1.14.34: Internal Pressure

According to the Thermodynamic Equation of State,

\[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\mathrm{T} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}-\mathrm{p}\]

The partial differential \((\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}}\) (with units, \(\mathrm{N m}^{-2}\)) is the internal pressure \(\pi_{\mathrm{int}}\).

\[\pi_{\mathrm{int}}=\mathrm{T} \, \beta_{\mathrm{V}}-\mathrm{p}\]

\(\pi_{\mathrm{int}\) describes the sensitivity of energy \(\mathrm{U}\) to a change in volume. A high \(\phi_{\mathrm{int}\) implies strong inter-molecular cohesion [1-8]. For many liquids, \(\mathrm{T} \, \boldsymbol{\beta}_{\mathrm{V}}>>\mathrm{p}\) such that

\[(\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}} \cong \mathrm{T} \, \beta_{\mathrm{V}}\]

\(\mathrm{T} \, \boldsymbol{\beta}_{\mathrm{V}}\) is sometimes called the thermal pressure. By definition, for \(\mathrm{n}\) moles of a perfect gas,

\[\mathrm{p} \, \mathrm{V}=\mathrm{n} \, \mathrm{R} \, \mathrm{T}\]

Then

\[\mathrm{V} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\mathrm{n} \, \mathrm{R}\]

Or,

\[\mathrm{T} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\mathrm{n} \, \mathrm{R} \, \mathrm{T} / \mathrm{V}=\mathrm{p}\]

From equation (a), for a perfect gas, \(\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}\) is zero.

The internal pressure for water(\(\ell\)) presents an interesting puzzle [9]. From equations (a) and (c), it follows that [1]

\[\pi_{\mathrm{int}}=\mathrm{T} \,\left(\frac{\alpha_{\mathrm{p}}}{\kappa_{\mathrm{T}}}\right)-\mathrm{p}\]

But at the temperature of maximum density (TMD), \(\alpha_{p}\) is zero. So near the TMD, \(\pi_{\mathrm{int}\) is zero. We understand this pattern if we think about hydrogen bonding. In order to form a strong hydrogen bond between two neighboring water molecules the O-H---O link has to be close to if not actually linear. In other words the molar volume for water(\(\ell\)) is larger than the molar volume of a system comprising close-packed water molecules. Consequently hydrogen bonding has a strong ‘repulsive’ component to intermolecular interaction. However once formed hydrogen bonding has a strong cohesive contribution to intermolecular forces. Hence for water between \(273\) and \(298 \mathrm{~K}\) cohesive and repulsive components of hydrogen bonding play almost competitive roles.

Footnotes

[1] Using a calculus operation, \(\left(\frac{\partial p}{\partial T}\right)_{V}=-\left(\frac{\partial V}{\partial T}\right)_{p} \,\left(\frac{\partial p}{\partial V}\right)_{T}\) For equilibrium properties,

\[\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\frac{\alpha_{\mathrm{p}}}{\kappa_{\mathrm{T}}}\]

[2] Some authors use the term ‘isochoric thermal pressure coefficient’ for the property, \(\left(\frac{\partial p}{\partial T}\right)_{V}\)

[3] For details of original proposals concerning internal pressures see the following references.

1. W. Westwater, H. W. Frantz and J. H. Hildebrand, Phys.Rev.,1928, 31,135.
2. J. H. Hildebrand, Phys.Rev.,1929,34,649 and 984.
3. See also S. E. Wood, J.Phys.Chem.,1962,66, 600.

[4] Internal pressures are quoted in the literature using many units. Here we use \(\mathrm{N m}^{-2}\). We list some internal pressures and relative permitivities at \(298.15 \mathrm{~K}\).

liquid	\(\varepsilon_{\mathrm{r}}\(	\(\pi_{\mathrm{int}} / 10^{5} \mathrm{~N m}^{-2}\)
water	78.5	1715
methanol	32.63	2849
ethanol	24.30	2908
propanone	20.7	3368
diethyl ether	4.3	2635
tetrachloromethane	2.24	3447
dioxan	2.2	4991

The above details are taken from M. R. J. Dack, J.Chem.Educ.,1974,51,231;see also

1. Aust. J. Chem.,1976, 29,771 and 779.
2. D. D. MacDonald and J. B. Hyne, Can.J.Chem.,1971, 49,2636

[5] For a discussion of effects of solvents on rates of chemical reactions with reference to internal pressures, see

1. K. Colter and M. L. Clemens, J.Phys.Chem.,1964,68,651.
2. A. P. Stefani, J. Am. Chem.Soc.,1968,90,1694.

[6] For comments on solvent polarity and internal pressures see J. E. Gordon, J. Phys. Chem.,1966,70,2413.

[7] For comments on internal pressures of binary aqueous mixtures see D. D. Macdonald, Can. J Chem.,1976,54,3559; and references therein.

[8] For comments on effect of internal pressure on conformational equilibria see R. J. Ouellette and S. H. Williams, J. Am. Chem.Soc.,1971,93,466.

[9] For details concerning the dependence of internal pressure of water and \(\mathrm{D}_{2}\mathrm{O}\), see M. J. Blandamer, J. Burgess and A.W.Hakin, J. Chem. Soc. Faraday Trans. 1, 1987, 83, 1783.

[10] For comments on the calculation of excess internal pressures for binary liquid mixtures using equation (h) see W. Marczak, Phys.Chem.Chem.Phys.,2002,4,1889.