Equations

Jacobus Henricus van’t Hoff (1852–1911), the winner of the first Nobel prize in chemistry, explained in his 1884 book Etudes de dynamique chimique the relation between chemical equilibrium and temperature [1, 2]. (Nomenclature is summarized in Table 1.) The two forms of the van’t Hoff equation in Table 2 quantify the (van’t Hoff) Le Chatelier principle, the displacement of the equilibrium to reactants or products when the equilibrium conditions vary:

  • An equilibrium constant \(K>1\) corresponds to an equilibrium with formation of products from the initial amounts of substance. On the contrary, for \(K<1\) there is formation of reactants.

  • In an endothermic reaction the enthalpy change \(\Delta H\) is positive and a temperature increase displaces the equilibrium towards the formation of products, vice versa for an exothermic reaction.

In a separate development, B. P. E. Clapeyron (1799–1864) related the variation of the equilibrium pressure with temperature in terms of the entropy and volume differences between the two phases in equilibrium [3]. R. J. E. Clausius (1822–1888) elaborated on this and obtained the Clausius–Clapeyron equations in Table 2 for the equilibrium between a condensed phase, liquid or solid, of a pure substance, and its vapor, in this case \(\Delta H\) is the evaporation or sublimation enthalpy, a positive quantity [4].

Table 1 Nomenclature
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Table 2 van’t Hoff (VH) and Clausius–Clapeyron (CC) equations. C is an integration constant
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The enthalpies of reaction and vaporization are functions of temperature that can be calculated with the differential forms of van’t Hoff (VH) and Clausius–Clapeyron (CC) equations in Table 2 at any T using numerical estimates of \(\text {d}\ln K/\text {d}T\) or \(\text {d}\ln P/\text {d}T\) to obtain \(\Delta H\). On the other hand, the integrated versions of VH and CC equations are linear models based on a constant \(\Delta H\) approximation that yield a single value of it from a linear regression of multiple \(K\left( T\right)\) or \(P\left( T\right)\) values over a temperature range. Not only do these equations have this inaccuracy but also the overuse of linearization in the teaching of chemistry has been questioned on statistical grounds in Refs. [5, 6]. Despite that, the integrated forms prevail in textbooks and research works, as they are easy to apply, usually fit very well the experimental data, and give consistent results.

The current article is based on the integrated forms because they make it easy to explain the connection between VH and CC models, but anyway the application examples were checked as follows. For the gas-phase equilibrium of the dehydrogenation reaction of methylcyclohexane to toluene the VH data reduction shown in Fig. 1 yields \(\Delta H=212.6\,\textrm{kJ}\,\textrm{mol}^{-1}\), in agreement with the standard value of \(\Delta H^{\circ }=205\,\textrm{kJ}\,\textrm{mol}^{-1}\) [7, 8]. Also, in the ebullition and sublimation examples of Figs. 2, 3, and 4, the CC equation fits very well the saturation pressure data, and \(\Delta H\) in Fig. 3 differs by less than 10% from measured experimental values.

Fig. 1
figure 1

van’t Hoff plot for the dehydrogenation of methylcyclohexane to toluene (\(\textrm{C}_{7}\textrm{H}_{14}\rightleftarrows \textrm{C}_{7}\textrm{H}_{8}+3\textrm{H}_{2}\)) in the experimental data range \(570\,\textrm{K} \le T \le 645\,\textrm{K}\) [8]. The slope of linear regression yields \(\Delta H=212.6\,\textrm{kJ}\,\textrm{mol}^{-1}\), with a determination coefficient \(r^{2}=0.9857\)

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Fig. 2
figure 2

Clausius–Clapeyron reduction of experimental isooctane (2,2,4-trimethylpentane) vapor–liquid \(P-T\) equilibrium measurements [9]. The linear regression slope, Eq. (12), yields a vaporization enthalpy \(\Delta H=34.0\,\textrm{kJ}\,\textrm{mol}^{-1}\), with a determination coefficient \(r^{2}=0.9998\)

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Fig. 3
figure 3

Vaporization enthalpy of isooctane (2,2,4-trimethylpentane). The fitted value, \(\Delta H=34.0\,\textrm{kJ}\,\textrm{mol}^{-1}\), was obtained from vapor pressure data [9]. Experimental measurements were taken from the NIST Chemistry WebBook [10]

