SAMPL4, a blind challenge for computational solvation free energies: the compounds considered

Abstract

For the fifth time I have provided a set of solvation energies (1 M gas to 1 M aqueous) for a SAMPL challenge. In this set there are 23 blind compounds and 30 supplementary compounds of related structure to one of the blind sets, but for which the solvation energy is readily available. The best current values of each compound are presented along with complete documentation of the experimental origins of the solvation energies. The calculations needed to go from reported data to solvation energies are presented, with particular attention to aspects which are new to this set. For some compounds the vapor pressures (VP) were reported for the liquid compound, which is solid at room temperature. To correct from VP_{subcooled liquid} to VP_sublimation requires ΔS_fusion, which is only known for mannitol. Estimated values were used for the others, all but one of which were benzene derivatives and expected to have very similar values. The final compound for which ΔS_fusion was estimated was menthol, which melts at 42 °C so that modest errors in ΔS_fusion will have little effect. It was also necessary to look into the effects of including estimated values of ΔCp on this correction. The approximate sizes of the effects of inclusion of ΔCp in the correction from VP_{subcooled liquid} to VP_sublimation were estimated and it was noted that inclusion of ΔCp invariably makes ΔG_S more positive. To extend the set of compounds for which the solvation energy could be calculated we explored the use of boiling point (b.p.) data from Reaxys/Beilstein as a substitute for studies of the VP as a function of temperature. B.p. data are not always reliable so it was necessary to develop a criterion for rejecting outliers. For two compounds (chlorinated guaiacols) it became clear that inclusion represented overreach; for each there were only two independent pressure, temperature points, which is too little for a trustworthy extrapolation. For a number of compounds the extrapolation from lowest temperature at which the VP was reported to 25 °C was long (sometimes over 100°) so that it was necessary to consider whether ΔCp might have significant effects. The problem is that there are no experimental values and possible intramolecular hydrogen bonds make estimation uncertain in some cases. The approximate sizes of the effects of ΔCp were estimated, and it was noted that inclusion of ΔCp in the extrapolation of VP down to room temperature invariably makes ΔGs more negative.

Appendix

No claim is made for originality of the derivations given here; they are included to help the reader understand where the final equations used in the calculations actually came from.

Derivation of ideal solubility for mannitol

Robinson and Stokes [15] have reported the osmotic, ϕ. and activity, γ, coefficients for mannitol (species C in what follows) as functions of concentration (molal)

$$\upphi^{{^\circ }} = 1 + .0034{\text{m}}_{\text{c}} + 0.0042{\text{m}}_{\text{c}}^{2}$$

$${\text{Log}}_{10}\upgamma_{\text{C}} = 0.00295{\text{m}}_{\text{c}} + 0.00274{\text{m}}_{\text{c}}^{2}$$

The solubility of mannitol is 1.186 m [14].ΔG for mannitol going from saturated solution to 0.01 m (assumed ideal; γ_c = 1.000) can be calculated from:

$${\text{M}} = 1.186, \upgamma_{\text{c}} = 10^{(0.00295 \times 1.186 + 0.00274 \times 1.1862)} = 1.017$$

$${\text{M}} = 0.01, \upgamma_{\text{c}} = 10^{(0.00295 \times 0.01 + 0.00274 \times 0.012)} = 1.000$$

$$\Delta {\text{G}} = {\text{RT}}\ln \left( {\left( {0.01 \times 1.000} \right)/\left( {1.184 \times 1.017} \right)} \right) = - 2.84$$

ΔG for mannitol going from 0.01 m solution to 1.00 m solution, assuming ideality

$$\Delta {\text{G}} = {\text{RT}}\ln \left( {1.000/0.01} \right) = + 2.73$$

No literature values of the densities for mannitol solutions were found. A sucrose solution with the same g/100 g concentration had density = 1.08 [271] while a glycerol solution with the same g/100 g concentration had density = 1.05 [271].

