I got distracted looking for Shomate parameters for ethane today, and came across this website on predicting the boiling point of water using the Shomate equations. The basic idea is to find the temperature where the Gibbs energy of water as a vapor is equal to the Gibbs energy of the liquid.
clear all; close all
Hf_liq = -285.830; % kJ/mol S_liq = 0.06995; % kJ/mol/K shomateL = [-203.6060 1523.290 -3196.413 2474.455 3.855326 -256.5478 -488.7163 -285.8304];
Interestingly, these parameters are listed as valid only above 500K. That means we have to extrapolate the values down to 298K. That is risky for polynomial models, as they can deviate substantially outside the region they were fitted to.
Hf_gas = -241.826; % kJ/mol S_gas = 0.188835; % kJ/mol/K shomateG = [30.09200 6.832514 6.793435 -2.534480 0.082139 -250.8810 223.3967 -241.8264];
T = linspace(0,200)' + 273.15; % temperature range from 0 to 200 degC t = T/1000; % normalize sTT by 1/1000 so entropies are in kJ/mol/K sTT = [log(t) t t.^2/2 t.^3/3 -1./(2*t.^2) 0*t.^0 t.^0 0*t.^0]/1000; hTT = [t t.^2/2 t.^3/3 t.^4/4 -1./t 1*t.^0 0*t.^0 -1*t.^0]; Gliq = Hf_liq + hTT*shomateL - T.*(sTT*shomateL); Ggas = Hf_gas + hTT*shomateG - T.*(sTT*shomateG);
The boiling point is where Gliq = Ggas, so we solve Gliq(T)-Ggas(T)=0 to find the point where the two free energies are equal.
f = @(t) interp1(T,Gliq-Ggas,t); bp = fzero(f,373)
bp = 373.2045
You can see the intersection occurs at approximately 100 degC.
plot(T-273.15,Gliq,T-273.15,Ggas) line([bp bp]-273.15,[min(Gliq) max(Gliq)]) legend('liquid water','steam') xlabel('Temperature \circC') ylabel('\DeltaG (kJ/mol)') title(sprintf('The boiling point is approximately %1.2f \\circC', bp-273.15))
The answer we get us 0.05 K too high, which is not bad considering we estimated it using parameters that were fitted to thermodynamic data and that had finite precision and extrapolated the steam properties below the region the parameters were stated to be valid for.
% tags: thermodynamics