Fugacity

The fugacity or is a quantity from physics that is defined differently depending on the subject. ${\displaystyle z}$ ${\displaystyle f}$

Statistical Physics

In statistical physics , which forms the basis of thermodynamics, fugacity is defined as a dimensionless function of chemical potential and temperature : ${\displaystyle z}$ ${\displaystyle \mu }$ ${\displaystyle T}$

${\displaystyle z=\mathrm {e} ^{\tfrac {\mu }{k_{\mathrm {B} }\cdot T}}}$

With

• the Boltzmann constant . ${\displaystyle k_{\mathrm {B} }}$

The fugacity is therefore equal to the absolute activity . Fugacity appears as a factor in the transition from the canonical sum of states , which describes systems with a constant number of particles , to the grand-canonical sum of states , which is suitable for describing systems with a variable number of particles: ${\displaystyle \lambda }$ ${\displaystyle Z(T,V,N)}$ ${\displaystyle N}$ ${\displaystyle \Xi (T,V,\mu )}$

${\displaystyle \Xi (T,V,\mu )=\sum _{N=0}^{\infty }{z^{N}\cdot Z(T,V,N)}}$

thermodynamics

In thermodynamics , fugacity is an intense state variable that has the unit of pressure (e.g. Pascal ). It was first introduced by Gilbert Newton Lewis as the "escaping tendency"; the name was shortened to "fugacity" by himself. It describes the tendency of a substance to leave a phase (fugare, Latin for "flee"). ${\displaystyle f}$

definition

The fugacity is introduced via the pressure dependence of the specific Gibbs energy . ${\displaystyle g}$

For an ideal gas applies (because of its equation of state and the fundamental equation of the Gibbs energy) at a isothermal state change from a print to : ${\displaystyle p^{0}}$ ${\displaystyle p}$

${\displaystyle g^{id}(T,p)-g^{id}(T,p^{0})=RT\cdot \ln {\frac {p}{p^{0}}}}$

With

• the gas constant ${\displaystyle R}$
• the natural logarithm . ${\displaystyle \ln }$

The fugacity is defined in such a way that the following applies to a real fluid (with any reference fugacity ): ${\displaystyle f^{0}}$

${\displaystyle g(T,p)-g(T,p^{0})=RT\cdot \ln {\frac {f}{f^{0}}}}$

Subtracting the first equation from the second gives:

${\displaystyle g(T,p)-g^{id}(T,p)-(g(T,p^{0})-g^{id}(T,p^{0}))=RT\cdot \ln \left({\frac {f}{f^{0}}}\cdot {\frac {p^{0}}{p}}\right)}$

If the reference pressure (index 0) is now allowed to approach zero, the difference between real and ideal Gibbs energy disappears; on the right-hand side, reference fugacity and pressure merge:

${\displaystyle {\begin{aligned}p^{0}\to 0:&g(T,p^{0})\approx g^{id}(T,p^{0});\quad f^{0}\approx p^{0}\\\Rightarrow &g(T,p)-g^{id}(T,p)=RT\cdot \ln {\frac {f}{p}}\end{aligned}}}$

Instead of the fugacity, the dimensionless fugacity coefficient is used more often :

${\displaystyle \phi ={\frac {f}{p}},}$

which is defined in multi- substance systems via the partial pressure ( is the proportion of the amount of substance ): ${\displaystyle y_{i}\cdot p}$ ${\displaystyle y_{i}}$

${\displaystyle \phi _{i}={\frac {f_{i}}{y_{i}\cdot p}}}$

About the relationship

${\displaystyle \int _{0}^{p}\left(v-{\frac {RT}{\hat {p}}}\right)\mathrm {d} {\hat {p}}=RT\cdot \ln \phi }$

With

• the specific volume ${\displaystyle v}$

the fugacity can be calculated from measured values ​​or with an equation of state .

Criterion for phase equilibria

The fugacity, like the chemical potential, is a criterion for a phase equilibrium : if the fugacity of a component is the same in all phases present (but not the fugacity of different components in the same phase), then these phases are in equilibrium: ${\displaystyle i}$

${\displaystyle f_{i}^{\mathrm {Ph1} }=f_{i}^{\mathrm {Ph2} }}$

From this condition the following relationship for vapor-liquid equilibria can be derived, with which z. B. Calculate phase diagrams for the design of rectification columns , which is therefore of great importance in process engineering :

${\displaystyle x_{i}\cdot \gamma _{i}\cdot \phi _{i}^{s}\cdot p_{i}^{s}\cdot \exp {\left(\int _{{p}_{i}^{s}}^{p}{{\frac {{v}_{i}^{L}}{RT}}\operatorname {d} \!p}\right)}=\underbrace {\phi _{i}^{v}\cdot y_{i}\cdot p} _{f_{i}^{\mathrm {Dampf} }}}$

Stand by

• ${\displaystyle x_{i}}$for the mole fraction of a substance in the liquid phase
• ${\displaystyle y_{i}}$for the mole fraction of a substance in the vapor phase
• ${\displaystyle \gamma _{i}}$for the activity coefficient . This can be calculated from g ^E models for the excess Gibbs energy, for example with UNIFAC .
• ${\displaystyle p_{i}^{s}}$for the vapor pressure of the pure substance .

With the exponential term, the Poynting factor , the deviation from the vapor pressure is taken into account; it is often very close to one and is then neglected. The fugacity coefficient on the right takes into account the non-ideality of the vapor phase.

See also

• volatility

literature

• Peter W. Atkins: Physical Chemistry . 2nd Edition. Wiley-VCH, Weinheim 1996, ISBN 3-527-29275-6 .
• W. Nolting : Basic Course Theoretical Physics 6, Statistical Physics . 4th edition. ISBN 3-540-41918-7 .
• J. Gmehling , B. Kolbe, M. Kleiber, J. Rarey: Chemical Thermodynamics for Process Simulation . 1st edition. Wiley-VCH, Weinheim 2012, ISBN 978-3-527-31277-1 .

Individual evidence

1. ^ GN Lewis: The Law of Physico-Chemical Change. In: Proceedings of the American Academy of Arts and Sciences. American Academy of Arts & Sciences, Vol. 37, No. 3 (Jun., 1901), pp. 49-69, doi : 10.2307 / 20021635 .