Definitions for molal and molar concentrations were taken from Alberty et al. [272]

$${\text{m}} = 1000{\text{n}}_{2} /{\text{n}}_{1} {\text{M}}_{1} = 1000{\text{n}}_{2} /1000\;{\text{for}}\;{\text{water}}$$

$${\text{M}} = 1000\uprho{\text{n}}_{2} /\left( {{\text{n}}_{1} {\text{M}}_{1} + {\text{n}}_{2} {\text{M}}_{2} } \right) = 1000\uprho{\text{n}}_{2} /\left( {1000 + 215.7} \right) =\uprho{\text{m}}/\left( {1 + 215.7/1000} \right)$$

$${\text{If}}\,\uprho = 1.08, \,{\text{M}} = 1.08*1.186/\left( {1 + .2157} \right) = 1.05$$

$${\text{If}}\,\uprho = 1.05,\;{\text{M}} = 1.05*1.186/\left( {1 + .2157} \right) = 1.02$$

For conversion of a 0.01 M to saturated solution (M scale), if ρ = 1.05, ΔG = .593*ln(1.05/.01) = +2.76 kcal. If instead ρ = 1.02, ΔG = .593*ln(1.02/.01) = +2.74.

The average = 2.75. Thus ΔG for solid to 1 M aqueous is −2.84 + 2.75 = −0.09 kcal/mol. A simple conclusion is that in the case of mannitol, using m or M makes very little difference to the value of ΔG for solid to 1 M aqueous.

A full error analysis was carried out for the calculation of the ideal saturated molarity for Mannitol.

$${\text{M}} =\uprho*\upphi*{\text{n}}_{2} /\left( {1 + {\text{n}}_{2} {\text{M}}_{2} /1000} \right)$$

$$\begin{aligned}\upsigma_{\text{M}} & = {\text{sqrt}}((\upsigma_{\uprho} *\upphi*{\text{n}}_{2} /\left( {1 + {\text{n}}_{2} {\text{M}}_{2} /1000} \right))^{2} + \, (\upsigma_{\upphi} *\uprho*{\text{n}}_{2} /\left( {1 + {\text{n}}_{2} {\text{M}}_{2} /1000} \right))^{2} \\ & \quad + (\upsigma_{{{\text{n}}2}} *\uprho*\upphi/\left( {{\text{M}}_{2} {\text{n}}_{2} /1000 + 1} \right)^{2} )^{2} ) \\ \end{aligned}$$

With ρ = 1.015, σ_ρ* = 0.02, ϕ = 1.017, σ_ϕ = 0.02, n₂ = 1.184, and σ_n2 = 0.01 this formula leads to σ_M = 0.029

Derivation of effect of finite ΔC_p on extrapolated vapor pressure

Constant ΔH_v

The data are centered on T = T_x. By fitting to the Clausius–Clapeyron equation we get $\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{x}} }}$.On this assumption, the VP at T = θ is given by

$$\ln ({\text{p}}^{\uptheta} ) = \ln \left( {{\text{p}}^{{{\text{T}}_{\text{x}} }} } \right) - \left( {\frac{{\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{x}} }} }}{\text{R}}} \right)\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{x}} }}} \right)$$

Constant ΔC_p, variable ΔH_v

If we know or have estimated ΔC_p, at T = θ, ΔC ^θ_p , then we can calculate $\Delta {\text{H}}_{\text{v}}^{\uptheta}$ from $\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{x}} }}$:

$$\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{x}} }} = \Delta {\text{H}}_{\text{v}}^{\uptheta} + \Delta {\text{C}}_{\text{p}}^{\uptheta} \left( {{\text{T}}_{\text{x}} -\uptheta} \right)$$

On this assumption, the VP at T = θ is given by

$$\ln \left( {{\text{p}}^{\uptheta} } \right) = \ln \left( {{\text{p}}^{{{\text{T}}_{\text{x}} }} } \right) - \left( {\frac{{\Delta {\text{H}}_{\text{v}}^{\uptheta} }}{\text{R}}} \right)\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{x}} }}} \right) + \frac{{\Delta {\text{C}}_{\text{p}}^{\uptheta} }}{\text{R}}\ln \left( {\frac{\uptheta}{{{\text{T}}_{\text{x}} }}} \right) + \frac{{\Delta {\text{C}}_{\text{p}}^{\uptheta}\uptheta}}{\text{R}}\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{x}} }}} \right)$$

which can be simplified to

$$\ln ({\text{p}}^{\uptheta} ) = \ln \left( {{\text{p}}^{{{\text{T}}_{\text{x}} }} } \right) - \left( {\frac{{\Delta {\text{H}}_{\text{v}}^{\uptheta} }}{\text{R}}} \right)\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{x}} }}} \right) + \frac{{\Delta {\text{C}}_{\text{p}}^{\uptheta} }}{\text{R}}\ln \left( {\frac{\uptheta}{{{\text{T}}_{\text{x}} }}} \right) + \frac{{\Delta {\text{C}}_{\text{p}}^{\uptheta} }}{\text{R}}\left( {1 - \frac{\uptheta}{{{\text{T}}_{\text{x}} }}} \right)$$

