PC-SAFT

The equation of state is organized into terms that account for different types of intermolecular interactions, including terms for

• the hard chain reference
• dispersion
• association
• polar interactions
• ions

The equation is most often expressed in terms of the residual Helmholtz energy because all other thermodynamic properties can be easily found by taking the appropriate derivatives of the Helmholtz energy.^[2]

${\displaystyle a=a^{\text{hc}}+a^{\text{disp}}+a^{\text{assoc}}+a^{\text{dipole}}+a^{\text{ion}}}$

Here ${\displaystyle a}$ is the molar residual Helmholtz energy.

${\displaystyle {\frac {a^{\text{hc}}}{kT}}={\overline {m}}\cdot a^{\text{hs}}-\sum _{i=1}^{NC}{x_{i}\cdot (m_{i}-1)\cdot \ln(g_{i,i}^{\text{hs}})}}$

where

• ${\displaystyle NC}$ is the number of compounds;
• ${\displaystyle x_{i}}$ is the mole fraction;
• ${\displaystyle {\overline {m}}=\sum _{i=1}^{NC}{x_{i}m_{i}}}$ is the average number of segments in the mixture;
• ${\displaystyle a^{\text{hs}}}$ is the Boublík-Mansoori-Leeland-Carnahan-Starling hard sphere equation of state,^[9][10][11] defined as:

${\displaystyle a^{\text{hs}}=\zeta _{0}\left({\frac {3\zeta _{1}\zeta _{2}}{1-\zeta _{3}}}+{\frac {\zeta _{2}^{3}}{\zeta _{3}(1-\zeta _{3})^{2}}}+{\frac {\zeta _{2}^{3}}{\zeta _{3}^{2}-\zeta _{0}}}\log {(1-\zeta _{3})}\right)}$

${\displaystyle \zeta _{k}=\sum _{i}{x_{i}m_{i}d_{i}^{k}}}$

${\displaystyle d_{i}=\sigma _{i}\left(1-0.12\exp {\left({\frac {-3\epsilon _{i}}{kT}}\right)}\right)}$, where ${\displaystyle k}$ is the Boltzmann constant. ${\displaystyle g_{i,i}^{\text{hs}}}$ is the hard sphere radial distribution function at contact.^[9]:$${\displaystyle g_{i,j}^{\text{hs}}={\frac {1}{1-\zeta _{3}}}+{\frac {3\zeta _{2}}{(1-\zeta _{3})^{2}}}\left({\frac {d_{i}d_{j}}{d_{i}+d_{j}}}\right)+{\frac {2\zeta _{2}^{2}}{(1-\zeta _{3})^{3}}}\left({\frac {d_{i}d_{j}}{d_{i}+d_{j}}}\right)^{2}}$$

${\displaystyle a^{\text{disp}}=-2\pi \mathbb {N} _{A}\rho I_{1}{\overline {m^{2}\epsilon \sigma ^{3}}}-{\overline {m}}\mathbb {N} _{A}\rho I_{1}{\overline {m^{2}\epsilon ^{2}\sigma ^{3}}}}$

${\displaystyle {\overline {m^{2}\epsilon \sigma ^{3}}}=\sum _{i}\sum _{j}{x_{i}x_{j}m_{i}m_{j}{\frac {\epsilon _{ij}}{kT}}\sigma _{ij}^{3}}}$

${\displaystyle {\overline {m^{2}\epsilon ^{2}\sigma ^{3}}}=\sum _{i}\sum _{j}{x_{i}x_{j}m_{i}m_{j}\left({\frac {\epsilon _{ij}}{kT}}\right)^{2}\sigma _{ij}^{3}}}$

Where ${\displaystyle N_{A}}$ is the Avogadro constant, ${\displaystyle \rho }$ is the molar density, and ${\displaystyle I_{1}}$, ${\displaystyle I_{2}}$ are the analytical functions representing the integrals of the radial distribution function in 1st and 2nd order perturbation terms:^[12]

${\displaystyle I_{1}=\sum _{i=0}^{i=6}{\left(a_{0i}+{\frac {{\overline {m}}-1}{\overline {m}}}a_{1i}+{\frac {{\overline {m}}-1}{\overline {m}}}{\frac {{\overline {m}}-2}{\overline {m}}}a_{2i}\right)\zeta _{3}^{i}}}$

