gsw_gibbs

`Gibbs energy and its derivatives`

USAGE:

`gibbs = gsw_gibbs(ns,nt,np,SA,t,p)`

DESCRIPTION:

```Calculates specific Gibbs energy and its derivatives up to order 3 for
seawater.  The Gibbs function for seawater is that of TEOS-10
(IOC et al., 2010), being the sum of IAPWS-08 for the saline part and
IAPWS-09 for the pure water part.  These International Association
for the Properties of Water and Steam (IAPWS) releases are the
officially blessed IAPWS descriptions of Feistel (2008) and the pure
water part of Feistel (2003).  Absolute Salinity, SA, in all of the GSW
routines is expressed on the Reference-Composition Salinity Scale of
2008 (RCSS-08) of Millero et al. (2008).```
 `Click for a more detailed description of Gibbs energy.`

INPUT:

```ns  =  order of SA derivative                     [ integers 0, 1 or 2 ]
nt  =  order of t derivative                      [ integers 0, 1 or 2 ]
np  =  order of p derivative                      [ integers 0, 1 or 2 ]
SA  =  Absolute Salinity                                        [ g/kg ]
t   =  in-situ temperature (ITS-90)                            [ deg C ]
p   =  sea pressure                                             [ dbar ]
(ie. absolute pressure - 10.1325 dbar)```
`SA, t and p need to have the same dimensions.`

OUTPUT:

`gibbs  =  Specific Gibbs energy or its derivatives.`
```          The Gibbs energy (when ns = nt = np = 0) has units of:
[ J/kg ]
The Absolute Salinity derivatives are output in units of:
[ (J/kg) (g/kg)^(-ns) ]
The temperature derivatives are output in units of:
[ (J/kg) (K)^(-nt) ]
The pressure derivatives are output in units of:
[ (J/kg) (Pa)^(-np) ]
The mixed derivatives are output in units of:
[ (J/kg) (g/kg)^(-ns) (K)^(-nt) (Pa)^(-np) ]```
```Note. The derivatives are taken with respect to pressure in Pa, not
withstanding that the pressure input into this routine is in dbar.```

AUTHOR:

`David Jackett, Paul Barker and Trevor McDougall     [ help@teos-10.org ]`

VERSION NUMBER:

`3.05 (16th February, 2015)`

REFERENCES:

```Feistel, R., 2003: A new extended Gibbs thermodynamic potential of
seawater,  Progr. Oceanogr., 58, 43-114.```
```Feistel, R., 2008: A Gibbs function for seawater thermodynamics
for -6 to 80°C and salinity up to 120 g kg–1, Deep-Sea Res. I,
55, 1639-1671.```
```IAPWS, 2008: Release on the IAPWS Formulation 2008 for the
Thermodynamic Properties of Seawater. The International Association
for the Properties of Water and Steam. Berlin, Germany, September
2008, available from http://www.iapws.org.
This Release is referred to as IAPWS-08.```
```IAPWS, 2009: Supplementary Release on a Computationally Efficient
Thermodynamic Formulation for Liquid Water for Oceanographic Use.
The International Association for the Properties of Water and Steam.
Doorwerth, The Netherlands, September 2009, available from
http://www.iapws.org.
This Release is referred to as IAPWS-09.```
```IOC, SCOR and IAPSO, 2010: The international thermodynamic equation of
seawater - 2010: Calculation and use of thermodynamic properties.
Intergovernmental Oceanographic Commission, Manuals and Guides No. 56,
UNESCO (English), 196 pp.  Available from  http://www.TEOS-10.org
See section 2.6 and appendices A.6,  G and H of this TEOS-10 Manual.```
```Millero, F. J., R. Feistel, D. G. Wright, and T. J. McDougall, 2008:
The composition of Standard Seawater and the definition of the
Reference-Composition Salinity Scale, Deep-Sea Res. I, 55, 50-72.```
`The software is available from http://www.TEOS-10.org`
```function gibbs = gsw_gibbs(ns,nt,np,SA,t,p)

% gsw_gibbs                                Gibbs energy and its derivatives
%==========================================================================
%
% USAGE:
%   gibbs = gsw_gibbs(ns,nt,np,SA,t,p)
%
% DESCRIPTION:
%  Calculates specific Gibbs energy and its derivatives up to order 3 for
%  seawater.  The Gibbs function for seawater is that of TEOS-10
%  (IOC et al., 2010), being the sum of IAPWS-08 for the saline part and
%  IAPWS-09 for the pure water part.  These IAPWS releases are the
%  officially blessed IAPWS descriptions of Feistel (2008) and the pure
%  water part of Feistel (2003).  Absolute Salinity, SA, in all of the GSW
%  routines is expressed on the Reference-Composition Salinity Scale of
%  2008 (RCSS-08) of Millero et al. (2008).
%
% INPUT:
%  ns  =  order of SA derivative                     [ integers 0, 1 or 2 ]
%  nt  =  order of t derivative                      [ integers 0, 1 or 2 ]
%  np  =  order of p derivative                      [ integers 0, 1 or 2 ]
%  SA  =  Absolute Salinity                                        [ g/kg ]
%  t   =  in-situ temperature (ITS-90)                            [ deg C ]
%  p   =  sea pressure                                             [ dbar ]
%         (ie. absolute pressure - 10.1325 dbar)
%
%  SA, t and p need to have the same dimensions.
