1.14.14: Cohesive Energy Density

The molar enthalpy of vaporisation \(\Delta_{\text {vap }} \mathrm{H}^{*}\) is the change in enthalpy for one mole of chemical substance \(j\) on going from the liquid to the (perfect) gaseous state. The properties of a given liquid-\(j\) are determined by \(j-j\) intermolecular forces. By definition, there are no intermolecular forces in a perfect gas. Hence \(\Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}(\ell)\) offers an insight into the strength of intermolecular forces in the liquid state. We have to be careful not to use the word ‘energy’. By definition enthalpy \(\mathrm{H}\) equals (\(\mathrm{U} + \mathrm{p} \, \mathrm{V}\)). For phase I at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\),

\[\mathrm{U}_{\mathrm{j}}^{*}(\mathrm{I})=\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{I})-\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{I})\]

\[\text { Similarly for phase II, } \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{II})=\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{II})-\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{II})\]

\[\text { Hence, } \quad U_{j}^{*}(\text { II })-U_{j}^{*}(I)=H_{j}^{*}(\text { II })-H_{j}^{*}(I)-p \,\left[V_{j}^{*}(\text { II })-V_{j}^{*}(I)\right]\]

If phase II is a perfect gas and phase I is the corresponding liquid, \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{II})>>\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{I})\); for one mole of chemical substance \(j\), \(\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{II})=\mathrm{R} \, \mathrm{T}\).

\[\text { Consequently } \Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})=\Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T})-\mathrm{R} \, \mathrm{T}\]

\(\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})\) is the molar thermodynamic energy of vaporisation for liquid \(j\) at temperature \(\mathrm{T}\). Having calculated \(\Delta_{\text {vap }} \mathrm{H}_{j}^{*}(\mathrm{~T})\) from experimental data we obtain \(\Delta_{\text {vap }} U_{j}^{*}(T)\), a measure of the strength of inter-molecular interactions in the liquid.

The differential quantity \((\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}}\) defines the internal pressure \(\pi_{int}(j)\) of chemical substance \(j\). For liquid \(j\),

\[\pi_{\mathrm{int}}^{*}(\ell ; \mathrm{j})=\left[\partial \mathrm{U}_{\mathrm{j}}^{*}(\ell) / \partial \mathrm{V}_{\mathrm{j}}^{*}(\ell)\right]_{\mathrm{T}}\]

The internal pressure for liquids, of the order \(10^{8} \mathrm{~Pa}\), is an indicator of the strength of intermolecular forces [1]. The structure of the terms in equation (e) prompts a slight rewrite using properties that are either readily measured or calculated, namely \(\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})\) and the molar volume of the liquid at temperature \(\mathrm{T}\), \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T})\) [2,3]. The result is the cohesive energy density, c.e.d., a measure of the cohesion within a liquid.

\[\text { By definition, c.e.d. }=\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T}) / \mathrm{V}_{\mathrm{j}}^{*}(\ell)\]

Intuitively, \(\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})\) is a measure of cohesive interactions in the liquid whereas volume is a measure of the repulsive interactions, keeping the molecules in the liquid apart. At constant \(\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})\), c.e.d decreases with increase in molar volume; c.f. repulsion. But at constant \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T})\), c.e.d. increases with increase in \(\Delta_{\text {vap }} \mathrm{U}_{j}^{*}(\mathrm{~T})\), the attractive part.

\[\text { If the vapour is a perfect gas, c.e.d. }=\left[\Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}(\ell)-\mathrm{R} \, \mathrm{T}\right] / \mathrm{V}_{\mathrm{j}}^{*}(\ell)\]

If the molar mass of the liquid \(j\) equals \(\mathrm{M}_{j}\) and the density equals \(\rho_{\mathrm{j}}^{*}(\ell)\)

\[\text { c.e.d. }=\left[\Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}(\ell)-\mathrm{R} \, \mathrm{T}\right] \, \rho_{\mathrm{j}}^{*}(\ell) / \mathrm{M}_{\mathrm{j}}\]

