irrespective of their particular sizes, and the TEOS-10 description of the thermal properties of the brine apply up to an Absolute Salinity of 120 g kg^{−1} [see section 2.6 of IOC et al. (2010)].

The upwelling of very cold seawater (colder than the surface freezing temperature) can lead to supercooling and the formation of small ice crystals called frazil ice, and this process is also examined using the TEOS-10 Gibbs functions of ice Ih and of seawater. Under the assumption that the relative vertical velocity (the Stokes velocity) of frazil can be ignored, we derive expressions for the rate at which the Absolute Salinity and the Conservative Temperature of seawater vary with pressure when frazil is present. Because of their tiny size, frazil ice crystals remain in thermodynamic equilibrium with the surrounding seawater when the parcel undergoes pressure excursions. From the thermodynamic perspective, a frazil ice parcel differs from a sea ice parcel only quantitatively, namely, by their opposite liquid–solid ratios. Properties such as the adiabatic lapse rates of these composite systems can formally be derived from a Gibbs function of sea ice (Feistel et al. 2010). Strictly speaking, the mixture of frazil ice with seawater is a metastable state. It still undergoes a slow process known as Ostwald ripening that minimizes the interface energy between ice and seawater, typically by finally forming a single piece of ice (or a single large brine pocket in the case of sea ice). In the TEOS-10 Gibbs function of sea ice, the interface energy is neglected.

A mixture of seawater and frazil ice has two important properties that are quite different from ice-free seawater. First, the second law of thermodynamics requires that the Gibbs function of seawater is a convex function of salinity, that is, the second derivative of the Gibbs function with respect to Absolute Salinity is positive, *g*^{SI}; see Eq. (54) below] is linear in sea ice salinity (Feistel and Hagen 1998), that is, ^{−1}) seawater possesses a temperature of maximum density where the adiabatic lapse rate changes its sign (McDougall and Feistel 2003), while sea ice exhibits a density minimum (Feistel and Hagen 1998). Ice has a much larger specific volume than water or seawater, and the freezing process is accompanied by volume expansion, that is, by a large negative thermal expansion coefficient (and lapse rate) of sea ice. This effect is strongest at low salinities and in fact the thermal expansion coefficient of pure water has a singularity at the freezing point. With decreasing temperature and increasing brine salinity, the rate of formation of ice in sea ice gradually decreases to the point where the volume increase caused by the newly formed ice (i.e., by the transfer of water from the liquid to the solid phase) is outweighed by the thermal contraction of the pure phases, ice and brine, so that the total thermal expansion coefficient of sea ice changes its sign and turns positive.

The thermodynamic interactions between ice and seawater described in this paper are first derived as equations between the various quantities and are illustrated graphically in the figures. In addition, the thermodynamic properties of ice Ih and the results from the equations of this paper are available as computer algorithms in the Gibbs SeaWater (GSW) Oceanographic Toolbox (McDougall and Barker 2011) and can be downloaded online (from www.TEOS-10.org).

## 2. The adiabatic lapse rate and the potential temperature of ice Ih

*t*experienced when pressure is changed while keeping entropy

*η*(and salinity) constant. This definition applies separately to both ice and seawater (where one needs to keep not only entropy but also Absolute Salinity constant during the pressure change). In terms of the Gibbs functions of seawater and of ice Ih, the adiabatic lapse rates of seawater

The adiabatic lapse rates of seawater and ice are numerically substantially different from each other. The thermal expansion coefficient of ice does not change sign as does that of seawater when it is cooler than the temperature of maximum density, and the specific heat capacity of ice

This substantial difference between the adiabatic lapse rates is also illustrated in Fig. 1b as the difference in the potential temperature of seawater and of ice *p* diagram of Fig. 1b, the in situ freezing temperature *p* = 0 dbar) as illustrated in Fig. 1b.