The free energy of vaporization (pure compound to vapor at 1 M, p = p_1M atm) is given by

$$\Delta {\text{G}}_{\text{v}} = - {\text{RT}}\ln ({\text{p}}_{{1{\text{M}}}} / {\text{VP}})$$

$${\text{p}}_{{1{\text{M}}}} = \left( {\frac{\text{n}}{\text{V}}} \right){\text{R}}^{{\prime }} {\text{T}} = {\text{R}}^{{\prime }} {\text{T}}$$

where R is the gas constant in kcal mol⁻¹ K⁻¹ and R′ is the gas constant in L atm mol⁻¹ K⁻¹.

Thus

$$\Delta {\text{G}}_{\text{v}} = - {\text{RT}}\ln ({\text{R}}^{{\prime }} {\text{T}}) + {\text{RT}}\ln \left( {{\text{p}}^{\uptheta} } \right)$$

The effect on the free energy of solvation of changing from the constant ΔH_v assumption to the constant ΔC_p assumption is then

$$\Delta \Delta {\text{G}}_{\text{v}} = {\text{R}}\uptheta\ln \left( {{\text{p}}_{\text{C}}^{\uptheta} } \right) - {\text{R}}\uptheta\ln \left( {{\text{p}}_{\text{H}}^{\uptheta} } \right)$$

where p ^θ_C is the VP at temperature θ calculated assuming constant ΔC_p and temperature dependent ΔH_v and p ^θ_H is the VP at temperature θ calculated assuming constant ΔH_v.

$$\Delta \Delta {\text{G}}_{\text{v}} = {\text{R}}\uptheta\left( {\ln \left( {{\text{p}}^{{{\text{T}}_{\text{x}} }} } \right) - \left( {\frac{{\Delta {\text{H}}_{\text{v}}^{\uptheta} }}{\text{R}}} \right)\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{x}} }}} \right) + \frac{{\Delta {\text{C}}_{\text{p}}^{\uptheta} }}{\text{R}}\ln \left( {\frac{\uptheta}{{{\text{T}}_{\text{x}} }}} \right) + \frac{{\Delta {\text{C}}_{\text{p}}^{\uptheta} }}{\text{R}}\left( {1 - \frac{\uptheta}{{{\text{T}}_{\text{x}} }}} \right)} \right) - {\text{R}}\uptheta\left( {\ln \left( {{\text{p}}^{{{\text{T}}_{\text{x}} }} } \right) - \left( {\frac{{\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{x}} }} }}{\text{R}}} \right)\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{x}} }}} \right)} \right)$$

which simplifies to:

$$\Delta \Delta {\text{G}}_{\text{v}} = {\text{R}}\uptheta\left( {\left( {\frac{{\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{x}} }} }}{\text{R}}} \right)\left( {\frac{1}{\theta } - \frac{1}{{{\text{T}}_{\text{x}} }}} \right) - \left( {\frac{{\Delta {\text{H}}_{\text{v}}^{\uptheta} }}{\text{R}}} \right)\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{x}} }}} \right) + \frac{{\Delta {\text{C}}_{\text{p}}^{\uptheta} }}{\text{R}}\ln \left( {\frac{\uptheta}{{{\text{T}}_{\text{x}} }}} \right) + \frac{{\Delta {\text{C}}_{\text{p}}^{\uptheta} }}{\text{R}}\left( {1 - \frac{\uptheta}{{{\text{T}}_{\text{x}} }}} \right)} \right)$$