%
% OUTPUT:
%  gibbs  =  Specific Gibbs energy or its derivatives.
%
%            The Gibbs energy (when ns = nt = np = 0) has units of:
%                                                                  [ J/kg ]
%            The Absolute Salinity derivatives are output in units of:
%                                                   [ (J/kg) (g/kg)^(-ns) ]
%            The temperature derivatives are output in units of:
%                                                      [ (J/kg) (K)^(-nt) ]
%            The pressure derivatives are output in units of:
%                                                     [ (J/kg) (Pa)^(-np) ]
%            The mixed derivatives are output in units of:
%                              [ (J/kg) (g/kg)^(-ns) (K)^(-nt) (Pa)^(-np) ]
%  Note. The derivatives are taken with respect to pressure in Pa, not
%    withstanding that the pressure input into this routine is in dbar.
%
% AUTHOR:
%  David Jackett, Paul Barker and Trevor McDougall     [ help@teos-10.org ]
%
% VERSION NUMBER: 3.05 (27th January 2015)
%
% REFERENCES:
%  Feistel, R., 2003: A new extended Gibbs thermodynamic potential of
%   seawater,  Progr. Oceanogr., 58, 43-114.
%
%  Feistel, R., 2008: A Gibbs function for seawater thermodynamics
%   for -6 to 80°C and salinity up to 120 g kg–1, Deep-Sea Res. I,
%   55, 1639-1671.
%
%  IAPWS, 2008: Release on the IAPWS Formulation 2008 for the
%   Thermodynamic Properties of Seawater. The International Association
%   for the Properties of Water and Steam. Berlin, Germany, September
%   2008, available from http://www.iapws.org.  This Release is referred
%   to as IAPWS-08.
%
%  IAPWS, 2009: Supplementary Release on a Computationally Efficient
%   Thermodynamic Formulation for Liquid Water for Oceanographic Use.
%   The International Association for the Properties of Water and Steam.
%   Doorwerth, The Netherlands, September 2009, available from
%   http://www.iapws.org.  This Release is referred to as IAPWS-09.
%
%  IOC, SCOR and IAPSO, 2010: The international thermodynamic equation of
%   seawater - 2010: Calculation and use of thermodynamic properties.
%   Intergovernmental Oceanographic Commission, Manuals and Guides No. 56,
%   UNESCO (English), 196 pp.  Available from http://www.TEOS-10.org
%    See section 2.6 and appendices A.6,  G and H of this TEOS-10 Manual.
%
%  Millero, F.J., R. Feistel, D.G. Wright, and T.J. McDougall, 2008:
%   The composition of Standard Seawater and the definition of the
%   Reference-Composition Salinity Scale, Deep-Sea Res. I, 55, 50-72.
%
%  The software is available from http://www.TEOS-10.org
%
%==========================================================================

% This line ensures that SA is non-negative.
SA(SA < 0) = 0;

% Set the upper and lower limits where the Gibbs function is defined.
SA(SA > 120 | t < -12 | t > 80 | p > 12000) = NaN;
t(SA > 120 | t < -12 | t > 80 | p > 12000) = NaN;
p(SA > 120 | t < -12 | t > 80 | p > 12000) = NaN;

sfac = 0.0248826675584615;                   % sfac = 1/(40*(35.16504/35)).

x2 = sfac.*SA;
x = sqrt(x2);
y = t.*0.025;
z = p.*1e-4; %Note.The input pressure (p) is sea pressure in units of dbar.

if ns==0 & nt==0 & np==0

g03 = 101.342743139674 + z.*(100015.695367145 + ...
z.*(-2544.5765420363 + z.*(284.517778446287 + ...
z.*(-33.3146754253611 + (4.20263108803084 - 0.546428511471039.*z).*z)))) + ...
y.*(5.90578347909402 + z.*(-270.983805184062 + ...
z.*(776.153611613101 + z.*(-196.51255088122 + (28.9796526294175 - 2.13290083518327.*z).*z))) + ...
y.*(-12357.785933039 + z.*(1455.0364540468 + ...
z.*(-756.558385769359 + z.*(273.479662323528 + z.*(-55.5604063817218 + 4.34420671917197.*z)))) + ...
y.*(736.741204151612 + z.*(-672.50778314507 + ...
z.*(499.360390819152 + z.*(-239.545330654412 + (48.8012518593872 - 1.66307106208905.*z).*z))) + ...
y.*(-148.185936433658 + z.*(397.968445406972 + ...
z.*(-301.815380621876 + (152.196371733841 - 26.3748377232802.*z).*z)) + ...
y.*(58.0259125842571 + z.*(-194.618310617595 + ...
z.*(120.520654902025 + z.*(-55.2723052340152 + 6.48190668077221.*z))) + ...
y.*(-18.9843846514172 + y.*(3.05081646487967 - 9.63108119393062.*z) + ...
z.*(63.5113936641785 + z.*(-22.2897317140459 + 8.17060541818112.*z))))))));

g08 = x2.*(1416.27648484197 + z.*(-3310.49154044839 + ...
z.*(384.794152978599 + z.*(-96.5324320107458 + (15.8408172766824 - 2.62480156590992.*z).*z))) + ...