## 3. Pure ice Ih melting into seawater

### a. The freezing temperature

*t*, and the pressure of a seawater parcel

*p*. The Gibbs function for ice Ih

*S*

_{A}= 35.165 04 g kg

^{−1}.

The freezing in situ temperatures derived from Eq. (4) were converted to the Conservative Temperature at which air-free seawater freezes and are shown in Fig. 2a as a function of pressure and Absolute Salinity. To compare these TEOS-10 freezing temperatures to those of EOS-80, the conversion between the practical salinity of EOS-80 and Absolute Salinity of TEOS-10 was made using the conversion factor *u*_{PS} ≡ (35.165 04/35) g kg^{−1} (Millero et al. 2008; IOC et al. 2010). It was assumed that the EOS-80 freezing temperatures of Millero and Leung (1976) were of air-saturated seawater. Having calculated the air-free in situ freezing temperature of EOS-80 in this manner, the Conservative Temperature is calculated from the TEOS-10 algorithm gsw_CT_from_t. The resulting differences between the freezing Conservative Temperatures from EOS-80 and TEOS-10 are illustrated in Fig. 2b and are very small at 0 dbar, rising to approximately 10 mK at 1000 dbar and 120 mK at 3000 dbar. We have developed a polynomial approximation for the freezing Conservative Temperature (see appendix D), and the error in using this computationally efficient polynomial is seen in Fig. 2c to be very small, being no larger than 0.05 mK at the sea surface and no larger than approximately 0.25 mK at other pressures.

### b. Finite-amplitude expressions for melting

We now turn our attention to the quantities that are conserved when a certain amount of ice melts into a known mass of seawater. In the following section, we will consider the melting of sea ice that contains pockets of brine, but in this section we consider the melting of pure ice Ih that contains no brine pockets. This section of the paper is appropriate when considering the melting of ice from glaciers or icebergs, because these types of ice are formed from compacted snow and hence do not contain the trapped seawater that is typical of ice formed at the sea surface, namely, sea ice.

The general case we consider in this section has the seawater temperature above its freezing temperature, while the ice, in order to be the stable phase ice Ih, needs to be at or below the freezing temperature of pure water (i.e., seawater having zero Absolute Salinity) at the given pressure level, typically at the sea surface. Note that this condition permits situations in which the initial ice temperature is higher than or equal to that of seawater. In other words, the general case we are considering is not an equilibrium situation in which certain amounts of ice and seawater coexist without further melting or freezing. During the melting of ice Ih into seawater at fixed pressure, entropy increases while three quantities are conserved: mass, salt, and enthalpy. While this process is assumed to be adiabatic it is not isentropic. Because of irreversibility, the freezing process is thermodynamically prohibited in a closed system. To form frazil ice in seawater at fixed pressure, more entropy must be exported from the sample than is produced internally; a typical example being an ice floe that is strongly cooled by the atmosphere.

*i*and

*f*stand for the initial and final values, that is, the values before and after the melting event, while the subscripts SW and Ih stand for seawater and ice Ih.

The use of Eqs. (8) and (9) is illustrated in Fig. 3, where the mass fraction of ice ^{−1}, *p* = 0 dbar. Note that these results apply for these finite-amplitude differences of temperature and salinity, and these calculations are accurate because of the existence of the TEOS-10 expressions for the specific enthalpies of seawater and ice Ih. We have not needed to resort to a linearization involving the specific heat capacities to obtain Eqs. (8)–(9) and the results of Fig. 3. Clearly, the salinity difference *p* = 0 dbar, Eq. (9) becomes simply

The conservation of Absolute Salinity and enthalpy when ice Ih melts into seawater is illustrated in Fig. 4a. The final values of Absolute Salinity

### c. The linearized expression for the ratio

Gade (1979) developed a mechanistic model of both the laminar and turbulent diffusion of heat and freshwater between ice and seawater, and using both this model and a much simpler linearized version of the conservation of “heat” [in the appendix of Gade (1979)] was able to derive an expression for the ratio of the changes in temperature and salinity in seawater due to the melting of a vanishingly small amount of ice into seawater. Here we have used the simpler “heat budget” approach, which is formally the conservation of enthalpy, and this led to Eqs. (8) and (9) that hold at finite amplitude when a finite mass fraction of ice melts into seawater. In this subsection, we linearize these equations to find the expressions (15)–(18) for the ratio of the changes in salinity and temperature when a vanishingly small mass fraction of ice melts into seawater. It is these Eqs. (15)–(18) that correspond to Gade’s key result for this ratio.