${\displaystyle I_{2}=\sum _{i=0}^{i=6}{\left(b_{0i}+{\frac {{\overline {m}}-1}{\overline {m}}}b_{1i}+{\frac {{\overline {m}}-1}{\overline {m}}}{\frac {{\overline {m}}-2}{\overline {m}}}b_{2i}\right)\zeta _{3}^{i}}}$

${\displaystyle b_{ji}}$, ${\displaystyle a_{ji}}$ are universal model parameters:

	${\displaystyle a_{0i}}$	${\displaystyle a_{1i}}$	${\displaystyle a_{2i}}$	${\displaystyle b_{0i}}$	${\displaystyle b_{1i}}$	${\displaystyle b_{2i}}$
0	0.9105631445	-0.3084016918	-0.0906148351	0.7240946941	-0.5755498075	0.0976883116
1	0.6361281449	0.1860531159	0.4527842806	2.2382791861	0.6995095521	-0.2557574982
2	2.6861347891	-2.5030047259	0.5962700728	-4.0025849485	3.8925673390	-9.1558561530
3	-26.547362491,	21.419793629	-1.7241829131	-21.003576815	-17.215471648	20.642075974
4	97.759208784	-65.255885330	-4.1302112531	26.855641363	192.67226447	-38.804430052
5	-159.59154087	83.318680481	13.776631870	206.55133841	-161.82646165	93.626774077
6	91.297774084	-33.746922930	-8.6728470368	-355.60235612	-165.20769346	-29.666905585

To obtain ${\displaystyle \epsilon _{ij}}$ and ${\displaystyle \sigma _{ij}}$, the following mixing rules are used:

${\displaystyle \epsilon _{ij}=(1-k_{ij}){\sqrt {\epsilon _{i}\epsilon _{j}}}}$

${\displaystyle \sigma _{ij}=(1-l_{ij}){\frac {\sigma _{i}+\sigma _{j}}{2}}}$

Were ${\displaystyle k_{ij}}$ and ${\displaystyle l_{ij}}$ interaction parameters, obtaining via fitting binary mixture data.

${\displaystyle a^{\text{assoc}}=\sum _{i=1}^{n_{c}}{x_{i}\sum _{a=1}^{n_{s}}{n_{i,a}\left(\log {X_{i,a}}-{\frac {X_{i,a}}{2}}+{\frac {1}{2}}\right)}}}$

Where ${\displaystyle X_{i,a}}$ is the fraction of non-bonded sites of type ${\displaystyle a}$ in component ${\displaystyle i}$, and ${\displaystyle n_{i,a}}$ is the number of sites of type ${\displaystyle a}$ in component ${\displaystyle i}$. ${\displaystyle X_{i,a}}$ is the solution of the following system of equations:

${\displaystyle X_{i,a}={\frac {1}{1+\sum _{j=1}^{n_{c}}{\sum _{b=1}^{n_{s}}{\rho x_{j}n_{j,b}X_{j,b}\Delta _{ij,ab}}}}}}$

Where ${\displaystyle \Delta _{ij,ab}}$ is the association strength between sites of type ${\displaystyle a}$ in component ${\displaystyle i}$, and sites of type ${\displaystyle b}$ in component ${\displaystyle j}$. In the specific case of PC-SAFT, ${\displaystyle \Delta _{ij,ab}}$ is defined as:

${\displaystyle \Delta _{ij,ab}=g_{i,j}^{\text{hs}}\sigma _{ij}^{3}\kappa _{ij,ab}\exp {\left({\frac {\epsilon _{ij,ab}^{\text{HB}}}{kT}}-1\right)}}$

Where ${\displaystyle \epsilon _{ij,ab}^{\text{HB}}}$ is the hydrogen bonding energy and ${\displaystyle \kappa _{ij,ab}}$ is the bonding volume, between sites of type ${\displaystyle a}$ in component ${\displaystyle i}$, and sites of type ${\displaystyle b}$ in component ${\displaystyle j}$.