x.*(-2432.14662381794 + x.*(2025.80115603697 + ...
y.*(543.835333000098 + y.*(-68.5572509204491 + ...
y.*(49.3667694856254 + y.*(-17.1397577419788 + 2.49697009569508.*y))) - 22.6683558512829.*z) + ...
x.*(-1091.66841042967 - 196.028306689776.*y + ...
x.*(374.60123787784 - 48.5891069025409.*x + 36.7571622995805.*y) + 36.0284195611086.*z) + ...
z.*(-54.7919133532887 + (-4.08193978912261 - 30.1755111971161.*z).*z)) + ...
z.*(199.459603073901 + z.*(-52.2940909281335 + (68.0444942726459 - 3.41251932441282.*z).*z)) + ...
y.*(-493.407510141682 + z.*(-175.292041186547 + (83.1923927801819 - 29.483064349429.*z).*z) + ...
y.*(-43.0664675978042 + z.*(383.058066002476 + z.*(-54.1917262517112 + 25.6398487389914.*z)) + ...
y.*(-10.0227370861875 - 460.319931801257.*z + y.*(0.875600661808945 + 234.565187611355.*z))))) + ...
y.*(168.072408311545 + z.*(729.116529735046 + ...
z.*(-343.956902961561 + z.*(124.687671116248 + z.*(-31.656964386073 + 7.04658803315449.*z)))) + ...
y.*(880.031352997204 + y.*(-225.267649263401 + ...
y.*(91.4260447751259 + y.*(-21.6603240875311 + 2.13016970847183.*y) + ...
z.*(-297.728741987187 + (74.726141138756 - 36.4872919001588.*z).*z)) + ...
z.*(694.244814133268 + z.*(-204.889641964903 + (113.561697840594 - 11.1282734326413.*z).*z))) + ...
z.*(-860.764303783977 + z.*(337.409530269367 + ...
z.*(-178.314556207638 + (44.2040358308 - 7.92001547211682.*z).*z))))));

g08(x>0) = g08(x>0) + x2(x>0).*(5812.81456626732 + 851.226734946706.*y(x>0)).*log(x(x>0));

gibbs = g03 + g08;

elseif ns==1 & nt==0 & np==0

g08 = 8645.36753595126 + z.*(-6620.98308089678 + ...
z.*(769.588305957198 + z.*(-193.0648640214916 + (31.6816345533648 - 5.24960313181984.*z).*z))) + ...
x.*(-7296.43987145382 + x.*(8103.20462414788 + ...
y.*(2175.341332000392 + y.*(-274.2290036817964 + ...
y.*(197.4670779425016 + y.*(-68.5590309679152 + 9.98788038278032.*y))) - 90.6734234051316.*z) + ...
x.*(-5458.34205214835 - 980.14153344888.*y + ...
x.*(2247.60742726704 - 340.1237483177863.*x + 220.542973797483.*y) + 180.142097805543.*z) + ...
z.*(-219.1676534131548 + (-16.32775915649044 - 120.7020447884644.*z).*z)) + ...
z.*(598.378809221703 + z.*(-156.8822727844005 + (204.1334828179377 - 10.23755797323846.*z).*z)) + ...
y.*(-1480.222530425046 + z.*(-525.876123559641 + (249.57717834054571 - 88.449193048287.*z).*z) + ...
y.*(-129.1994027934126 + z.*(1149.174198007428 + z.*(-162.5751787551336 + 76.9195462169742.*z)) + ...
y.*(-30.0682112585625 - 1380.9597954037708.*z + y.*(2.626801985426835 + 703.695562834065.*z))))) + ...
y.*(1187.3715515697959 + z.*(1458.233059470092 + ...
z.*(-687.913805923122 + z.*(249.375342232496 + z.*(-63.313928772146 + 14.09317606630898.*z)))) + ...
y.*(1760.062705994408 + y.*(-450.535298526802 + ...
y.*(182.8520895502518 + y.*(-43.3206481750622 + 4.26033941694366.*y) + ...
z.*(-595.457483974374 + (149.452282277512 - 72.9745838003176.*z).*z)) + ...
z.*(1388.489628266536 + z.*(-409.779283929806 + (227.123395681188 - 22.2565468652826.*z).*z))) + ...
z.*(-1721.528607567954 + z.*(674.819060538734 + ...
z.*(-356.629112415276 + (88.4080716616 - 15.84003094423364.*z).*z)))));

g08(x>0) = g08(x>0) + (11625.62913253464 + 1702.453469893412.*y(x>0)).*log(x(x>0));
g08(x==0) = nan;

gibbs = 0.5.*sfac.*g08;

elseif ns==0 & nt==1 & np==0

g03 = 5.90578347909402 + z.*(-270.983805184062 + ...
z.*(776.153611613101 + z.*(-196.51255088122 + (28.9796526294175 - 2.13290083518327.*z).*z))) + ...
y.*(-24715.571866078 + z.*(2910.0729080936 + ...
z.*(-1513.116771538718 + z.*(546.959324647056 + z.*(-111.1208127634436 + 8.68841343834394.*z)))) + ...
y.*(2210.2236124548363 + z.*(-2017.52334943521 + ...
z.*(1498.081172457456 + z.*(-718.6359919632359 + (146.4037555781616 - 4.9892131862671505.*z).*z))) + ...