*p*= 0 dbar,

*p*= 0 dbar, these equations becomeandwhere the potential temperatures of seawater

*p*= 0 dbar. Note that the potential enthalpy of seawater referenced to

*p*= 0 dbar,

^{−1}K

^{−1}.

Equation (17) is very similar to Eq. (25) of Gade (1979). If we associate Gade’s temperature *L*, while the difference between the enthalpy of seawater and its enthalpy at the freezing temperature is approximately equal to the term

The use of Conservative Temperature rather than potential temperature means that the slope of the melting process on the *p* = 0 dbar [Eq. (18)], where (i)

The very simple Eqs. (16) and (18) for the slope of the melting process on the

We first illustrate these equations for the ratio of the changes of Conservative Temperature to those of Absolute Salinity by considering the melting to occur very close to thermodynamic equilibrium conditions. If both the seawater and the ice were exactly at the freezing temperature at the given values of Absolute Salinity and pressure, then no melting or freezing would occur. In Fig. 5, we consider the limit as the temperatures of both the seawater and the ice approach the freezing temperature. The ratio

The corresponding result for the ratio of the changes of in situ temperature and Absolute Salinity near equilibrium conditions

Equation (16) for *p* = 0 dbar, when Eq. (16) reduces to Eq. (18); this equation is illustrated in Fig. 6a, which applies at all values of Absolute Salinity. The contoured values of Fig. 6a,

### d. The influence of pressure on the melting ratio

Considering now the melting process at a gauge pressure larger than 0 dbar, the right-hand side of Eq. (16) is evaluated at *p* = 500 dbar and ^{−1}, with the differences between these values and the corresponding values at *p* = 0 dbar contoured in Fig. 6b. That is, this figure is the difference between the right-hand sides of Eqs. (16) and (18), with the in situ temperature of the ice being converted into the potential temperature of ice

*p*compared with using the expression (18), which involves the potential enthalpies of seawater and of ice, but is only 100% accurate for melting at

*p*= 0 dbar. The first term after unity on the right-hand side of the last expression of Eq. (19) is

^{−1}. This term is responsible for less than one-tenth of the 0.15% differences that we see in Fig. 6b between Eqs. (16) and (18) at 500 dbar. The last term in Eq. (19) involves a combination of enthalpy differences that we can express as follows (with the primed variables being the variable of integration and the use of upper case

*υ*[this is true of both seawater and ice, that is,

The conclusion from this comparison between Eqs. (16) and (18) is that as far as evaluating the slope on the *p*, very little error is made if the melting is assumed to occur at *p* = 0 dbar and taking the relevant enthalpy difference to be the difference between the potential enthalpies of seawater and of ice Ih, as in Eq. (18). The error in the *p* = 500 dbar and 0.9% at *p* = 3000 dbar.