\(\mathrm{M}_{j}\) is expressed in \(\mathrm{kg mol}^{-1}\) and \(\rho_{\mathrm{j}}^{*}(\ell)\) in \(\mathrm{kg m}^{-3}\), consistent with c.e.d. being expressed in (\(\mathrm{J mol}^{–1} \mathrm{~m}^{–3}\)). At \(298.2 \mathrm{~K}\), \(\mathrm{R} \, \mathrm{T}=2.48 \mathrm{~kJ} \mathrm{~mol}^{-1}\). The ratio of internal pressure \(\pi_{\text {int }}(\mathrm{j})\) to c.e.d. defines a property \(\mathrm{n}\) using equation (i).

\[\mathrm{n}=\left[\partial \mathrm{U}_{\mathrm{j}}^{*}(\ell) / \partial \mathrm{V}_{\mathrm{j}}^{*}(\ell)\right]_{\mathrm{T}} /\left[\Delta_{\mathrm{vap}} \mathrm{U}_{\mathrm{j}}^{*}(\ell) / \mathrm{V}_{\mathrm{j}}^{*}(\ell)\right]\]

The dimensionless ratio \(\mathrm{n}\) has been used to comment on the strength of intermolecular forces in a liquid [4].

In the context of the properties of liquid mixtures, using the definition of enthalpy \(\mathrm{H}(=\mathrm{U}+\mathrm{p} \, \mathrm{V})\) we can write the following equation for a given phase I containing \(\mathrm{n}_{1}\) moles of substance 1 and \(\mathrm{n}_{2}\) moles of substance 2.

\[\mathrm{U}\left(\mathrm{I}, \mathrm{n}_{1}+\mathrm{n}_{2}\right)=\mathrm{H}\left(\mathrm{I} ; \mathrm{n}_{1}+\mathrm{n}_{2}\right)-\mathrm{p} \, \mathrm{V}\left(\mathrm{I} ; \mathrm{n}_{1}+\mathrm{n}_{2}\right)\]

We assert that phase I is an ideal binary liquid mixture. Then,

\[\begin{aligned}
&\mathrm{U}\left(\mathrm{I} ; \mathrm{n}_{1}+\mathrm{n}_{2} ; \mathrm{mix} ; \mathrm{id}\right)= \\
&\quad \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{H}_{2}^{*}(\ell)-\mathrm{p} \, \mathrm{V}\left(\mathrm{I} ; \mathrm{n}_{1}+\mathrm{n}_{2} ; \text { mix } ; \mathrm{id}\right)
\end{aligned}\]

We assert that phase II is a perfect gas comprising \(\mathrm{n}_{1}\) moles of substance 1 and \(\mathrm{n}_{2}\) moles of substance 2. Then

\[\begin{aligned}
\mathrm{U}\left(\mathrm{II} ; \mathrm{n}_{1}+\right.&\left.\mathrm{n}_{2} ; \mathrm{pfg}\right)=\\
& \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\mathrm{pfg})+\mathrm{n}_{2} \, \mathrm{H}_{2}^{*}(\mathrm{pfg})-\mathrm{p} \, \mathrm{V}\left(\mathrm{II} ; \mathrm{n}_{1}+\mathrm{n}_{2} ; \mathrm{pfg}\right)
\end{aligned}\]

\[\text { For a perfect gas, } \mathrm{p} \, \mathrm{V}\left(\mathrm{II} ; \mathrm{n}_{1}+\mathrm{n}_{2} ; \mathrm{pfg}\right)=\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right) \, \mathrm{R} \, \mathrm{T}\]

We express the thermodynamic energy of vaporisation for (\(\mathrm{n}_{1} + \mathrm{n}_{2}\)) moles passing from phase I to phase II.

