Chemical (engineering) thermodynamics — how do we define the component fugacity?

Question

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1. What the question is really asking

The student is trying to understand how the fugacity of a component in a mixture is defined and why the textbook definition

[ RT\,d\ln \hat f_i=(d\mu_i)_T\tag{1} ]

does not seem to give a unique value (any factor that depends only on temperature could be multiplied onto (\hat f_i) and still satisfy the differential relation).
The student also sees another definition that ties fugacity to the chemical potential of a real pure gas by

[ \mu_i^{\text{real}}=\mu_i^{\ominus}+RT\ln\frac{f_i}{P^{\ominus}}\tag{2} ]

and wonders how these two ideas are related, what the role of the reference (standard) state is, and how the fugacity coefficient (\phi_i) enters.

In short:

What is the precise definition of the component fugacity (f_i) (or (\hat f_i)) in a mixture?
Why does Eq. (1) not lead to an arbitrary constant?
How does the “pressure that an ideal gas would need to have to give the same μ” definition fit in?

2. Full derivation – from the differential definition to an explicit expression

2.1. Starting point: the differential relation

For any component (i) in a multicomponent system the chemical potential (\mu_i) is a state function.
At constant temperature the exact differential of (\mu_i) can be written as

[ \left(d\mu_i\right)_T = RT\, d\ln f_i\tag{3} ]

where (f_i) is called the fugacity of component (i). Equation (3) is the definition of fugacity; it tells us that the natural logarithm of fugacity is, up to the factor (RT), the thermodynamic potential conjugate to the amount of component (i) at fixed (T).

Why a differential?
Because (\mu_i) is only defined up to an additive constant. By relating changes in (\mu_i) to changes in (\ln f_i) we avoid having to guess that constant. The constant will be fixed later by specifying a reference (standard) state.

2.2. Integration – introducing the reference state

Integrate Eq. (3) from a convenient reference state “0” (denoted by a superscript (0)) to the actual state of interest:

[ \int_{\mu_i^{0}}^{\mu_i} d\mu_i = RT\int_{f_i^{0}}^{f_i} \frac{d f_i’}{f_i’}
\quad\Longrightarrow\quad \mu_i-\mu_i^{0}=RT\ln\frac{f_i}{f_i^{0}}.\tag{4} ]

Equation (4) is the integrated definition of fugacity.
The two quantities that still need to be chosen are:

Symbol	Meaning
(\mu_i^{0})	Chemical potential of component (i) in a standard state at the same temperature (and, for gases, the same pressure).
(f_i^{0})	Fugacity of component (i) in that same standard state.

If we pick the standard state to be an ideal gas at the same temperature and pressure as the real fluid, then by definition

[ f_i^{0}=P \qquad\text{(ideal‑gas fugacity equals the pressure)}\tag{5} ]

and the standard‑state chemical potential is the ideal‑gas chemical potential (\mu_i^{\text{ig}}). Substituting (5) into (4) gives the more familiar expression

[ \boxed{\;\mu_i = \mu_i^{\text{ig}} + RT\ln\frac{f_i}{P}\;}\tag{6} ]

which is exactly the relation you saw in the LibreTexts article (they write (P^{\ominus}) for the chosen standard pressure, often 1 bar).

2.3. Uniqueness – why we cannot multiply by an arbitrary function of (T)

Suppose we replace (f_i) by (c(T)f_i) where (c(T)) depends only on temperature.
Equation (3) would indeed still hold because at constant (T),

[ d\ln[c(T)f_i]=d\ln f_i+\underbrace{d\ln c(T)}_{=0}=d\ln f_i . ]

However, after integration we must also replace the reference fugacity:

[ \mu_i-\mu_i^{0}=RT\ln\frac{c(T)f_i}{c(T)f_i^{0}}=RT\ln\frac{f_i}{f_i^{0}} . ]

The factor (c(T)) cancels provided we use the same factor for the reference state. Consequently, the absolute value of fugacity is not determined until we declare a standard state (i.e., we fix (\mu_i^{0}) and (f_i^{0})). The usual convention is the ideal‑gas reference at the same (T) and (P); this makes the fugacity unique for a given real state.