### e. An illustration from the Amery Ice Shelf

Figure 7 shows oceanographic data obtained under the Amery Ice Shelf that illustrate the ratio of the changes in Absolute Salinity and Conservative Temperature, as given by Eq. (16), when the melting of ice occurs. The vertical profile named AM06 begins under the ice at a pressure of 546 dbar and the uppermost 175 m of the vertical profile is shown. The data in the uppermost 50–100 dbar are closely aligned with the ratio given by Eq. (16) (as shown by the dashed line) evaluated at this pressure and with the ice temperature being the freezing temperature at this salinity and pressure. Two freezing lines are shown in Fig. 7b, for pressures of 0 and 578 dbar. Any observations cooler than the freezing temperature appropriate to 0 dbar is evidence of the influence of melting of ice or of heat lost by conduction through the ice. AM06 is located on the eastern side of the ice shelf in an area that is melting, as can be inferred by the presence of ocean water at AM06 that is well above the in situ freezing temperature at the base of the ice shelf. This water is thought to be flowing in a primarily southward direction from the open ocean as it enters the underice cavity. The other CTD profile was taken from borehole AM05, located on the western side of the ice shelf in an area that is refreezing [as is drawn in Fig. 7a] and represents flow that has likely come from deeper in the cavity below the ice shelf than at AM06 (Post et al. 2014) and hence has been in contact with the ice for longer. The upper 50 m or so of this cast is at the freezing temperature of seawater at this pressure. For both casts the data near the upper part of the water column have the ratio of the changes of

## 4. Sea ice melting into seawater

### a. Finite-amplitude expressions for melting

Now we consider the situation where the ice contains a certain fraction of salt, such as occurs when ice is formed by freezing from seawater. We reserve the name “sea ice” for this mixture of pure ice Ih and a small amount of trapped brine that is in thermodynamic equilibrium with the ice Ih at the temperature of the ice Ih

^{−1}but is more commonly around 3–5 g kg

^{−1}, is defined to be the mass fraction of sea salt in sea ice so that

^{−6}°C.

The use of Eqs. (25) and (26) is illustrated in Fig. 9 where the mass fraction of sea ice ^{−1}, *p* = 0 dbar, and the sea ice salinity is taken to be ^{−1}. Note that these finite-amplitude calculations are accurate because of the existence of the TEOS-10 expressions for the specific enthalpies of seawater and ice Ih. Clearly, the salinity difference

We have not contoured values of ^{−1}), larger mass fractions of sea ice are admissible because the ratio ^{−1}, *p* = 0 dbar can be calculated implicitly from Eq. (26), and this is shown in Fig. 10. Values of

The conservation of Absolute Salinity and enthalpy when sea ice melts into seawater is illustrated in Fig. 11a. The final values of Absolute Salinity

### b. The linearized expression for the ratio of changes in and

The first property of this melting

Now we restrict attention to the case where the melting of sea ice into seawater is occurring at *p* = 0 dbar, corresponding to the trapping of brine into an ice matrix only occurring due to the rapid freezing of ice at the sea surface. The melting ratio ^{−1}, *p* = 0 dbar. For sea ice that is more than 1°C cooler than the freezing temperature *p* = 0 dbar is

*p*= 0 dbar with Eq. (18), namely,

^{−1},

*p*= 0 dbar. It is seen that the melting ratio

## 5. Frazil ice formation through adiabatic uplift of seawater

When seawater at the freezing temperature undergoes upward vertical motion so that its pressure decreases, frazil forms, primarily due to the increase in the freezing temperature as a result of the reduction in pressure. When this mixture of seawater and frazil continues to rise to lower pressures (assisted by the buoyancy provided by the presence of the ice), the frazil will experience a larger change in their in situ temperature than the seawater, simply because the adiabatic lapse rate of ice is much larger than that of seawater (see section 2 above). We will here consider this situation under the assumption that the frazil and the seawater moves together, so ignoring the tendency of the frazil to rise faster than the seawater, driven by the buoyancy of the individual ice crystals. We further assume that the uplift rate is sufficiently small that the in situ temperature of the ice and the seawater are the same at each pressure, this temperature being the freezing temperature. Under these conditions no entropy is produced during the freezing process, that is, this freezing process is reversible and can be reversed by increasing the pressure, leading to the related reversible ice melt. The lack of entropy production occurs as long as the exchange of heat and water between seawater and ice is conducted at mutual equilibrium, that is, at equal temperatures and equal chemical potentials of the two phases.