\[\begin{aligned}
&\Delta_{\text {vap }} \mathrm{U}\left(\mathrm{id}, \mathrm{n}_{1}+\mathrm{n}_{2}\right)= \\
&\begin{aligned}
\mathrm{n}_{1} \, \Delta_{\text {vap }} \mathrm{H}_{1}^{*}+\mathrm{n}_{2} \, \Delta_{\mathrm{vap}} \mathrm{H}_{2}^{*}-\left(\mathrm{n}_{1}\right.&\left.+\mathrm{n}_{2}\right) \, \mathrm{R} \, \mathrm{T} \\
&+\mathrm{p} \, \mathrm{V}\left(\mathrm{I} ; \mathrm{n}_{1}+\mathrm{n}_{2} ; \text { mix; id }\right)
\end{aligned}
\end{aligned}\]

Therefore for one mole,

\[\Delta_{\text {vap }} \mathrm{U}_{\mathrm{m}}(\mathrm{id})=\mathrm{x}_{1} \, \Delta_{\text {vap }} \mathrm{H}_{1}^{*}+\mathrm{x}_{2} \, \Delta_{\text {vap }} \mathrm{H}_{2}^{*}-\mathrm{R} \, \mathrm{T}+\mathrm{p} \, \mathrm{V}_{\mathrm{m}}(\mathrm{I} ; \mathrm{mix} ; \mathrm{id})\]

Suppose however that the thermodynamic properties of the liquid mixture are not ideal. We rewrite equation (k) in the following form (for one mole of mixture) where \(\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\) and \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) are the excess molar enthalpies and excess molar volumes of mixing.

\[\begin{aligned}
&\mathrm{U}_{\mathrm{m}}(\mathrm{I}, \operatorname{mix})= \\
&\quad\left[\mathrm{x}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{H}_{2}^{*}(\ell)+\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\right]-\mathrm{p} \,\left[\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)+\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\right]
\end{aligned}\]

\[\operatorname{Or} \mathrm{U}_{\mathrm{m}}(\mathrm{I}, \operatorname{mix})=\left[\mathrm{x}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{H}_{2}^{*}(\ell)+\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\right]-\mathrm{p} \, \mathrm{V}_{\mathrm{m}}(\operatorname{mix})\]

Therefore the molar thermodynamic energy of vaporisation on going from the real mixture to the perfect gas in given by equation (r).

\[\Delta_{\text {vap }} \mathrm{U}_{\mathrm{m}}=\left[\mathrm{x}_{1} \, \Delta_{\text {vap }} \mathrm{H}_{1}^{*}(\mathrm{~T})+\mathrm{x}_{2} \, \Delta_{\text {vap }} \mathrm{H}_{2}^{*}(\mathrm{~T})-\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\right]-\mathrm{R} \, \mathrm{T}+\mathrm{p} \, \mathrm{V}_{\mathrm{m}}(\mathrm{mix})\]

The cohesive energy density, c.e.d., for a real binary liquid mixture is given by equation (s).

\[\begin{aligned}
\text { c.e.d. }=\left\{\left[\mathrm{x}_{1} \, \Delta_{\mathrm{vap}} \mathrm{H}_{1}^{*}(\mathrm{~T})+\mathrm{x}_{2} \,\right.\right.&\left.\left.\Delta_{\mathrm{vap}} \mathrm{H}_{2}^{*}(\mathrm{~T})-\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\right] / \mathrm{V}_{\mathrm{m}}(\mathrm{mix})\right\} \\
&-\left\{\mathrm{R} \, \mathrm{T} / \mathrm{V}_{\mathrm{m}}(\mathrm{mix})\right\}+\mathrm{p}
\end{aligned}\]

The c.e.d. for a given binary mixture is given by the molar enthalpies of vaporisation of the pure components, the excess molar enthalpy of mixing and the molar volume of the mixture. For the corresponding ideal binary mixture, c.e.d.(id) is given by equation (t).