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Fig. 4
figure 4

Clausius–Clapeyron reduction of phenoxazine sublimation \(P-T\) data [11]. The slope of linear regression, Eq. (12) yields a sublimation enthalpy \(\Delta H=103.83\,\textrm{kJ}\,\textrm{mol}^{-1}\), with a determination coefficient \(r^{2}=0.9986\)

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There is an obvious similarity between the van’t Hoff (VH) and Clausius–Clapeyron (CC) equations in Table 2: in CC the \(\ln \left( \right)\) vs. 1/T pattern of VH reappears, with ebullition or sublimation \(\Delta H\) in the place of reaction \(\Delta H\), and K is replaced by P. This analogy between VH and CC models has been overlooked in textbooks and articles, except perhaps in the applications of the van’t Hoff equation to liquid–vapor equilibria proposed in Ref. [12], and in the treatment of Henry’s law constant dependency on temperature in Ref. [13].

In order to show the VH–CC equivalence the current work puts aside the usual Clausius–Clapeyron deduction and instead it explains in the next section how to deduce it from the van’t Hoff equation treating the phases in vapor–liquid and solid–vapor equilibria as reacting species. In the next two sections the application of this procedure for the estimation of enthalpies of dissolution is explained for gas–liquid, and solid–liquid equilibria.

Transformation

The van’t Hoff equation comes from the fundamental relation

$$\begin{aligned} \frac{\partial }{\partial T}\left( \frac{G}{RT}\right) _{P} =-\frac{H}{RT^{2}} \end{aligned}$$
(1)

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

$$\begin{aligned} \ln {K}=-\frac{\Delta G}{RT}, \end{aligned}$$
(2)

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

$$\begin{aligned} \frac{\text {d}\ln K}{\text {d}T}=\frac{\Delta H}{RT^{2}} \end{aligned}$$
(3)

which integrated yields

$$\begin{aligned} \ln K=-\frac{\Delta H}{RT}+C, \end{aligned}$$
(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

$$\begin{aligned} K = \prod _{i}\left( \frac{{\bar{f}}_{i}}{f_{i}^{\circ }}\right) ^{\nu _{i}} = \prod _{i}\left( a_{i}\right) ^{\nu _{i}} \end{aligned}$$
(5)

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

$$\begin{aligned} \textrm{C}_{7}\textrm{H}_{14} \rightleftarrows \textrm{C}_{7}\textrm{H}_{8}+3\textrm{H}_{2}, \end{aligned}$$
(6)

Equation (5) becomes

$$\begin{aligned} K = \frac{ P_{ \textrm{C}_{7}\textrm{H}_{8} } \left( P_{\textrm{H}_{2}}\right) ^{3} }{ P_{ \textrm{C}_{7}\textrm{H}_{14} } \left( P^{\circ }\right) ^{3} } \cdot \end{aligned}$$
(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

$$\begin{aligned} \textrm{L}\rightleftarrows \textrm{V}, \end{aligned}$$
(8)

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

$$\begin{aligned} K = \frac{a_{\textrm{V}}}{a_{\textrm{L}}} \cdot \end{aligned}$$
(9)

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

$$\begin{aligned} a_{\textrm{V}} = \frac{f\left( T,P\right) }{f^{\circ }\left( T,P^{\circ }\right) } \approx \frac{P}{1\,\textrm{bar}} \end{aligned}$$
(10)

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

$$\begin{aligned} K=\frac{P}{1\,\textrm{bar}}, \end{aligned}$$

which leads to

$$\begin{aligned} \frac{\text {d}\ln P}{\text {d}T}=\frac{\Delta H}{RT^{2}}, \end{aligned}$$
(11)