y.*(-592.743745734632 + z.*(1591.873781627888 + ...
z.*(-1207.261522487504 + (608.785486935364 - 105.4993508931208.*z).*z)) + ...
y.*(290.12956292128547 + z.*(-973.091553087975 + ...
z.*(602.603274510125 + z.*(-276.361526170076 + 32.40953340386105.*z))) + ...
y.*(-113.90630790850321 + y.*(21.35571525415769 - 67.41756835751434.*z) + ...
z.*(381.06836198507096 + z.*(-133.7383902842754 + 49.023632509086724.*z)))))));

g08 = x2.*(168.072408311545 + z.*(729.116529735046 + ...
z.*(-343.956902961561 + z.*(124.687671116248 + z.*(-31.656964386073 + 7.04658803315449.*z)))) + ...
x.*(-493.407510141682 + x.*(543.835333000098 + x.*(-196.028306689776 + 36.7571622995805.*x) + ...
y.*(-137.1145018408982 + y.*(148.10030845687618 + y.*(-68.5590309679152 + 12.4848504784754.*y))) - ...
22.6683558512829.*z) + z.*(-175.292041186547 + (83.1923927801819 - 29.483064349429.*z).*z) + ...
y.*(-86.1329351956084 + z.*(766.116132004952 + z.*(-108.3834525034224 + 51.2796974779828.*z)) + ...
y.*(-30.0682112585625 - 1380.9597954037708.*z + y.*(3.50240264723578 + 938.26075044542.*z)))) + ...
y.*(1760.062705994408 + y.*(-675.802947790203 + ...
y.*(365.7041791005036 + y.*(-108.30162043765552 + 12.78101825083098.*y) + ...
z.*(-1190.914967948748 + (298.904564555024 - 145.9491676006352.*z).*z)) + ...
z.*(2082.7344423998043 + z.*(-614.668925894709 + (340.685093521782 - 33.3848202979239.*z).*z))) + ...
z.*(-1721.528607567954 + z.*(674.819060538734 + ...
z.*(-356.629112415276 + (88.4080716616 - 15.84003094423364.*z).*z)))));

g08(x>0) = g08(x>0) + 851.226734946706.*x2(x>0).*log(x(x>0));

gibbs = (g03 + g08).*0.025;

elseif ns==0 & nt==0 & np==1

g03 = 100015.695367145 + z.*(-5089.1530840726 + ...
z.*(853.5533353388611 + z.*(-133.2587017014444 + (21.0131554401542 - 3.278571068826234.*z).*z))) + ...
y.*(-270.983805184062 + z.*(1552.307223226202 + ...
z.*(-589.53765264366 + (115.91861051767 - 10.664504175916349.*z).*z)) + ...
y.*(1455.0364540468 + z.*(-1513.116771538718 + ...
z.*(820.438986970584 + z.*(-222.2416255268872 + 21.72103359585985.*z))) + ...
y.*(-672.50778314507 + z.*(998.720781638304 + ...
z.*(-718.6359919632359 + (195.2050074375488 - 8.31535531044525.*z).*z)) + ...
y.*(397.968445406972 + z.*(-603.630761243752 + (456.589115201523 - 105.4993508931208.*z).*z) + ...
y.*(-194.618310617595 + y.*(63.5113936641785 - 9.63108119393062.*y + ...
z.*(-44.5794634280918 + 24.511816254543362.*z)) + ...
z.*(241.04130980405 + z.*(-165.8169157020456 + 25.92762672308884.*z)))))));

g08 = x2.*(-3310.49154044839 + z.*(769.588305957198 + ...
z.*(-289.5972960322374 + (63.3632691067296 - 13.1240078295496.*z).*z)) + ...
x.*(199.459603073901 + x.*(-54.7919133532887 + 36.0284195611086.*x - 22.6683558512829.*y + ...
(-8.16387957824522 - 90.52653359134831.*z).*z) + ...
z.*(-104.588181856267 + (204.1334828179377 - 13.65007729765128.*z).*z) + ...
y.*(-175.292041186547 + (166.3847855603638 - 88.449193048287.*z).*z + ...
y.*(383.058066002476 + y.*(-460.319931801257 + 234.565187611355.*y) + ...
z.*(-108.3834525034224 + 76.9195462169742.*z)))) + ...
y.*(729.116529735046 + z.*(-687.913805923122 + ...
z.*(374.063013348744 + z.*(-126.627857544292 + 35.23294016577245.*z))) + ...
y.*(-860.764303783977 + y.*(694.244814133268 + ...
y.*(-297.728741987187 + (149.452282277512 - 109.46187570047641.*z).*z) + ...
z.*(-409.779283929806 + (340.685093521782 - 44.5130937305652.*z).*z)) + ...
z.*(674.819060538734 + z.*(-534.943668622914 + (176.8161433232 - 39.600077360584095.*z).*z)))));

gibbs = (g03 + g08).*1e-8;
% Note. This pressure derivative of the gibbs function is in units of (J/kg) (Pa^-1) = m^3/kg

elseif ns==1 & nt==1 & np==0

g08 = 1187.3715515697959 + z.*(1458.233059470092 + ...
z.*(-687.913805923122 + z.*(249.375342232496 + z.*(-63.313928772146 + 14.09317606630898.*z)))) + ...
x.*(-1480.222530425046 + x.*(2175.341332000392 + x.*(-980.14153344888 + 220.542973797483.*x) + ...
y.*(-548.4580073635929 + y.*(592.4012338275047 + y.*(-274.2361238716608 + 49.9394019139016.*y))) - ...