Substituting for $\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{x}} }}$:

$$\Delta \Delta {\text{G}}_{\text{v}} = {\text{R}}\uptheta\left( {\left( {\frac{{\Delta {\text{H}}_{\text{v}}^{\uptheta} }}{\text{R}} + \frac{{\Delta {\text{C}}_{\text{p}}^{\uptheta} }}{\text{R}}\left( {{\text{T}}_{\text{x}} -\uptheta} \right)} \right)\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{x}} }}} \right) - \left( {\frac{{\Delta {\text{H}}_{\text{v}}^{\uptheta} }}{\text{R}}} \right)\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{x}} }}} \right) + \frac{{\Delta {\text{C}}_{\text{p}}^{\uptheta} }}{\text{R}}\ln \left( {\frac{\uptheta}{{{\text{T}}_{\text{x}} }}} \right) + \frac{{\Delta {\text{C}}_{\text{p}}^{\uptheta} }}{\text{R}}\left( {1 - \frac{\uptheta}{{{\text{T}}_{\text{x}} }}} \right)} \right)$$

which simplifies to:

$$\Delta \Delta {\text{G}}_{\text{v}} = \Delta {\text{C}}_{\text{p}}^{\uptheta} \left( {{\text{T}}_{\text{x}} -\uptheta} \right) + \Delta {\text{C}}_{\text{p}}^{\uptheta}\uptheta\ln \left( {\frac{\uptheta}{{{\text{T}}_{\text{x}} }}} \right)$$

The solvation free energy is given by

$$\Delta \Delta {\text{G}}_{\text{s}} = - {\text{R}}\uptheta\ln \left( {\frac{\text{c}}{\text{p}}} \right) - {\text{R}}\uptheta\ln ({\text{R}}^{{\prime }}\uptheta)$$

where c is the concentration of the compound in water. The effect of changing from a VP ${\text{p}}_{\text{H}}^{\uptheta}$ based on an assumed constant ΔH_v to a VP ${\text{p}}_{\text{C}}^{\uptheta}$ based on an assumed constant ΔC_p is given by

$$\Delta \Delta {\text{G}}_{\text{s}} = - {\text{R}}\uptheta\ln \left( {\frac{\text{c}}{{{\text{p}}_{\text{C}}^{\uptheta} }}} \right) + {\text{R}}\uptheta\ln \left( {\frac{\text{c}}{{{\text{p}}_{\text{H}}^{\uptheta} }}} \right)$$

$$\Delta \Delta {\text{G}}_{\text{s}} = {\text{R}}\uptheta\ln \left( {{\text{p}}_{\text{C}}^{\uptheta} } \right) - {\text{R}}\uptheta\ln \left( {{\text{p}}_{\text{H}}^{\uptheta} } \right) = \Delta \Delta {\text{G}}_{\text{v}} = \Delta\Delta {\text{C}}_{\text{p}}^{\uptheta} \left( {\left( {{\text{T}}_{\text{x}} -\uptheta} \right) +\uptheta\ln \left( {\frac{\uptheta}{{{\text{T}}_{\text{x}} }}} \right)} \right)$$

The function $\left( {{\text{T}}_{\text{x}} -\uptheta} \right) +\uptheta\ln \left( {\frac{\uptheta}{{{\text{T}}_{\text{x}} }}} \right)$ is invariably positive, and $\Delta {\text{C}}_{\text{p}}^{\uptheta}$ is always negative so $\Delta \Delta {\text{G}}_{\text{s}}$ is always negative. It seems that $\Delta {\text{C}}_{\text{p}}^{\uptheta}$ must always be negative for normal liquids because $\Delta {\text{H}}_{\text{v}}^{\uptheta}$ has a finite value at room temperature but is zero at the critical point.

Derivation of the correction from VP_{sub-cooled liquid} to VP_solid

At the outset, for most of the compounds which are solid at room temperature but for which the VP refers to the liquid or sub-cooled liquid, all that is known are: the VP at θ; the melting point, T_fus; that the ln(p) versus 1/T lines meet at 1/T_fus and the entropy of fusion (measured or estimated). In the first treatment we are implicitly assuming that ΔCp can be ignored for both liquid and solid; this is a reasonable assumption for sublimation, where it is common to use a general value of ΔCp = −12 cal/K/mol, but less so for evaporation of a liquid where for the sizes of molecules considered here ΔCp is in the range −30 to −40. Next we will examine the effects of going to the next approximation, namely that ΔH varies with temperature but ΔCp is constant

• (a) First assumption (ΔH terms are temperature independent)

Let ln(p) = a + b/T for the liquid at any temperature and ln(p) = a′ + b′/T for the solid at temperatures at or below the m.p., T_fus.

