# 维里展开

（重定向自维里状态方程

${\displaystyle Z={\frac {Pv}{RT}}=1}$

T是绝对温度，R是通用气体常数，v是摩尔体积。对于真正的气体和液体，Z不等于1，偏差取决于温度，压力和摩尔体积。其偏差可以用维里状态方程表示：

${\displaystyle Z={\frac {P}{RT\rho }}=A+B\rho +C\rho ^{2}+D\rho ^{3}+E\rho ^{4}+F\rho ^{5}+\cdots }$

## 将状态方程改寫為维里方程

1873年范德华提出了他著名的状态方程[5]

${\displaystyle P={\frac {RT}{\left(v-b\right)}}-{\frac {a}{v^{2}}}}$

${\displaystyle Z=1+\left(b-{\frac {a}{RT}}\right)\rho +b^{2}\rho ^{2}+b^{3}\rho ^{3}+\cdots }$

${\displaystyle p={\frac {RT}{v^{2}}}\left(1-{\frac {c}{vT^{3}}}\right)(v+B)-{\frac {A}{v^{2}}}}$

${\displaystyle A=A_{0}\left(1-{\frac {a}{v}}\right)}$
${\displaystyle B=B_{0}\left(1-{\frac {b}{v}}\right)}$

${\displaystyle Z=1+\left(B_{0}-{\frac {A_{0}}{RT}}-{\frac {c}{RT^{3}}}\right)\rho -\left(B_{0}b-{\frac {A_{0}a}{RT^{3}}}\right)\rho ^{2}+\left({\frac {B_{0}bc}{T^{3}}}\right)\rho ^{3}}$

1940年的Benedict-Webb-Rubin状态方程[11]就有显着的改善，特別是在低于临界温度的等温线:

${\displaystyle Z=1+\left(B_{0}-{\frac {A_{0}}{RT}}-{\frac {C_{0}}{RT^{3}}}\right)\rho +\left(b-{\frac {a}{RT}}\right)\rho ^{2}+\left({\frac {\alpha a}{RT}}\right)\rho ^{5}+{\frac {c\rho ^{2}}{RT^{3}}}\left(1+\gamma \rho ^{2}\right)\exp \left(-\gamma \rho ^{2}\right)}$

Starling [12] 在1972年提出了更多的改进：

${\displaystyle Z=1+\left(B_{0}-{\frac {A_{0}}{RT}}-{\frac {C_{0}}{RT^{3}}}+{\frac {D_{0}}{RT^{4}}}-{\frac {E_{0}}{RT^{5}}}\right)\rho +\left(b-{\frac {a}{RT}}-{\frac {d}{RT^{2}}}\right)\rho ^{2}+\alpha \left({\frac {a}{RT}}+{\frac {d}{RT^{2}}}\right)\rho ^{5}+{\frac {c\rho ^{2}}{RT^{3}}}\left(1+\gamma \rho ^{2}\right)\exp \left(-\gamma \rho ^{2}\right)}$

${\displaystyle Z=1+b\rho _{r}+c\rho _{r}^{2}+f\rho _{r}^{5}}$

${\displaystyle b=b_{0}+{\frac {b_{1}}{t_{r}}}+{\frac {b_{2}}{t_{r}^{2}}}+{\frac {b_{3}}{t_{r}^{3}}}}$
${\displaystyle c=c_{0}+{\frac {c_{1}}{t_{r}}}+{\frac {c_{2}}{t_{r}^{2}}}+{\frac {c_{3}}{t_{r}^{3}}}:}$
${\displaystyle f=f_{0}+{\frac {f_{1}}{t_{r}}}}$

Fluid ${\displaystyle b_{0}}$  ${\displaystyle b_{1}}$  ${\displaystyle b_{2}}$  ${\displaystyle b_{3}}$  ${\displaystyle c_{0}}$  ${\displaystyle c_{1}}$  ${\displaystyle c_{2}}$  ${\displaystyle c_{3}}$  ${\displaystyle f_{0}}$  ${\displaystyle f_{1}}$
Methane 0.440 -1.171 -0.236 -0.210 0.364 -0.275 -0.014 0.396 0.0319 1.71E-03
Ethane 0.330 -0.806 -0.363 -0.378 0.553 -0.675 -0.038 0.680 0.0461 2.63E-03
Propanr 0.288 -0.706 -0.245 -0.575 0.532 -0.546 -0.308 0.843 0.0334 1.89E-02
n-butane 0.377 -0.916 -0.115 -0.610 0.547 -0.519 -0.347 0.871 0.0305 2.04E-02
i-butane 0.438 -1.051 -0.172 -0.401 0.483 -0.342 -0.021 0.538 0.0194 1.19E-03
n-pentane 0.481 -1.056 -0.166 -0.560 0.668 -0.720 -0.204 0.841 0.0411 1.17E-02
i-pentane 0.242 -0.674 -0.306 -0.520 0.815 -0.943 -0.194 0.868 0.0484 9.99E-03
n-heane 0.435 -0.636 -0.358 -0.759 0.848 -1.275 -0.105 1.120 0.0604 4.98E-03
n-heptane 0.493 -0.798 -0.636 -0.428 0.589 -0.738 -0.017 0.814 0.0508 1.21E-03
n-octane 0.600 -0.744 -0.456 -0.763 0.174 -0.197 -0.272 0.919 0.0144 1.99E-02
nitrogen 0.502 -1.380 0.092 -0.333 0.400 -0.276 -0.027 0.322 0.0279 2.72E-03
CO2 0.178 -0.044 -1.517 0.039 0.428 -0.422 -0.008 0.687 0.0490 9.52E-04
H2S 0.191 -0.927 -0.078 -0.366 1.093 -1.227 -0.001 0.577 0.0578 8.37E-05