90.6734234051316.*z) + z.*(-525.876123559641 + (249.57717834054571 - 88.449193048287.*z).*z) + ...
y.*(-258.3988055868252 + z.*(2298.348396014856 + z.*(-325.1503575102672 + 153.8390924339484.*z)) + ...
y.*(-90.2046337756875 - 4142.8793862113125.*z + y.*(10.50720794170734 + 2814.78225133626.*z)))) + ...
y.*(3520.125411988816 + y.*(-1351.605895580406 + ...
y.*(731.4083582010072 + y.*(-216.60324087531103 + 25.56203650166196.*y) + ...
z.*(-2381.829935897496 + (597.809129110048 - 291.8983352012704.*z).*z)) + ...
z.*(4165.4688847996085 + z.*(-1229.337851789418 + (681.370187043564 - 66.7696405958478.*z).*z))) + ...
z.*(-3443.057215135908 + z.*(1349.638121077468 + ...
z.*(-713.258224830552 + (176.8161433232 - 31.68006188846728.*z).*z))));

g08(x>0) = g08(x>0) + 1702.453469893412.*log(x(x>0));
g08(SA==0) = nan;

gibbs = 0.5.*sfac.*0.025.*g08;

elseif ns==1 & nt==0 & np==1

g08 =  -6620.98308089678 + z.*(1539.176611914396 + ...
z.*(-579.1945920644748 + (126.7265382134592 - 26.2480156590992.*z).*z)) + ...
x.*(598.378809221703 + x.*(-219.1676534131548 + 180.142097805543.*x - 90.6734234051316.*y + ...
(-32.65551831298088 - 362.10613436539325.*z).*z) + ...
z.*(-313.764545568801 + (612.4004484538132 - 40.95023189295384.*z).*z) + ...
y.*(-525.876123559641 + (499.15435668109143 - 265.347579144861.*z).*z + ...
y.*(1149.174198007428 + y.*(-1380.9597954037708 + 703.695562834065.*y) + ...
z.*(-325.1503575102672 + 230.7586386509226.*z)))) + ...
y.*(1458.233059470092 + z.*(-1375.827611846244 + ...
z.*(748.126026697488 + z.*(-253.255715088584 + 70.4658803315449.*z))) + ...
y.*(-1721.528607567954 + y.*(1388.489628266536 + ...
y.*(-595.457483974374 + (298.904564555024 - 218.92375140095282.*z).*z) + ...
z.*(-819.558567859612 + (681.370187043564 - 89.0261874611304.*z).*z)) + ...
z.*(1349.638121077468 + z.*(-1069.887337245828 + (353.6322866464 - 79.20015472116819.*z).*z))));

gibbs = g08.*sfac.*0.5e-8;
% Note. This derivative of the Gibbs function is in units of (m^3/kg)/(g/kg) = m^3/g,
% that is, it is the derivative of specific volume with respect to Absolute
% Salinity measured in g/kg.

elseif ns==0 & nt==1 & np==1

g03 = -270.983805184062 + z.*(1552.307223226202 + z.*(-589.53765264366 + ...
(115.91861051767 - 10.664504175916349.*z).*z)) + ...
y.*(2910.0729080936 + z.*(-3026.233543077436 + ...
z.*(1640.877973941168 + z.*(-444.4832510537744 + 43.4420671917197.*z))) + ...
y.*(-2017.52334943521 + z.*(2996.162344914912 + ...
z.*(-2155.907975889708 + (585.6150223126464 - 24.946065931335752.*z).*z)) + ...
y.*(1591.873781627888 + z.*(-2414.523044975008 + (1826.356460806092 - 421.9974035724832.*z).*z) + ...
y.*(-973.091553087975 + z.*(1205.20654902025 + z.*(-829.084578510228 + 129.6381336154442.*z)) + ...
y.*(381.06836198507096 - 67.41756835751434.*y + z.*(-267.4767805685508 + 147.07089752726017.*z))))));

g08 = x2.*(729.116529735046 + z.*(-687.913805923122 + ...
z.*(374.063013348744 + z.*(-126.627857544292 + 35.23294016577245.*z))) + ...
x.*(-175.292041186547 - 22.6683558512829.*x + (166.3847855603638 - 88.449193048287.*z).*z + ...
y.*(766.116132004952 + y.*(-1380.9597954037708 + 938.26075044542.*y) + ...
z.*(-216.7669050068448 + 153.8390924339484.*z))) + ...
y.*(-1721.528607567954 + y.*(2082.7344423998043 + ...
y.*(-1190.914967948748 + (597.809129110048 - 437.84750280190565.*z).*z) + ...
z.*(-1229.337851789418 + (1022.055280565346 - 133.5392811916956.*z).*z)) + ...
z.*(1349.638121077468 + z.*(-1069.887337245828 + (353.6322866464 - 79.20015472116819.*z).*z))));

gibbs = (g03 + g08).*2.5e-10;
% Note. This derivative of the Gibbs function is in units of (m^3/(K kg)),
% that is, the pressure of the derivative in Pa.