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

$$\begin{aligned} \ln P=-\frac{\Delta H}{RT}+C, \end{aligned}$$
(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

$$\begin{aligned} \textrm{S}\rightleftarrows \textrm{V} \end{aligned}$$

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

Gas–liquid equilibrium

The dissolution of a gas (i) in a non-volatile solvent can be written as the “reaction”

$$\begin{aligned} i\left( \textrm{G}\right) \rightleftarrows i\left( \textrm{L}\right) , \end{aligned}$$
(13)

where the gas becomes a solute i in the liquid solution (G and L represent the gas and liquid phases, respectively). For this case the equilibrium constant is written as an infinite dilution limit,

$$\begin{aligned} K &= \lim _{x_{i} \rightarrow 0} \frac{\left( {\bar{f}}_{i}/f_{i}^{\circ }\right) _{\textrm{L}}}{\left( {\bar{f}}_{i}/f_{i}^{\circ }\right) _{\textrm{G}}} \nonumber \\ &= \lim _{x_{i} \rightarrow 0} \frac{\left( f_{i}^{\circ }\right) _{\textrm{G}}}{\left( f_{i}^{\circ }\right) _{\textrm{L}}}\nonumber \frac{\left( {\bar{f}}_{i}\right) _{\textrm{L}}}{\left( {\bar{f}}_{i}\right) _{\textrm{G}}} \end{aligned}$$
(14)

because the concentration of the dissolved gas in the liquid tends to 0. Also,

  • For the gas phase \(\left( f_{i}^{\circ }\right) _{\textrm{G}}=P^{\circ }=1\,\textrm{bar}\).

  • Equilibrium between gaseous and dissolved i implies equal fugacities, i.e., \(\left( {\bar{f}}_{i}\right) _{\textrm{L}}=\left( {\bar{f}}_{i}\right) _{\textrm{G}}\).

  • The solute’s standard fugacity in the liquid phase, \(f_{i}^{\circ }\), cannot be set with the Lewis–Randall rule because the gas i does not exist as pure liquid at the equilibrium conditions. Instead, following Henry’s law

    $$\begin{aligned} \left( f_{i}^{\circ }\right) _{\textrm{L}}={\mathcal {H}}_{ij}, \end{aligned}$$

    where \({\mathcal {H}}_{ij}\) is the Henry’s constant at equilibrium temperature (not to be confused with the enthalpy).

Replacing all the above in Eq. (14) it becomes

$$\begin{aligned} K = \frac{1\,\textrm{bar}}{{\mathcal {H}}_{ij}}, \end{aligned}$$

which replaced in Eqs. (3) and (4) yields

$$\begin{aligned} \frac{\text {d}\ln {\mathcal {H}}_{ij}}{\text {d}\left( 1/T\right) } = \frac{\Delta H}{R} \end{aligned}$$
$$\begin{aligned} \ln {\mathcal {H}}_{ij} = \frac{\Delta H}{RT} + C, \end{aligned}$$
(15)

where \(\Delta H\) is the enthalpy of dissolution of i, from the gas to the liquid phase [13,14,15,16].

Fig. 5
figure 5

Experimental solubilities of \(\mathrm {CO_{2}}\) in water [18]. Lines represent the least-squares fit for each temperature

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Table 3 Henry’s law constants of \(\mathrm {CO_{2}}\) in water from experimental data [18]
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Fig. 6
figure 6

van’t Hoff equation reduction for \(\mathrm {CO_{2}}\) solubility in water data [18]. Dissolution enthalpy, obtained from the slope, is \(\Delta H=-11.75\,\textrm{kJ}\,\textrm{mol}^{-1}\)

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Equation (15) follows the pattern of van’t Hoff and Clausius–Clapeyron (Eqs. 4 and 12, but with a positive sign in the right-hand side), allowing the estimation of \(\Delta H\) from experimental data, in fact semiempirical representations of \(\ln {\mathcal {H}}_{ij}\) include 1/T terms [13, 17]. Here, for example, data of carbon dioxide (i) concentration in water was reduced to Henry’s law (Ref. [18], Figs. 5, 6, Table 3)

$$\begin{aligned} P_{i}=x_{i}{\mathcal {H}}_{ij} \end{aligned}$$
(16)

by linear regression at each temperature to obtain the enthalpy of dissolution of the gas from the slope of the linear fit in Fig. 6. The negative value, \(\Delta H=-11.75\,\textrm{kJ}\,\textrm{mol}^{-1}\), is explained with the Le Chatelier principle: the solubility of the gas decreases with the temperature because the dissolution is exothermic.