## 三次维里状态方程

${\displaystyle Z=1+B\rho +C\rho ^{2}}$

${\displaystyle {\frac {dP}{dv}}=0}$  and ${\displaystyle {\frac {d^{2}P}{dv^{2}}}=0}$

${\displaystyle B=-v_{c}}$ ${\displaystyle C={\frac {v_{c}^{2}}{3}}}$  and :${\displaystyle Z_{c}={\frac {P_{c}v_{c}}{RT_{c}}}=1/3}$

${\displaystyle Z_{c}}$ 为0.333，可以与Van del Waals状态方程解出的0.375相比。

${\displaystyle P_{sat}=RT_{sat}\left(1+B\rho +C\rho ^{2}\right)\rho }$

${\displaystyle 1-{\frac {RT_{sat}}{P_{sat}}}\left(1+B\rho +C\rho ^{2}\right)\rho =0}$

${\displaystyle RT_{sat}/P_{sat}}$ 因子实际上是根据理想气体定律的饱和气体体积，它可以命名為：

${\displaystyle v_{ideal}={\frac {RT_{sat}}{P_{sat}}}}$

${\displaystyle \left(1-v_{l}\rho \right)\left(1-v_{m}\rho \right)\left(1-v_{g}\rho \right)=0}$

${\displaystyle 1-\left(v_{l}+v_{g}+v_{m}\right)\rho +\left(v_{l}v_{g}+v_{g}v_{m}+v_{m}v_{l}\right)\rho ^{2}-v_{l}v_{g}v_{m}\rho ^{3}=0}$

${\displaystyle v_{m}}$ ${\displaystyle v_{l}}$ ${\displaystyle v_{g}}$ 之间的中介体积，这些三次维里方程式完全相等。根据这些方程的一次方项，可以解出${\displaystyle v_{m}}$

${\displaystyle v_{m}=v_{ideal}-v_{l}-v_{g}}$

${\displaystyle B=-{\frac {\left(v_{l}v_{g}+v_{g}v_{m}+v_{m}v_{l}\right)}{v_{ideal}}}}$

${\displaystyle C={\frac {v_{l}v_{g}v_{m}}{v_{ideal}}}}$

## 气液固体三相平衡

${\displaystyle P={\frac {RT}{V}}\left(1+{\frac {B}{V}}+{\frac {C}{V^{2}}}+{\frac {U}{V^{n}}}+{\frac {W}{V^{2n}}}\right)}$

${\displaystyle p={\frac {t}{vZ_{c}}}\left(1-{\frac {b}{v}}+{\frac {c}{v^{2}}}-\left({\frac {v_{u}}{v}}\right)^{n}+\left({\frac {v_{w}}{v}}\right)^{2n}\right)}$

${\displaystyle V_{w}}$ 必须略大于固体的体积0.33，${\displaystyle v_{u}}$ 的体积必须在液体和固体之间。首先我们将${\displaystyle v_{w}}$ 设置为0.335 ，使等温线在固体时急剧上升。然后我们必须估计指数n，使得n-2n位能曲线中的深谷必须落在固体（0.33）和液体（0.378）的体积之间。在确定了指数n之后，我们再调整${\displaystyle v_{u}}$ 的值以满足吉布斯规则(Gibbs Rule)，它要求在三相点温度和压力下，液相和固相的吉布斯自由能(Gibbs free energy)必须相等。

${\displaystyle p_{1}={\frac {t_{t}}{vZ_{c}}}\left(1-{\frac {b}{v}}+{\frac {c}{v^{2}}}\right)}$
${\displaystyle p_{2}={\frac {t_{t}}{vZ_{c}}}\left({\frac {v_{u}}{v}}\right)^{n}}$
${\displaystyle p={\frac {t_{t}}{vZ_{c}}}\left({\frac {v_{w}}{v}}\right)^{2n}}$
${\displaystyle p=p_{1}-p_{2}+p_{3}}$

${\displaystyle p={\frac {t_{t}}{vZ_{c}}}\left(1-{\frac {3.424}{v}}+{\frac {1.152}{v^{2}}}-\left({\frac {0.3443}{v}}\right)^{30}+\left({\frac {0.3350}{v}}\right)^{60}\right)}$

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