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I'm studying equilibria and thermodynamics and came across these two terms.

My problem is, unlike other thermodynamic properties that I can understand physically like volume, pressure, enthalpy etc. do these two quantities have physical significance or, is it just that we define fugacity (without any physical meaning) just to make sure the equation

$$\mu - \mu_0 = RT\ln\frac{p}{p_0}$$

looks the same for real gases? If yes, then what is the motivation for the name “fugacity” meaning literally “escaping tendency”?

Similarly, do we just define chemical potential randomly, since it's extremely useful in various calculations when the composition varies, or does it have some physical significance?

Also, why do we call it chemical potential?

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    $\begingroup$ I have written a very detailed reply as to why fugacity got this name in a previous Q&A post [What is fugacity?][1]. I think the concept of chemical potential can also looked from a historical point of view. Wikipedia has it en.wikipedia.org/wiki/Chemical_potential#History $\endgroup$
    – M. Farooq
    Jan 20 at 3:38
  • $\begingroup$ The definition of chemical potential is not arbitrary (random). $\endgroup$
    – Buck Thorn
    Feb 19 at 13:12
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The chemical potential is the partial derivative of the Gibbs free energy with respect to the number of moles of the specified species at constant T, P, and numbers of moles of all other species. The equation you wrote is the what this partial derivative reduces to for a species in an ideal gas mixture, where P is the partial pressure of the species in the mixture. For a real gas, the fugacity of the species replaces the partial pressure of the species in the same equation for the chemical potential.

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  • $\begingroup$ Chet, Still what is the physical consequence or meaning of a partial derivative of Gibbs free energy with respect to moles. $\endgroup$
    – M. Farooq
    Jan 19 at 15:52
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    $\begingroup$ @M.Farooq I really don't ascribe any physical significance to it, although, in combination with the chemical potentials of the other species in the mixture, it determines the driving force for interphase mass transfer of a species and for chemical reaction. $\endgroup$ Jan 19 at 15:58
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    $\begingroup$ @ChetMiller thanks for the answer, but do you mean fugacity and chemical potential have no physical meaning by themselves and we just define them as mathematical terms like $(dG)/(d{n_i})$ since they are important/handy tools?? $\endgroup$ Jan 19 at 18:54
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    $\begingroup$ Saying chemical potential has no physical meaning is IMHO like saying molar heat capacity or molar conductivity have no physical meaning. $\endgroup$
    – Poutnik
    Jan 19 at 20:29
  • $\begingroup$ Well @Poutnik, molar heat capacity is the partial derivative of either molar internal energy or molar enthalpy with respect to temperature. And, as I said, chemical potential is the partial derivative of Gibbs free energy with respect to the number of moles of a species. Is that the kind of physical meaning you were referring to? $\endgroup$ Jan 20 at 3:27
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In less mathematical terms, the chemical potential of a component in a mixture at a specific temperature and pressure is the amount of free energy that can be ascribed to one mole of that component under the conditions found in the mixture at that T and p. Alternately it is how much the free energy of the system, if it were scaled to an infinitely large size, would increase if you added one mole of that component.

The chemical potential is conceptually useful in that it can inform you on whether a species in a sample is content to be where it is located or in the chemical form it is in. For instance, if the chemical potential of an atom is lower in one chemical form than in others, it will prefer to be in the lower-potential form, thereby driving a reaction to that form. Minimization of the sum of the chemical potentials of all species (which is just the Gibbs free energy) drives processes such as chemical reactions (at constant T and p).

Another example is that if the chemical potential is lower in one phase of a multiphase system, then the system may spontaneously undergo a phase transformation, the driving force dissipating when equilibrium is reached and the species has the same chemical potential in all available phases. We say that at equilibrium each species in a system has the same chemical potential in all phases.

The fugacity of a species describes an effective partial pressure in the gas phase. At low pressure of the system the fugacity of a species tends to be equal to its partial pressure. However, as the partial pressures and thereby intermolecular interactions increase in importance, a simple relation between pressure and chemical potential breaks down and fugacity becomes useful when discussing the state (chemical potential).

The mathematical relation between fugacity and chemical potential was chosen to resemble that between partial pressure and chemical potential in the ideal state. This allows one formulation to describe both real scenarios and the limiting ideal case. A fugacity greater than the partial pressure would suggest that a gas has a greater chemical potential than is suggested by extrapolation from low pressures where it behaves ideally. A greater chemical potential suggests greater reactivity or a higher tendency to escape to regions of lower chemical potential (therefore escaping tendency is an appropriate term).

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The internal energy of a closed system is $dU=TdS-PdV $. In an isolated system $dU=0, dV=0$ which would imply that $dS=0$, However, the entropy can increase due to chemical reaction or diffusion mixing different substances which were initially separate. Similarly $dG=-SdT+VdP$ and if $T$ and $P$ are held constant this does not mean that $G$ cannot change as this is an extensive quantity and thus depends on the amount of material. For example in a galvanic cell more useful work can be done if more material is present, e.g. adding copper sulphate to a Daniel cell, so that $G$ must increase. This means that in a chemical system work can be done, i.e. lifting a weight etc., without there being any change in volume between the initial and final states, i.e. work done other than $PV$ work.

The difficulties with $dU=TdS-PdV $ and $dG=-SdT+VdP$ are due to the fact that we ignore the amount of material present. This changes the energy by an amount $\sum_i \mu_idn_i$ where $\mu$ is the chemical potential. It is called this by analogy to gravitational and electrical potential. In a gravitational field a mass $m$ has energy $mgh$ where $gh$ is the potential and $m$ the quantity. A charge $q$ has energy $q\varphi$ where $\varphi$ is the potential and $q$ the quantity. In chemical energy, $dn$ is the quantity and $\mu$ the potential so the energy is $\mu dn$ for each species.

In a homogeneous phase containing several different substances with $n_1,n_2..$ moles of species $1,2...$ the total differential of $U$ in terms of $S, T$ and $n_1,n-2..$ is

$$\displaystyle dU=TdS-PdV +\sum\mu_idn_i,\qquad \mu_i \equiv \left(\frac{\partial U}{\partial n_i} \right)_{S,V,n_j}$$

where subscript $j$ indicates all other species except $i$ are constant. Similarly for the Gibbs energy

$$\displaystyle dG=-SdT+VdP+\sum\mu_idn_i,\qquad \mu_i \equiv \left(\frac{\partial G}{\partial n_i} \right)_{T,P,n_j}$$

and this last definition provides the chemical potential shorthand label as the Gibbs energy /mole or more properly the gradient of the Gibbs energy/mole for a given species.

With reversible change in a closed system using $dU=TdS-PdV +\sum\mu_idn_i$ the heat absorbed by the system is $TdS$, thus the remaining terms represent the total work done. Thus $\sum\mu_idn_i$ is a form of work/energy in the absence of a change in volume.

The fugacity is used as a proxy for pressure for example when an imperfect gas is studied. It is defined as $\mu = \mu^\text{o} +RT\ln(f),\quad f/P\to 1 ; P\to 0$ so that fugacity equals the pressure when the ideal gas law is obeyed.

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