Solid–liquid equilibrium

In this case the precipitation of the pure components of a binary liquid mixture can be described as the equilibrium

$$\begin{aligned} i\left( \textrm{S}\right) \rightleftarrows i\left( \textrm{L}\right) \end{aligned}$$
(17)

between the pure solid phase and i dissolved in the liquid, with

$$\begin{aligned} K = \frac{\left( a_{i}\right) _{\textrm{L}}}{\left( a_{i}\right) _{\textrm{S}}} = \left( x\gamma \right) _{i,\textrm{L}} \end{aligned}$$
(18)

because the activity for the pure solid is \(a_{i,\textrm{S}}=1\). The estimation of the activity, that is \(x\gamma\), from the data is commonly described in textbooks starting with the equality of liquid and solid fugacities

$$\begin{aligned} {\bar{f}}_{i}^{\textrm{L}} = x_{i}\gamma _{i}f_{i}^{\textrm{L}} = f_{i}^{\textrm{S}} \end{aligned}$$

where the symbol f represents the fugacity of pure component, while \({\bar{f}}\) is used for the fugacity of the component in solution. It is tempting to simplify this expression by canceling out \(f_{i}^{\textrm{L}}\) and \(f_{i}^{\textrm{S}}\), but it would only be valid at \(T_{\textrm{fus}}\), the fusion temperature of pure i; instead

$$\begin{aligned} \ln \left( x\gamma \right) _{i,\textrm{L}} = -\ln \left( \frac{f_{i}^{\textrm{L}}}{f_{i}^{\textrm{S}}} \right) \end{aligned}$$

where the ratio \(\left( f_{i}^{\textrm{L}}/f_{i}^{\textrm{S}}\right)\) for the pure solid is deduced by applying a three-stage integration path to \(\text {d}G=RT\text {d}\ln f\), \(T\rightarrow T_{\textrm{fus}}\) as solid, solid\(\rightarrow\)liquid at \(T_{\textrm{fus}}\), and \(T_{\textrm{fus}}\rightarrow T\) as liquid to obtain

$$\begin{aligned}{} & {} \ln \left( x\gamma \right) _{i,\textrm{L}} = -\left\{ \frac{\Delta _{\textrm{fus}}H}{RT} \left[ 1-\frac{T}{T_{\textrm{fus}}}\right] \right. \nonumber \\{} & {} \left. + \frac{\Delta C}{R} \left[ 1-\frac{T_{\textrm{fus}}}{T} + \ln \left( \frac{T_{\textrm{fus}}}{T}\right) \right] \right\} _{i}, \end{aligned}$$
(19)

where \(T_{\textrm{fus}}\) and \(\Delta _{\textrm{fus}}H\) are the fusion temperature and enthalpy of the pure solid. The term \(\Delta C=C_{\textrm{L}}-C_{\textrm{S}}\) is the difference of the specific heat capacities of liquid and solid phases. The enthalpy of dilution of pure solid i in the mixture is obtained by replacing this K in the van’t Hoff differential equation (Eq. 3)

$$\begin{aligned} \Delta H = -R \cdot \frac{\text {d}\ln \left( x\gamma \right) _{i,\textrm{L}}}{\text {d}\left( 1/T\right) }, \end{aligned}$$
(20)

which is a function of the mole fraction and the equilibrium temperature.

Fig. 7
figure 7

Solid–liquid equilibrium for the mixture naphthalene (1) + m-dinitrobenzene (2) [19]. \(\Delta H\) is the dilution enthalpy of pure solid i to the \(x_{1}\) concentration in the liquid. \(K=x\gamma\). Lines were obtained from polynomial fit

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An example of the application of van’t Hoff equation to solid–liquid equilibrium data is given in Fig. 7 for the mixture naphthalene (1) + m-dinitrobenzene (2) [19]. First, the data was divided into two series, one for the equilibrium with solid m-dinitrobenzene (left side, \(i=2\)), and other for the naphthalene (right side, \(i=1\)), while \(\ln \left( x\gamma \right)\) was calculated from Eq. (19) using the properties of each pure solid i. Then, these values were fitted to polynomial functions of \(1000\,\textrm{K}/T\), which provided the derivatives in the right-hand side of Eq. (20). In agreement with the Le Chatelier principle it was found that the dilution is endothermic, i.e., \(\Delta H>0\), as the solubility of both components increase with temperature; also, as expected \(\Delta H \rightarrow \Delta _{\textrm{fus}}H\) in the pure component limit.