$${\text{b}} = - \Delta {\text{H}}_{\text{v}} /{\text{R}}$$

$${\text{b}}^{{\prime }} = - \left( {\Delta {\text{H}}_{\text{v}} + \Delta {\text{H}}_{\text{fus}} } \right)/{\text{R}}$$

$${\text{At}}\;{\text{T}} = {\text{T}}_{\text{fus}}$$

$${\text{a}} + {\text{b}}/{\text{T}}_{\text{fus}} = {\text{a}}^{{\prime }} + {\text{b}}^{{\prime }} /{\text{T}}_{\text{fus}} ,{\text{b}}^{{\prime }}$$

$${\text{At}}\;{\text{T}} =\uptheta$$

a + b/θ = a′ + b′/θ + c_H where c_H is the amount which must be added to ln(vp _sublimation) to get ln(vp _{subcooled liquid}). The subscript c_H denotes the assumption of temperature independent ΔH values.

Take the difference between the equations for T = T_fus and, substitute for b’ and simplify

$$\left( {{\text{b}}/{\text{T}}{-}{\text{b}}/\uptheta} \right) = \left( {{\text{b}}^{{\prime }} /{\text{T}}_{\text{fus}} {-}{\text{b}}^{{\prime }} /\uptheta} \right){-}{\text{c}}_{\text{H}}$$

$${\text{c}}_{\text{H}} = \left( {\Delta {\text{S}}_{\text{fus}} /\left( {{\text{R}}\;\uptheta} \right)} \right)\left( {{\text{T}}_{\text{fus}} {-}\uptheta} \right)$$

• (b) Second assumption (The ΔCp terms are temperature independent):

Assume that we know $\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }}$ (It will cancel out in the end.), $\Delta {\text{C}}_{\text{p}}^{\text{vap}}$, $\Delta {\text{C}}_{\text{p}}^{\text{sub}}$ and ln(p_sub-cooled). We will derive an expression for c_C the correction to be added to ln(p_sublimation) to obtain ln(p_sub-cooled). The subscript c_C denotes the assumption of temperature independent ΔC_p values.

The VP at T_fus can be written:

$$\ln \left( {{\text{p}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }} } \right) - \ln \left( {{\text{p}}_{\text{v}}^{\uptheta} } \right) = \int\limits_{\uptheta}^{{{\text{T}}_{\text{fus}} }} {\frac{{\Delta\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }} + \Delta {\text{C}}_{\text{p}}^{\text{vap}} \left( {{\text{T}} - {\text{T}}_{\text{fus}} } \right)}}{{{\text{RT}}^{2} }}} {\text{dT}} = \left( {\frac{{\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }} }}{\text{R}}} \right)\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{fus}} }}} \right) + \left( {\frac{{\Delta {\text{C}}_{\text{p}}^{\text{vap}} }}{\text{R}}} \right)\left( {1 - \frac{{{\text{T}}_{\text{fus}} }}{\uptheta} + \ln \left( {\frac{{{\text{T}}_{\text{fus}} }}{\uptheta}} \right)} \right)$$

Or the VP at θ in terms of temperatures, $\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }}$, and $\Delta {\text{C}}_{\text{p}}^{\text{vap}}$ can be written as:

$$\ln \left( {{\text{p}}_{\text{v}}^{\uptheta} } \right) = \ln \left( {{\text{p}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }} } \right) - \left( {\frac{{\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }} }}{\text{R}}} \right)\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{fus}} }}} \right) - \left( {\frac{{\Delta {\text{C}}_{\text{p}}^{\text{vap}} }}{\text{R}}} \right)\left( {1 - \frac{{{\text{T}}_{\text{fus}} }}{\uptheta} + \ln \left( {\frac{{{\text{T}}_{\text{fus}} }}{\uptheta}} \right)} \right)$$

Now we will do the same for the sublimation pressure. First we note that at any temperature T (less than or equal to T_fus):

$$\Delta {\text{H}}_{\text{sub}}^{\text{T}} = \Delta {\text{H}}_{\text{v}}^{\text{T}} + \Delta {\text{H}}_{\text{fus}}^{\text{T}}$$

$$\Delta {\text{C}}_{\text{p}}^{\text{sub}} = \Delta {\text{C}}_{\text{p}}^{\text{vap}} + \Delta {\text{C}}_{\text{p}}^{\text{fus}}$$

and

$$\Delta {\text{C}}_{\text{p}}^{\text{fus}} = \Delta {\text{C}}_{\text{p}}^{\text{sub}} - \Delta {\text{C}}_{\text{p}}^{\text{vap}}$$