elseif ns==2 & nt==0 & np==0

g08 = 5812.814566267320 + 851.2267349467060.*y + x.*(-3648.219935726910 ...
+ x.*(8103.204624147880 + x.*(-8187.513078222526 + x.*(4495.214854534080 ...
- 850.3093707944657.*x))) + y.*(-740.1112652125230 + x.*(2175.341332000392 ...
+ x.*(-1470.212300173320 + 441.0859475949660.*x)) + y.*(-64.59970139670629 ...
- 274.2290036817964.*x + y.*(-15.03410562928125 + 197.4670779425016.*x ...
+ y.*(1.313400992713418 - 68.55903096791521.*x + 9.987880382780312.*x.*y)))) ...
+ z.*(299.1894046108515 + x.*(-219.1676534131548 + 270.2131467083145.*x) ...
+ y.*(-262.9380617798205 - 90.67342340513160.*x + y.*(+ 574.5870990037140 ...
+ y.*(-690.4798977018854 + 351.8477814170325.*y))) + z.*(-78.44113639220025 ...
- 16.32775915649044.*x + y.*(124.7885891702729 - 81.28758937756680.*y) ...
+ z.*(102.0667414089689 - 120.7020447884644.*x + y.*(-44.22459652414350 ...
+ 38.45977310848710.*y) - 5.118778986619230.*z))));

g08(x>0) = g08(x>0)./(x2(x>0));
g08(x==0) = NaN;

gibbs = 0.5.*sfac.*sfac.*g08;

elseif ns==0 & nt==2 & np==0

g03 = -24715.571866078 + z.*(2910.0729080936 + z.* ...
(-1513.116771538718 + z.*(546.959324647056 + z.*(-111.1208127634436 + 8.68841343834394.*z)))) + ...
y.*(4420.4472249096725 + z.*(-4035.04669887042 + ...
z.*(2996.162344914912 + z.*(-1437.2719839264719 + (292.8075111563232 - 9.978426372534301.*z).*z))) + ...
y.*(-1778.231237203896 + z.*(4775.621344883664 + ...
z.*(-3621.784567462512 + (1826.356460806092 - 316.49805267936244.*z).*z)) + ...
y.*(1160.5182516851419 + z.*(-3892.3662123519 + ...
z.*(2410.4130980405 + z.*(-1105.446104680304 + 129.6381336154442.*z))) + ...
y.*(-569.531539542516 + y.*(128.13429152494615 - 404.50541014508605.*z) + ...
z.*(1905.341809925355 + z.*(-668.691951421377 + 245.11816254543362.*z))))));

g08 = x2.*(1760.062705994408 + x.*(-86.1329351956084 + ...
x.*(-137.1145018408982 + y.*(296.20061691375236 + y.*(-205.67709290374563 + 49.9394019139016.*y))) + ...
z.*(766.116132004952 + z.*(-108.3834525034224 + 51.2796974779828.*z)) + ...
y.*(-60.136422517125 - 2761.9195908075417.*z + y.*(10.50720794170734 + 2814.78225133626.*z))) + ...
y.*(-1351.605895580406 + y.*(1097.1125373015109 + y.*(-433.20648175062206 + 63.905091254154904.*y) + ...
z.*(-3572.7449038462437 + (896.713693665072 - 437.84750280190565.*z).*z)) + ...
z.*(4165.4688847996085 + z.*(-1229.337851789418 + (681.370187043564 - 66.7696405958478.*z).*z))) + ...
z.*(-1721.528607567954 + z.*(674.819060538734 + ...
z.*(-356.629112415276 + (88.4080716616 - 15.84003094423364.*z).*z))));

gibbs = (g03 + g08).*0.000625;

elseif ns==0 & nt==0 & np==2

g03 = -5089.1530840726 + z.*(1707.1066706777221 + ...
z.*(-399.7761051043332 + (84.0526217606168 - 16.39285534413117.*z).*z)) + ...
y.*(1552.307223226202 + z.*(-1179.07530528732 + (347.75583155301 - 42.658016703665396.*z).*z) + ...
y.*(-1513.116771538718 + z.*(1640.877973941168 + z.*(-666.7248765806615 + 86.8841343834394.*z)) + ...
y.*(998.720781638304 + z.*(-1437.2719839264719 + (585.6150223126464 - 33.261421241781.*z).*z) + ...
y.*(-603.630761243752 + (913.178230403046 - 316.49805267936244.*z).*z + ...
y.*(241.04130980405 + y.*(-44.5794634280918 + 49.023632509086724.*z) + ...
z.*(-331.6338314040912 + 77.78288016926652.*z))))));

g08 = x2.*(769.588305957198 + z.*(-579.1945920644748 + (190.08980732018878 - 52.4960313181984.*z).*z) + ...
x.*(-104.588181856267 + x.*(-8.16387957824522 - 181.05306718269662.*z) + ...