Thus the “arbitrariness” is only apparent; it is removed by the conventional choice of standard state.

2.4. Component fugacity in a mixture

For a gas mixture the total pressure is (P) and the mole fraction of component (i) is (y_i).
If the mixture behaves ideally, the chemical potential of component (i) is

[ \mu_i^{\text{ig}} = \mu_i^{\ominus} + RT\ln\frac{y_i P}{P^{\ominus}} . ]

For a real mixture we define the fugacity coefficient (\phi_i) by

[ f_i \equiv \phi_i \, y_i P \tag{7} ]

so that (6) becomes

[ \mu_i = \mu_i^{\ominus} + RT\ln\frac{y_i P}{P^{\ominus}} + RT\ln\phi_i . ]

Equation (7) is the working definition of the component fugacity in a gas mixture.
The same idea works for liquids, where we use the liquid‑phase standard state (often the pure liquid at the system temperature and 1 bar) and write

[ f_i = \gamma_i \, x_i \, f_i^{\ast} \quad\text{(liquid)}\tag{8} ]

with (\gamma_i) the activity coefficient and (f_i^{\ast}) the fugacity of the pure liquid.

3. Final answer – concise statement

Definition (differential form)

[ RT\,d\ln f_i = (d\mu_i)_T ]

Integrated form (with a chosen standard state)

[ \mu_i = \mu_i^{0} + RT\ln\frac{f_i}{f_i^{0}} ]

– If the standard state is the ideal gas at the same (T) and (P), then (f_i^{0}=P) and

[ \boxed{\mu_i = \mu_i^{\text{ig}} + RT\ln\frac{f_i}{P}} ]

Component fugacity in a gas mixture

[ \boxed{f_i = \phi_i \, y_i P} ]

where (\phi_i) is the fugacity coefficient (dimensionless) that quantifies the deviation from ideal‑gas behaviour.

Uniqueness – The fugacity is uniquely defined once a reference (standard) state is specified; any temperature‑only factor cancels out when the same factor is used for the reference state, so the apparent arbitrariness disappears.

4. Common mistakes

Mistake	Why it is wrong	How to avoid it
Treating (1) as an absolute definition and forgetting to specify a reference state.	Without a reference, (\mu_i) and (f_i) are defined only up to an additive constant, leading to the “any (c(T)) works” confusion.	Always write the integrated form (4) and state the chosen standard state (ideal gas at the same (T,P) is the usual convention).
*Identifying fugacity with pressure for any* gas**.	Only true for an ideal gas; for real gases a fugacity coefficient (\phi_i\neq 1) is needed.	Use (f_i = \phi_i y_i P) for real gas mixtures; set (\phi_i=1) only after proving the gas is ideal.
Mixing up the pure‑component fugacity (f_i^{\ast}) and the mixture fugacity (f_i).	(f_i^{\ast}) is the fugacity of the pure component at the system temperature and a reference pressure; (f_i) is the effective pressure that the component experiences in the mixture.	Keep the notation clear: (f_i^{\ast}) (pure) vs. (f_i = \phi_i y_i P) (mixture).
*Using the pressure of the mixture instead of the partial* pressure** in Eq. (7).	The partial pressure of component (i) in an ideal mixture is (y_i P); inserting the total pressure would over‑estimate fugacity.	Remember the factor (y_i) belongs with the pressure when writing (f_i = \phi_i y_i P).
Neglecting temperature dependence of the standard chemical potential (\mu_i^{0}(T)).	(\mu_i^{0}) varies with (T); assuming it is constant gives wrong values for (\mu_i).	Use tabulated values or equations of state for (\mu_i^{0}(T)) (e.g., ideal‑gas reference).

Bottom line:
Fugacity is defined through the exact differential relation (1). By integrating from a well‑chosen standard state—most commonly the ideal gas at the same temperature and pressure—we obtain an explicit, unique expression that links the real‑fluid chemical potential to an “effective pressure” (f_i). In mixtures the component fugacity is simply the product of the mole fraction, the total pressure, and a dimensionless fugacity coefficient that measures non‑ideality.

Original question: Chemical (engineering) thermodynamics — how do we define the component fugacity? on Chemistry Stack Exchange, licensed CC BY-SA.