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Formelsammlung Physik: Thermodynamik 2

Definitionsgleichungen

${\displaystyle U=TS-PV\,}$
${\displaystyle H=U+PV\,}$
${\displaystyle G=H-TS\,}$
${\displaystyle F=U-TS\,}$

Fundamentalgleichungen

${\displaystyle dU=TdS-PdV+\sum _{i=1}^{k}\mu _{i}dn_{i}}$
${\displaystyle dH=TdS+VdP+\sum _{i=1}^{k}\mu _{i}dn_{i}}$
${\displaystyle dG=-SdT+VdP+\sum _{i=1}^{k}\mu _{i}dn_{i}}$
${\displaystyle dF=-SdT-PdV+\sum _{i=1}^{k}\mu _{i}dn_{i}}$

Kalorische Zustandsgröße

${\displaystyle C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{V}}$
${\displaystyle C_{P}=\left({\frac {\partial H}{\partial T}}\right)_{P}}$

Maxwell-Beziehungen

${\displaystyle -\left({\frac {\partial P}{\partial S}}\right)_{V}=\left({\frac {\partial T}{\partial V}}\right)_{S}}$
${\displaystyle \left({\frac {\partial V}{\partial S}}\right)_{P}=\left({\frac {\partial T}{\partial P}}\right)_{S}}$
${\displaystyle \left({\frac {\partial V}{\partial T}}\right)_{P}=-\left({\frac {\partial S}{\partial P}}\right)_{T}}$
${\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial P}{\partial T}}\right)_{V}}$

Gibbs-Duhem-Gleichungen

Freie Enthalpie:

${\displaystyle 0=SdT-VdP+\sum _{i=1}^{k}n_{i}d\mu _{i}}$

Gleichgewicht und Stabilität

S= maximum G= minimum

Gibbs-Phasenregel

${\displaystyle {\text{Freiheitsgrade}}={\text{Komponenten}}+2-{\text{Phasen}}\,}$

Clausius-Clapeyron

${\displaystyle {\frac {dP^{LV}}{dT}}={\frac {\Delta S^{LV}}{\Delta V^{LV}}}={\frac {\Delta H^{LV}}{T\Delta V^{LV}}}}$

August-Gleichung

${\displaystyle \ln P^{LV}=A-{\frac {B}{T}}}$

Antoine-Gleichung

${\displaystyle \ln P^{LV}=A-{\frac {B}{T+C}}}$

Gibbs-Helmholtz-Gleichung

${\displaystyle \left({\frac {\partial {\frac {G}{T}}}{\partial T}}\right)=-{\frac {H}{T^{2}}}}$

Fugazitätskoeffizient

Definition:

${\displaystyle \varphi _{0,i}={\frac {f_{0,i}(T,P)}{P}}}$ mit ${\displaystyle \lim _{P\to \infty }f_{0,i}(T,P)=P}$ und ${\displaystyle \lim _{P\to \infty }\varphi _{0,i}(T,P)=1}$

Reine Komponenten:

${\displaystyle \ln \varphi _{0,i}(T,P)=\int _{0}^{P}{\frac {z-1}{P}}dP}$
${\displaystyle \ln \varphi _{0,i}(T,P)=z-1-\ln z+\int _{V}^{\infty }{\frac {z-1}{V}}dV}$

Mischungen:

${\displaystyle \ln \varphi _{i}(T,P)=\int _{0}^{P}\left({\frac {v_{i}}{RT}}-{\frac {1}{P}}\right)dP}$
${\displaystyle \ln \varphi _{i}(T,P)=\int _{V}^{\infty }\left(-{\frac {1}{RT}}\left({\frac {\partial P}{\partial n_{i}}}\right)_{T,V,{n\neq j}}+{\frac {1}{V}}\right)dV-\ln z}$

Virialgleichung

Leidenform:

${\displaystyle z={\frac {Pv}{RT}}=1+B(T)\rho +C(T)\rho ^{2}+...=1+{\frac {B(T)}{v}}+{\frac {C(T)}{v^{2}}}+...}$

Berlinform:

${\displaystyle z={\frac {Pv}{RT}}=1+B'(T)P+C'(T)P^{2}+...}$

Umrechnungsformel:

${\displaystyle B'\approx {\frac {B}{RT}}}$
${\displaystyle C'\approx {\frac {C-B^{2}}{{(R_{0}T)}^{2}}}}$

Boyle-Temperatur

Bei Boyle-Temperatur ist das PVT-Verhalten ideal.

