## Thermodynamic Evidence for [PERSON]'s Rule
### Trouton's Rule for Vapor-Liquid and Vapor-Sorber Equilibria  
In his original paper, [PERSON] observed that for a collection of vapor-liquid equilibria for different fluids, the latent heat \(\Delta h\) is proportional to the normal boiling temperature \(T_{0}\) at \(p_{0}=1\) atm [14]. In fact, from \(g=h-Ts\) for both vapor and liquid in equilibrium at \(T=T_{0}\), we obtain:  
\[\Delta h=T_{0}\Delta s. \tag{8}\]  
This would imply the proportionality of \(\Delta h\) and \(T_{0}\) if \(\Delta s\) would be the same for different fluids \(j\) at the boiling temperature \(T_{0j}\) at reference pressure \(p_{0}\). [PERSON]'s rule may thus be formulated as:  
\[\Delta h_{j}=T_{0j}\Delta s. \tag{9}\]  
It has been argued from the principle of corresponding states that [PERSON]'s rule in the form of Equation (9) should not be valid, but that instead \(\Delta h\) should be proportional to the critical temperature \(T_{c}\)[17; 24]. According to the principle of corresponding states, different media obey the same equation of states when expressed in terms of reduced pressure, temperature and volume \(p/p_{c}\), \(T/T_{c}\), and \(v/v_{c}\), such as for van der Waals fluids. Indeed, we already noted that for van der Waals fluids, \(\Delta h\cong 27 RT_{c}/8\) for \(T<0.5T_{c}\). From this, [PERSON]'s rule (9) would additionally follow if the boiling temperature \(T_{0}\) at reference pressure \(p_{0}\) would be proportional to the critical temperature \(T_{c}\), which is not obvious. However, for experimental boiling temperatures \(T_{0}\) at \(p_{0}=1\) atm for common gases, it indeed roughly holds that \(T_{0}\cong 0.6T_{c}\)[17]. This is illustrated by Figure 5, showing \(p_{0}/p_{c}\) as a function of \(T_{0}/T_{c}\) for both van der Waals fluids and for experimental values for common gases [17]. The experimental values are off the pT-line of the van der Waals equation, but the rapid increase of \(p_{0}/p_{c}\) with \(T_{0}/T_{c}\) is similar, and we see that for the experimental values of \(p_{0}/p_{c}\), the reduced boiling temperatures \(T_{0}/T_{c}\) would vary with less than 10% around 0.5 for van der Waals fluids, somewhat below the experimental average of 0.6. Taking \(T_{0}\cong 0.5T_{c}\) for van der Waals fluids with \(\Delta h=27 RT_{c}/8\) we arrive at the following prediction:  
\[\Delta h_{j}\cong\frac{27 RT_{0j}}{4}. \tag{10}\]  
This can be regarded as [PERSON]'s rule for van der Waals fluids, with \(\Delta s\cong 27R/4=6.75R\). Although this is considerably off the experimental value of \(\Delta s\cong 9.0R\)[17], the van der Waals equation and the above argument do support the idea that \(\Delta s\) should be roughly the same for common fluids. To this, we may add that the numerical accuracy of the van de Waals equation is limited anyway, as, for instance, the van der Waals equation predicts \(RT_{c}/p_{c}v_{c}=8/3\cong 2.7\), while the experimental value for common gases is rather between 3.0 and 3.5 [19]. Figure 5: Van der Waals normalized vapor pressure lines \(p_{0}/p_{c}\) as a function of normalized boiling temperatures \(T_{0}/T_{c}\) at \(p_{0}=1\) atm, together with points representing experimental values for common fluids [17]. Although the points are well off the line, both the line and the points show a similar rapid increment of \(p_{0}/p_{c}\) with \(T_{0}/T_{c}\). Since the difference in critical pressures between common gases is limited between about 25 and 90 atm for the selected gases, and because \(p/p_{c}\) increases so rapidly with \(T/T_{c}\), it follows that the ratio of boiling temperature and critical temperature \(T_{0}/T_{c}\) is roughly constant at a given pressure of e.g., \(p_{0}=1\) atm. For vapor-sorbed equilibria, we cannot use the van der Waals equation, but we may adopt the idea that the entropy is a function of the free volume \(v-b\), where \(b\) represents the volume occupied by the molecules [21].