\[\text { c.e.d.(id) } \begin{aligned}
=\left\{\left[\mathrm{x}_{1} \, \Delta_{\text {vap }} \mathrm{H}_{1}^{*}(\mathrm{~T})+\mathrm{x}_{2} \,\right.\right.&\left.\left.\Delta_{\text {vap }} \mathrm{H}_{2}^{*}(\mathrm{~T})\right] / \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})\right\} \\
&-\left\{\mathrm{R} \, \mathrm{T} / \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})\right\}+\mathrm{p}
\end{aligned}\]

The difference between \(\Delta_{\text {vap }} \mathrm{U}_{\mathrm{m}} / \mathrm{V}_{\mathrm{m}}(\mathrm{mix})\) and \(\Delta_{\mathrm{vap}} \mathrm{U}_{\mathrm{m}} / \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})\) is the excess cohesive energy density, \((\text { c.e.d. })^{\mathrm{E}}\). The sign of \((\text { c.e.d. })^{\mathrm{E}}\) is controlled to a significant extent by the excess molar volume \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) and the excess molar enthalpy \(\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\) [6]. In fact equations (s) and (t) lead to m equation (u).

\[(\text { c.e.d. })^{\mathrm{E}}=\frac{-\left[\mathrm{x}_{1} \, \Delta_{\mathrm{vap}} \mathrm{H}_{1}^{*}(\mathrm{~T})+\mathrm{x}_{2} \, \Delta_{\mathrm{vap}} \mathrm{H}_{2}^{*}(\mathrm{~T})-\mathrm{R} \, \mathrm{T}\right]}{\mathrm{V}_{\mathrm{m}}(\operatorname{mix}) \, \mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})}-\frac{\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{V}_{\mathrm{m}}(\mathrm{mix})}\]

Recently the cohesive energy density of a liquid has been described as a ‘solvation pressure’ acting on, for example, ethanol in ethanol + water and ethanol + trichloromethane liquid mixtures [7].

Footnotes

[1] M. R. J. Dack, Aust. J. Chem.,1976,27,779.

[2] \(\text { c.e.d. }=\left[\mathrm{J} \mathrm{mol}^{-1}\right] /\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]=\left[\mathrm{J} \mathrm{m}^{-3}\right]=\left[\mathrm{N} \mathrm{m}^{-2}\right]\); the unit of pressure. \(\mathrm{R} \, \mathrm{T}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\)

[3] A variety of units are used for cohesive energy densities. Despite the fact that there are good grounds for using the unit \(\mathrm{J m}^{-3}\), the commonly used unit is calories per \(\mathrm{cm}^{3}\), \mathrm{cal cm}^{-3}\).

\[\begin{aligned}
&\text { For liquids } 298.15 \mathrm{~K} \text { and ambient pressure. }\\
&\begin{array}{lc}
\text { Liquid } & \text { c.e.d./ } \mathrm{cal} \mathrm{cm}^{-3} \\
\text { water } & 547 \\
\text { methanol } & 204 \\
\text { benzene } & 85 \\
\text { tetrachloromethane } & 74
\end{array}
\end{aligned}\]

[4] A. F. M. Barton, J.Chem.Educ.,1971, 48,156.

[5] For comments on the role of cohesive energy densities of solvents and rates of disproportionation, see A.P. Stefani, J. Am. Chem. Soc., 1968,90,1694.

[6] For comments on cohesive energy densities of binary aqueous mixtures see,

1. D. D. Macdonald and J. B. Hyne, Can. J. Chem.,1971, 49,611,2636.
2. D. D. Macdonald, Can. J. Chem.,1976, 54,3559.

[7] N.W. A.van Uden, H. Hubel, D. A. Faux, A. C. Tanczos, B. Howlin and D. J. Dunstan, J. Phys.: Condens. Mater,2003,15,1577.