Now writing $\Delta {\text{H}}_{\text{sub}}^{\text{T}}$ in terms of temperature independent quantities and T itself we obtain:

$$\Delta {\text{H}}_{\text{sub}}^{\text{T}} = \Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }} + \Delta {\text{C}}_{\text{p}}^{\text{vap}} \left( {{\text{T}} - {\text{T}}_{\text{fus}} } \right) + \Delta {\text{H}}_{\text{fus}}^{{{\text{T}}_{\text{fus}} }} + \left( {\Delta {\text{C}}_{\text{p}}^{\text{sub}} - \Delta {\text{C}}_{\text{p}}^{\text{vap}} } \right)\left( {{\text{T}} - {\text{T}}_{\text{fus}} } \right) = \Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }} + \Delta {\text{H}}_{\text{fus}}^{{{\text{T}}_{\text{fus}} }} + \Delta {\text{C}}_{\text{p}}^{\text{sub}} \left( {{\text{T}} - {\text{T}}_{\text{fus}} } \right)$$

$$\begin{aligned} \ln \left( {{\text{p}}_{\text{sub}}^{{{\text{T}}_{\text{fus}} }} } \right) - \ln \left( {{\text{p}}_{\text{sub}}^{\uptheta} } \right) & = \int\limits_{\uptheta}^{{{\text{T}}_{\text{fus}} }} {\frac{{\left( {\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }} + \Delta {\text{H}}_{\text{fus}}^{{{\text{T}}_{\text{fus}} }} + \Delta {\text{C}}_{\text{p}}^{\text{sub}} \left( {{\text{T}} - {\text{T}}_{\text{fus}} } \right)} \right)}}{{{\text{RT}}^{2} }}} {\text{dT}} \\ & = \frac{{\left( {\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }} + \Delta {\text{H}}_{\text{fus}}^{{{\text{T}}_{\text{fus}} }} } \right)}}{\text{R}}\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{fus}} }}} \right) + \frac{{\Delta {\text{C}}_{\text{p}}^{\text{sub}} }}{\text{R}}\left( {1 - \frac{{{\text{T}}_{\text{fus}} }}{\uptheta} + \ln \left( {\frac{{{\text{T}}_{\text{fus}} }}{\uptheta}} \right)} \right) \\ \end{aligned}$$

$$\ln \left( {{\text{p}}_{\text{sub}}^{\uptheta} } \right) = \ln \left( {{\text{p}}_{\text{sub}}^{{{\text{T}}_{\text{fus}} }} } \right) - \frac{{\left( {\Delta {\text{H}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }} + \Delta {\text{H}}_{\text{fus}}^{{{\text{T}}_{\text{fus}} }} } \right)}}{\text{R}}\left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{fus}} }}} \right) - \frac{{\Delta {\text{C}}_{\text{p}}^{\text{sub}} }}{\text{R}}\left( {1 - \frac{{{\text{T}}_{\text{fus}} }}{\uptheta} + \ln \left( {\frac{{{\text{T}}_{\text{fus}} }}{\uptheta}} \right)} \right)$$

Now we can write the revised definition of the correction c_C, the quantity to be added to $\ln \left( {{\text{p}}_{\text{sub}}^{\uptheta} } \right)$ to obtain $\ln \left( {{\text{p}}_{\text{v}}^{\uptheta} } \right)$, recalling that at T_fus, $\ln \left( {{\text{p}}_{\text{sub}}^{{{\text{T}}_{\text{fus}} }} } \right) = \ln \left( {{\text{p}}_{\text{v}}^{{{\text{T}}_{\text{fus}} }} } \right)$

$${\text{c}}_{\text{C}} = \left( {\frac{{\Delta {\text{C}}_{\text{p}}^{\text{sub}} }}{\text{R}}} \right)\left( {1 - \frac{{{\text{T}}_{\text{fus}} }}{\uptheta} + \ln \left( {\frac{{{\text{T}}_{\text{fus}} }}{\uptheta}} \right)} \right) - \left( {\frac{{\Delta {\text{C}}_{\text{p}}^{\text{vap}} }}{\text{R}}} \right)\left( {1 - \frac{{{\text{T}}_{\text{fus}} }}{\uptheta} + \ln \left( {\frac{{{\text{T}}_{\text{fus}} }}{\uptheta}} \right)} \right) + \Delta {\text{H}}_{\text{fus}}^{{{\text{T}}_{\text{fus}} }} \left( {\frac{1}{\uptheta} - \frac{1}{{{\text{T}}_{\text{fus}} }}} \right)$$