(408.2669656358754 - 40.95023189295384.*z).*z + ...
y.*(166.3847855603638 - 176.898386096574.*z + y.*(-108.3834525034224 + 153.8390924339484.*z))) + ...
y.*(-687.913805923122 + z.*(748.126026697488 + z.*(-379.883572632876 + 140.9317606630898.*z)) + ...
y.*(674.819060538734 + z.*(-1069.887337245828 + (530.4484299696 - 158.40030944233638.*z).*z) + ...
y.*(-409.779283929806 + y.*(149.452282277512 - 218.92375140095282.*z) + ...
(681.370187043564 - 133.5392811916956.*z).*z))));

gibbs = (g03 + g08).*1e-16;
% Note. This is the second derivative of the Gibbs function with respect to
% pressure, measured in Pa.  This derivative has units of (J/kg) (Pa^-2).

elseif ns==0 & nt==3 & np==0

g03 = 4420.44722490967251 + y.*(-3556.46247440779189 + y.*(3481.55475505542563 ...
+ y.*(-2278.12615817006417 + 640.671457624730749.*y))) + z.*(-4035.04669887042019 ...
+ y.*(9551.24268976732856 + y.*(-11677.0986370556993 + y.*(7621.36723970141975 ...
- 2022.52705072543023.*y))) + z.*(2996.16234491491196 + y.*(-7243.56913492502372 ...
+ y.*(7231.23929412149937 - 2674.76780568550794.*y)) ...
+ z.*(-1437.27198392647188 + y.*(3652.71292161218389 + y.*(-3316.33831404091188 ...
+ 980.472650181734480.*y)) + z.*(292.807511156323187 + y.*(-632.996105358724890 ...
+ 388.914400846332569.*y) - 9.97842637253430098.*z))));

g08 = x2.*(-1351.605895580406 + x.*(-60.1364225171250 + 296.200616913752.*x) ...
+ y.*(2194.22507460302 + y.*(-1299.61944525187 + 255.620365016620.*y) ...
+ x.*(21.0144158834147 + x.*(-411.354185807491 + 149.818205741705.*y))) ...
+ z.*(4165.46888479961 - 2761.91959080754.*x + y.*(-7145.48980769249 ...
+ 5629.56450267252.*x) + z.*(-1229.33785178942 + 1793.42738733014.*y ...
+ z.*(681.370187043564 - 875.695005603811.*y - 66.7696405958478.*z))));

gibbs = (g03 + g08).*0.000625.*0.025;

elseif ns==0 & nt==2 & np==1

g03 = 2910.07290809360 + y.*(-4035.04669887042 + y.*(4775.62134488366  ...
+ y.*(-3892.36621235190 + y.*(1905.34180992535 - 404.505410145086.*y))))  ...
+ z.*(-3026.23354307744 + y.*(5992.32468982982 + y.*(-7243.56913492502  ...
+ y.*(4820.82619608100 - 1337.38390284275.*y))) + z.*(1640.87797394117  ...
+ y.*(-4311.81595177942 + y.*(5479.06938241828 + y.*(-3316.33831404091  ...
+ 735.354487636301.*y)))+ z.*(-444.483251053774 + y.*(1171.23004462529  ...
+ y.*(-1265.99221071745 + 518.552534461777.*y))  ...
+ z.*(43.4420671917197 - 49.8921318626715.*y))));

g08 = x2.*(-1721.52860756795 + 766.116132004952.*x - 2761.91959080754.*x.*y  ...
+ y.*(4165.46888479961 + y.*(-3572.74490384624 + 2814.78225133626.*x))  ...
+ z.*(1349.63812107747 - 216.766905006845.*x + y.*(-2458.67570357884 ...
+ 1793.42738733014.*y) + z.*(-1069.88733724583 + 153.839092433949.*x  ...
+ y.*(+ 2044.11056113069 - 1313.54250840572.*y) + z.*(353.632286646400  ...
- 267.078562383391.*y - 79.2001547211682.*z))));

gibbs = (g03 + g08).*2.5e-10.*0.025;

elseif ns==1 & nt==1 & np==1

g08 =  1458.23305947009 + x.*(-525.876123559641 - 90.6734234051316.*x ...
+ y.*(2298.34839601486 + y.*(-4142.87938621131 + 2814.78225133626.*y))) ...
+ y.*(-3443.05721513591 + y.*(4165.46888479961 - 2381.82993589750.*y)) ...
+ z.*(-1375.82761184624 + x.*(499.154356681091 - 650.300715020534.*y) ...
+ y.*(2699.27624215494 + y.*(-2458.67570357884 + 1195.61825822010.*y)) ...
+ z.*(748.126026697488 + x.*(-265.347579144861 + 461.517277301845.*y) ...
+ y.*(-2139.77467449166 + y.*(2044.11056113069 - 875.695005603811.*y)) ...
+ z.*(-253.255715088584 + y.*(707.264573292800 - 267.078562383391.*y) ...
+ z.*(70.4658803315449 - 158.400309442336.*y))));

gibbs = g08.*sfac.*0.5e-8.*0.025;

elseif ns==2 & nt==0 & np==1

g08 = 299.1894046108515 + x.*(-219.1676534131548 + 270.2131467083145.*x) ...
+ y.*(-262.9380617798205 - 90.67342340513160.*x + y.*(+ 574.5870990037140 ...
+ y.*(-690.4798977018854 + 351.8477814170325.*y))) + z.*(- 156.8822727844005 ...
- 32.65551831298088.*x + y.*( 249.5771783405457 - 162.5751787551336.*y) ...