The previous two sections of this paper considered the irreversible melting of ice into seawater at constant pressure, and thermodynamic equilibrium was not assumed except between the pockets of brine and the surrounding ice Ih of the sea ice matrix. This section is different because (i) it considers the seawater phase and the small frazil ice crystals to be in thermodynamic equilibrium, and (ii) we study the consequences of a change in pressure of the seawater/frazil ice mixture.

We will study the thermodynamics of this process of adiabatic uplift of a seawater–ice mixture via a two-step thought process composed of three stages (Fig. 13). First, we imagine the mixture of preexisting ice and seawater to undergo a reduction in pressure but without any exchange of heat, water, or salt between the two phases. That is, during this first part of the process the mass of ice and the mass of seawater remain constant, and the change in the enthalpy of the ice and the change in the enthalpy of the seawater are only due to the pressure change. During this adiabatic process a (infinitesimal) contrast in in situ temperature will develop between the ice phase and the seawater phase because the adiabatic lapse rate of ice is much larger (by about an order of magnitude) than that of seawater (see Fig. 1a). During the second part of our thought experiment, the ice and seawater phases will be allowed to equilibrate their temperatures and further frazil ice will form so that the temperature of both the ice and seawater phases and the final Absolute Salinity of the seawater phase will be consistent with the freezing temperature at this pressure. This part of our thought experiment occurs at constant pressure and so, from the first law of thermodynamics, we know that enthalpy is conserved.

*υ*and

*υ*and

One key result is apparent from this equation already, namely, that as the mass fraction of frazil ice

^{−1}(actually

^{−1}. That is, Fig. 15a is simply 35.165 04 g kg

^{−1}times Fig. 14a, so that the quantity contoured in Fig. 15a is in temperature units. As the mass fraction of ice tends to zero, Eq. (47) tends to Eq. (16), so that values of Fig. 15a at

^{−1}. The dependence of

The variation of Conservative Temperature with pressure under frazil ice conditions ^{−1}. It is seen that ^{−1}. This figure follows, of course, as simply the ratio of the Figs. 14a and 14b.

When no frazil is present in seawater, its Conservative Temperature is unaffected by adiabatic and isohaline changes in pressure, but the in situ temperature changes with pressure according to the adiabatic lapse rate

^{−1}kg, while a typical value of

^{−1}kg. By contrast, we have seen that the variation of Conservative Temperature with pressure for frazil ice

## 6. Conclusions

We have exploited the thermodynamically consistent TEOS-10 definitions of seawater and ice Ih to derive the finite-amplitude relationships of Eqs. (8) and (9) that predict the final enthalpy and Absolute Salinity when a certain mass fraction of ice melts into seawater. We have not had to assume that the two components are in thermodynamic equilibrium during the melting process. In the limit as the mass fraction of melting ice tends to zero, the relative rate is derived at which the Conservative Temperature and Absolute Salinity vary when ice melts into seawater, essentially confirming the earlier linearized derivation of Gade (1979). These results have been illustrated graphically, and they have been extended to the case of sea ice that contains a small amount of trapped brine. The final section of the paper-derived results for the way Conservative Temperature and Absolute Salinity vary with pressure when frazil is present and in thermodynamic equilibrium with the seawater phase. These results of frazil ice assume that the ice crystals are small enough to not move relative to the seawater and to permanently remain in thermodynamic equilibrium with it. At some stage in the evolution of frazil ice, the individual ice crystals will become sufficiently large that the relative vertical motion of the ice crystals cannot be ignored and a more complicated analysis incorporating the Stokes drift of the ice crystals would be needed.

# APPENDIX A

## The TEOS-10 Gibbs Function of Ice Ih and Its Derivatives

Constants used in the TEOS-10 definition of ice Ih (IAPWS 2009a).