${\displaystyle B(T^{B})=0\,}$

Van der Waals

Allgemein:

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

nach Druck P:

${\displaystyle P={\frac {RT}{v-b}}-{\frac {a}{v^{2}}}}$

nach z:

${\displaystyle z={\frac {Pv}{RT}}={\frac {v}{v-b}}-{\frac {a}{vRT}}}$

Generalisierten Form:

${\displaystyle \left[{\frac {P}{P_{kr}}}+3\left({\frac {v_{kr}}{v}}\right)^{2}\right]\left[3{\frac {v}{v_{kr}}}-1\right]=8{\frac {T}{T_{kr}}}}$ mit ${\displaystyle v_{kr}=3b\,}$, ${\displaystyle T_{kr}={\frac {8a}{27bR}}}$ und ${\displaystyle P_{kr}={\frac {a}{27b^{2}}}}$

Reduzierte Form:

${\displaystyle \left(P_{r}+{\frac {3}{v_{r}^{2}}}\right)\left(3v_{r}-1\right)=8T_{r}}$ mit ${\displaystyle v_{r}={\frac {v}{v_{kr}}}}$, ${\displaystyle T_{r}={\frac {T}{T_{kr}}}}$ und ${\displaystyle P_{r}={\frac {P}{P_{kr}}}}$

Virialgleichung und van der Waals

${\displaystyle B(T)=b-{\frac {a}{RT}}}$
${\displaystyle T^{B}={\frac {a}{Rb}}}$

Korrespondenzprinzip

Zweiparameterkorrespondenzprinzip:

Stoffe sind vergleichbar, wenn sie aus vergleichbaren Punkten betrachtet werden.

Dreiparameterkorrespondenzprinzip:

${\displaystyle z=z_{0}+\omega _{0,i}z_{1}\,}$

mit Pitzerfaktor ${\displaystyle \omega _{0,i}=-1-\log \left({\frac {P_{0,i}^{LV}(T_{r}=0,7)}{P_{kr}}}\right)}$

Partielle molare Größe

${\displaystyle {\overline {o_{i}}}=\left({\frac {\partial O}{\partial n_{i}}}\right)_{T,P,n_{i\neq j}}}$

Mischungen

1. Differenzansatz:

${\displaystyle \Delta o=o_{ideal}-o_{real}\,}$

2. Partielle molare Ansatz:

${\displaystyle o=\sum _{i=1}^{k}{\overline {o}}_{i}x_{i}}$ oder ${\displaystyle O=\sum _{i=1}^{k}{\overline {o}}_{i}n_{i}}$

3. Exzessansatz:

${\displaystyle o=\sum _{i=1}^{k}x_{i}o_{i}+\Delta o^{ideal}+o^{E}}$

Aktivitätskoeffizienten

Definition:

${\displaystyle \gamma _{i}={\frac {\varphi _{i}^{L}}{\varphi _{0,i}^{L}}}}$

VLE

Allgemein:

Isofugazitätskriterium
${\displaystyle f_{i}^{V}=f_{i}^{L}}$ mit
${\displaystyle f_{i}^{V}=y_{i}\varphi _{i}P}$ und
${\displaystyle f_{i}^{L}=x_{i}\gamma _{i}f_{0,i}^{L}=x_{i}\gamma _{i}P_{0,i}^{LV}\varphi _{0,i}^{LV}\Pi }$
Poynting-Korrektur ${\displaystyle \Pi =\exp \left(\int _{P_{0,i}^{LV}}^{P}{\frac {v_{0,i}^{L}}{RT}}dP\right)}$

Vereinfacht:

${\displaystyle x_{i}\gamma _{i}P_{0,i}^{LV}=y_{i}P}$

Raoultsche Gesetz:

${\displaystyle x_{i}P_{0,i}^{LV}=y_{i}P}$

Least-Square-Methode

${\displaystyle \sum _{i=1}^{k}\left(f_{{\text{exp}},i}-f_{{\text{cal}},i}\right)^{2}{\stackrel {!}{=}}{\text{minimal}}}$