Or in terms of $\Delta {\text{S}}_{\text{fus}}^{{{\text{T}}_{\text{fus}} }}$

$${\text{c}}_{\text{C}} = \left( {\frac{{\Delta {\text{C}}_{\text{p}}^{\text{sub}} }}{\text{R}}} \right)\left( {1 - \frac{{{\text{T}}_{\text{fus}} }}{\uptheta} + \ln \left( {\frac{{{\text{T}}_{\text{fus}} }}{\uptheta}} \right)} \right) - \left( {\frac{{\Delta {\text{C}}_{\text{p}}^{\text{vap}} }}{\text{R}}} \right)\left( {1 - \frac{{{\text{T}}_{\text{fus}} }}{\uptheta} + \ln \left( {\frac{{{\text{T}}_{\text{fus}} }}{\uptheta}} \right)} \right) + \left( {\frac{{\Delta {\text{S}}_{\text{fus}}^{{{\text{T}}_{\text{fus}} }} }}{{{\text{R}}\uptheta}}} \right)\left( {{\text{T}}_{\text{fus}} -\uptheta} \right)$$

If the heat capacity terms are set to zero, then this becomes the equation deduced above for the first hypothesis.

In the challenge dataset the worst case was 2,6-syringealdehyde, which is the highest melting compound of those where VP for the sub-cooled liquid must be reduced to VP for the solid. For this compound inclusion of the estimated ΔCp values lowers ln(vp) by 2.99. What is of interest is the effect on ΔG_s. From the derivation of the effect of including heat capacity terms on the extrapolated VP, the effect on ΔG_s can be written as:

$$\Delta \Delta {\text{G}}_{\text{s}} = {\text{R}}\uptheta\ln \left( {{\text{p}}_{\text{C}}^{\uptheta} } \right) - {\text{R}}\uptheta\ln \left( {{\text{p}}_{\text{H}}^{\uptheta} } \right)$$

where p ^θ_C is the VP at temperature θ calculated assuming constant ΔC_p values and temperature dependent ΔH_v values and p ^θ_H is the VP at temperature θ calculated assuming constant ΔH_v values.

$${\text{R}}\uptheta\ln \left( {{\text{p}}_{\text{C}}^{\uptheta} } \right) = {\text{R}}\uptheta\ln \left( {{\text{p}}_{\text{sub - cooled}}^{\uptheta} } \right) - {\text{R}}\uptheta{\text{c}}_{\text{C}}$$

$${\text{R}}\uptheta\ln \left( {{\text{p}}_{\text{H}}^{\uptheta} } \right) = {\text{R}}\uptheta\ln \left( {{\text{p}}_{\text{sub - cooled}}^{\uptheta} } \right) - {\text{R}}\uptheta{\text{c}}_{\text{H}}$$

Thus

$$\Delta \Delta {\text{G}}_{\text{s}} = - {\text{R}}\uptheta{\text{c}}_{\text{C}} + {\text{R}}\uptheta{\text{c}}_{\text{H}} = {\text{R}}\uptheta\left( {{\text{c}}_{\text{H}} - {\text{c}}_{\text{C}} } \right)$$

In the case of 2,6-syringealdehyde, the change in ΔG_s on inclusion of ΔCp values relative to the value of the change in ΔG_s values assuming constant ΔH terms, is (at 25 °C) +0.593*1.57 = +0.93 kcal/mol, meaning that the correction from VP_{sub-cooled liquid} to VP_sublimation would be smaller with inclusion of ΔCp terms. This value is based on estimated ΔCp values, which may be in error.

The additivity scheme reported by Chickos et al. [16] has some unusual features which make it a bit tricky to apply until one gets on to it. Accordingly this table is included to help the interested reader who wishes to apply the method (Tables 4, 5).

Table 4 Estimation of entropy of fusion

Table 5 Estimated heat capacities of vaporization