If the sea pressure *P*_{t} = 0.061 165 7 dbar, so that

# APPENDIX B

## Summary of Various Seawater Thermodynamic Relationships

Here we list expressions for several key thermodynamic quantities of seawater, as functions of the TEOS-10 Gibbs function of seawater

*T*

_{0}= 273.15 K. The partial derivative of the specific enthalpy of seawater

# APPENDIX C

## The Variation of the Freezing Conservative Temperature with Salinity and Pressure

*p*= 0 dbar. The partial derivative of

# APPENDIX D

## A Polynomial Expression for the Freezing Conservative Temperature of Seawater

The polynomial expression presented here is a fit to the TEOS-10 freezing temperature over the range in *S*_{A}–*p* space between 0 and 120 g kg^{−1} and between 0 and 10 000 dbar (100 MPa). We have chosen to do the polynomial fit for the Conservative Temperature at which seawater freezes rather than the in situ freezing temperature because ocean models will have Conservative Temperature as their temperature variable.

The TEOS-10 Gibbs function for seawater is valid in the ranges 0 ≤ *S*_{A} ≤ 42 g kg ^{−1} and 0 ≤ *p* ≤ 10 000 dbar. Additionally, at *p* = 0 dbar TEOS-10 is valid for thermal and colligative properties for Absolute Salinity up to where a constituent of seawater first saturates and comes out of solution. This typically occurs at an Absolute Salinity of between 90 and 110 g kg^{−1} (Feistel and Marion 2007; Marion et al. 2009). Technically, we should restrict the range of applicability of our polynomial fit to this area of *S*_{A}–*p* space plus the line at *p* = 0 dbar up to the Absolute Salinity of saturation, but the work of Feistel and Marion (2007, see their Fig. 15) suggests that the freezing temperatures calculated using the TEOS-10 Gibbs function at high pressures beyond *S*_{A} = 42 g kg^{−1} will not contain gross errors.

In the region of validity of the TEOS-10 Gibbs function, the rms accuracy of the freezing temperature is estimated to be 1.5 mK [see section 6.3, Fig. 4, and Table 7 of Feistel (2008)]. The present polynomial fits the full TEOS-10 freezing Conservative Temperature to within ±0.6 mK over both the valid TEOS-10 *S*_{A}–*p* range and the extrapolated region. Hence, we conclude that the use of this polynomial is essentially as accurate as the full TEOS-10 approach for calculating the freezing temperature. There is a triangle of data in *S*_{A}–*p* space at the largest pressures and Absolute Salinities where TEOS-10 does not provide the freezing temperature. This is because the TEOS-10 code returns values for the freezing temperature down to about −12°C. This in situ freezing temperature corresponds approximately to the line in *S*_{A}–*p* space connecting (50 g kg^{-1}, 10 000 dbar) to (120 g kg^{−1}, 5000 dbar), and the polynomial should not be used if the input Absolute Salinity and pressure lie beyond this line in *S*_{A}–*p* space. In the GSW Oceanographic Toolbox, this polynomial for freezing Conservative Temperature is available as gsw_CT_freezing_poly.

*p*= 0 dbar is known very accurately (with an uncertainty of only 2

*μ*K) to be 0.002 519°C. In terms of Conservative Temperature

*gsw_CT_from_pt*(0,0.002 519) = 0.017 947 064 327 968 736°C], and this is the value of

Constants

*S*

_{A}= 0 g kg

^{−1}) at

*p*= 0 dbar is 2.4 mK, while for standard seawater with

*S*

_{A}= 35.165 04 g kg

^{−1}the in situ freezing temperature depression is 1.9 mK. The dependence of the freezing potential temperature

*p*= 0 dbar, and we note that

*a*= 0.502 500 117 621 and

*b*= 0.057 000 649 899 720. These two coefficients have been chosen so that the in situ temperature freezing point depression for air-saturated seawater at

*p*= 0 dbar is exactly 2.4 mK at

*S*

_{A}= 0 g kg

^{−1}and exactly 1.9 mK at

*S*

_{A}= 35.165 04 g kg

^{−1}.

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