+ z.*(306.2002242269066 - 362.1061343653932.*x + y.*(-132.6737895724305 ...
+ 115.3793193254613.*y) - 20.47511594647692.*z));

g08(x>0) = g08(x>0)./x(x>0);
g08(x==0) = nan;

gibbs = 0.5.*sfac.*sfac.*g08.*1e-8;

elseif ns==1 & nt==2 & np==0

g08 = 3520.125411988816 + x.*(-258.3988055868252 - 548.4580073635929.*x) ...
+ y.*(-2703.211791160812 + x.*(-180.409267551375 + 1184.8024676550094.*x) ...
+ y.*(2194.2250746030217 + x.*(31.52162382512202 + x.*(-822.70837161498247 ...
+ 199.7576076556064.*y)) + y.*(-866.4129635012441 + 127.8101825083098.*y))) ...
+ z.*(-3443.057215135908 + 2298.348396014856.*x + y.*(8330.937769599217 ...
- 8285.758772422625.*x + y.*(-7145.4898076924878 + 8444.34675400878.*x)) ...
+ z.*(1349.638121077468 - 325.1503575102672.*x + y.*(-2458.675703578836 ...
+ 1793.427387330144.*y) + z.*(-713.258224830552 + 153.8390924339484.*x ...
+ y.*(1362.740374087128 - 875.6950056038112.*y) + z.*(176.8161433232 ...
- 133.5392811916956.*y - 31.6800618884673.*z))));

g08(x==0) = nan;

gibbs = 0.5.*sfac.*0.025.*g08.*0.025;

elseif ns==2 & nt==1 & np==0

g08 = 851.22673494670596 + x.*(-740.11126521252299 + x.*(2175.3413320003920 ...
+ x.*(-1470.2123001733199 + 441.08594759496600.*x)) + y.*(-129.19940279341259 ...
- 548.45800736359286.*x + y.*(-45.102316887843749 + 592.40123382750477.*x ...
+ y.*(5.2536039708536704 + x.*(-274.23612387166082 + 49.939401913901600.*y)))) ...
+ z.*(-262.93806177982049 - 90.673423405131601.*x + y.*(1149.1741980074280 ...
+ y.*(- 2071.4396931056563 + 1407.3911256681299.*y)) + z.*( 124.78858917027286 ...
- 162.57517875513361.*y + z.*(-44.224596524143500 + 76.919546216974197.*y))));

g08(x>0) = g08(x>0)./x2(x>0);
g08(x==0) = nan;

gibbs = 0.5.*sfac.*sfac.*g08.*0.025;

elseif ns==1 & nt==0 & np==2

g08 = 1539.17661191439606 + x.*(-313.764545568801 - 32.6555183129809.*x) ...
+ y.*(-1375.827611846244 + 499.15435668109143.*x + y.*(1349.638121077468 ...
- 325.150357510267.*x + y.*(-819.558567859612 + 298.904564555024.*y))) ...
+ z.*(-1158.3891841289496 + x.*(1224.8008969076263 - 724.2122687307865.*x) ...
+ y.*(1496.252053394976 - 530.695158289722.*x + y.*(-2139.774674491656 ...
+ 461.5172773018452.*x + y.*(1362.740374087128 - 437.847502801906.*y))) ...
+ z.*(380.179614640378 - 122.850695678862.*x + y.*(-759.767145265752 ...
+ y.*(1060.8968599392 - 267.078562383391.*y)) + z.*(-104.992062636397 ...
+ y.*(281.86352132618 - 316.800618884673.*y))));

gibbs = g08.*sfac.*0.5e-8.*1e-8;

elseif ns==0 & nt==1 & np==2

g03 = 1552.307223226202 + y.*(-3026.233543077436 + y.*(2996.162344914912 ...
+ y.*(-2414.523044975008 + y.*(1205.20654902025 - 267.4767805685508.*y)))) ...
+ z.*(-1179.07530528732 + y.*(3281.755947882336 + y.*(-4311.815951779416 ...
+ y.*(3652.712921612184 + y.*(-1658.169157020456 + 294.141795054520.*y)))) ...
+ z.*(347.755831553010 + y.*(-1333.449753161323 + y.*(1756.84506693794 ...
+ y.*(-1265.99221071745 + 388.9144008463326.*y))) + z.*(-42.6580167036654 ...
+ y.*(173.7682687668788 - 99.7842637253430.*y))));

g08 = x2.*(-687.9138059231220 + 166.3847855603638.*x + y.*(1349.638121077468 ...
- 216.7669050068448.*x + y.*(-1229.337851789418 + 597.8091291100480.*y)) ...
+ z.*(748.1260266974880 - 176.8983860965740.*x + y.*(-2139.774674491656 ...
+ 307.6781848678968.*x + y.*(2044.110561130692 - 875.6950056038113.*y)) ...
+ z.*(-379.883572632876 + y.*(1060.89685993920 - 400.6178435750868.*y) ...
+ z.*(140.9317606630898 - 316.8006188846728.*y))));

gibbs = (g03 + g08).*2.5e-10.*1e-8;
% Note. This derivative of the Gibbs function is in units of (m^3/(K kg)),
% that is, the pressure of the derivative in Pa.

end

end

```