Digressions

The representation of chemical equilibrium experimental data in \(\ln K\) vs. 1/T coordinates is known as the van’t Hoff plot, where \(\Delta H / R\) is the slope of the best fit line. In practice this slope is the coefficient of the independent variable (1/T) obtained from the linear regression of the data, which in general purpose software is based on matrix operations where the small values of 1/T can lead to the formation of ill-conditioned matrix systems [20]. To prevent such problems, the current work uses the common transformation \(1000\,\textrm{K}/T\) as the independent variable in the regressions, as shown in the figures, although it is not mandatory. Another option is to replace the linear regression with a minimization of the sum of discrepancies between experimental and calculated data by means of a general non-linear optimization function [6, 21]. Also, a deep analysis of the regression results falls beyond the pedagogical scope of this work, although the statistics of linearization results have been studied in the context of chemistry applications in Refs. [5, 6, 21].

A general form of the equilibrium constant is given in Eq. (5) to ensure a dimensionless K, avoiding the well-known problems associated with the use of equilibrium constants with units; see, for example, Ref. [22]. Despite this, Eqs. (11, 12, 15) include logarithms of quantities with pressure units, \(\ln \left( P\right)\) and \(\ln \left( {\mathcal {H}}_{ij}\right)\), but in these cases units of P or \({\mathcal {H}}_{ij}\) are not relevant.

  • In the differential forms, such as Eq. (11), the differential eliminates any conversion factor, for example \(\text {d}\ln \left( P\times 100\,\textrm{kPa}/1\,\textrm{bar}\right) =\text {d}\ln \left( P\right)\).

  • In the integral forms, such as Eq. (12), any conversion factor is adsorbed by the integration constant C.

Also, the differential form \(\text {d}\ln P/\text {d}T\) makes it possible to obtain vaporization or sublimation \(\Delta H\) values from vapor pressure \(P\left( T\right)\) models, such as the August or Antoine equations, regardless of their pressure units.

A definite integration of Eq. (11) from a reference temperature \(T_{0}\) yields

$$\begin{aligned} \ln \frac{P}{P_{0}} = -\frac{\Delta H}{R} \left( \frac{1}{T}-\frac{1}{T_{0}}\right) \end{aligned}$$
(21)

where \(P_{0}\) is the equilibrium pressure at \(T_{0}\). Given that the entropy of the phase change at \(T_{0}\) is \(\Delta S=\Delta H/T_{0}\) , its value can be estimated from the intercept in the linear regression of \(\ln \left( P/P_{0}\right)\) vs. 1/T. Apparently, it can be resumed as [11]

$$\begin{aligned} \ln P = -\frac{\Delta H}{RT} + \frac{\Delta S}{R}, \end{aligned}$$
(22)

but there is a caveat: in this expression the value of \(\Delta S\) changes with the pressure units, which are included in the intercept. Hence, Eq. (21) is preferable, and the results should specify the values of \(T_{0}\) and \(P_{0}\).

Henry’s law is also written as \(x_{i}=P_{i}{\mathcal {H}}_{ij}\), or an equivalent expression [15]. In such cases the van’t Hoff analysis remains valid, but the values of Henry’s constant should be converted to the form of Eq. (16).

Conclusion

Thermal effects on both chemical and phase equilibria can be described in terms of the van’t Hoff equation, which leads to the Clausius–Clapeyron equation. However, this deduction route requires itself a formulation of the reaction equilibrium and therefore it is preferable to use it as a reinforcement of the usual explanation of the Clausius–Clapeyron equation. Analogously to the pure substance equilibria, the enthalpy changes implicit in the dissolution of gases or solids in liquids can be estimated by applying the van’t Hoff equation to the \(K\left( T\right)\) data. It is expected that the formulations proposed in this work will serve for both teaching and experimental analysis of phase equilibria.