The van’t Hoff equation comes from the fundamental relation

$$\begin \frac\left( \frac\right) _ =-\frac} \end$$

(1)

(see nomenclature in Table 1) applied to the definition of chemical equilibrium constant

$$\begin \ln =-\frac, \end$$

(2)

where \(\Delta G\) is the change of Gibbs energy associated with the reaction, to obtain

$$\begin \frac\ln K}T}=\frac} \end$$

(3)

which integrated yields

$$\begin \ln K=-\frac+C, \end$$

(4)

where \(\Delta H\) is the (constant) reaction enthalpy change; Eqs. (3) and (4) are the two forms of the van’t Hoff equation in Table 2.

The equilibrium constant K governs the composition of the system through the expression

$$\begin K = \prod _\left( \frac}_}^}\right) ^} = \prod _\left( a_\right) ^} \end$$

(5)

where \(\nu _\) represents the stoichiometric coefficient of species i. The terms \(a_=}_/f_^\) are the activities, while the fugacities, \(}_\), depend on the system composition. In the gas phase the common ideal gas simplification implies \(}_=P_=y_P\), while the standard fugacity is \(f_^=P^=1\,\textrm\). In this way, for the gas-phase equilibrium example, the dehydrogenation of methylcyclohexane to toluene

$$\begin \textrm_\textrm_ \rightleftarrows \textrm_\textrm_+3\textrm_, \end$$

(6)

Equation (5) becomes

$$\begin K = \frac_\textrm_ } \left( P__}\right) ^ }_\textrm_ } \left( P^\right) ^ } \cdot \end$$

(7)

The equilibrium constants K in Fig. 1 were calculated with the experimental partial pressures available in Refs. [7, 8].

The Clausius–Clapeyron equation appears when liquid and vapor phases are treated as reactant and product in a chemical reaction so that vapor–liquid equilibrium (VLE) of a pure substance becomes

$$\begin \textrm\rightleftarrows \textrm, \end$$

(8)

where L and V identify the liquid and vapor phases, respectively. Using Eq. (5) the equivalent equilibrium constant for this “reaction” is

$$\begin K = \frac}}}} \cdot \end$$

(9)

For the liquid, which is a condensed phase \(a_}\thickapprox 1\), while for the vapor phase

$$\begin a_} = \frac\left( T,P^\right) } \approx \frac} \end$$

(10)

where f is the fugacity of the ideal gas pure phase, and \(P^\) is the standard reference pressure, 1 bar. By replacing the activities \(a_}\)and \(a_}\) the equilibrium constant becomes

$$\begin K=\frac}, \end$$

which leads to

$$\begin \frac\ln P}T}=\frac}, \end$$

(11)

where \(\Delta H\) is the vaporization enthalpy. The integrated form of the Clausius–Clapeyron equation,

$$\begin \ln P=-\frac+C, \end$$

(12)

(with a constant \(\Delta H\) value approximation) is much more common. Both Clausius–Clapeyron forms are also valid for the vapor–solid equilibrium in the sublimation of a pure substance

$$\begin \textrm\rightleftarrows \textrm \end$$

(S represents the solid phase), given that \(a_}\thickapprox 1\) and Eq. (10) remains valid for the vapor phase, leading again to an equilibrium constant \(K=P/}\).