1.14.34: Internal Pressure
According to the Thermodynamic Equation of State,
\[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\mathrm{T} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}-\mathrm{p} \nonumber \]
The partial differential \((\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}}\) (with units, \(\mathrm{N m}^{-2}\)) is the internal pressure \(\pi_{\mathrm{int}}\).
\[\pi_{\mathrm{int}}=\mathrm{T} \, \beta_{\mathrm{V}}-\mathrm{p} \nonumber \]
\(\pi_{\mathrm{int}\) 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
\[(\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}} \cong \mathrm{T} \, \beta_{\mathrm{V}} \nonumber \]
\(\mathrm{T} \, \boldsymbol{\beta}_{\mathrm{V}}\) is sometimes called the thermal pressure. By definition, for \(\mathrm{n}\) moles of a perfect gas,
\[\mathrm{p} \, \mathrm{V}=\mathrm{n} \, \mathrm{R} \, \mathrm{T} \nonumber \]
Then
\[\mathrm{V} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\mathrm{n} \, \mathrm{R} \nonumber \]
Or,
\[\mathrm{T} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\mathrm{n} \, \mathrm{R} \, \mathrm{T} / \mathrm{V}=\mathrm{p} \nonumber \]
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]
\[\pi_{\mathrm{int}}=\mathrm{T} \,\left(\frac{\alpha_{\mathrm{p}}}{\kappa_{\mathrm{T}}}\right)-\mathrm{p} \nonumber \]
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,
\[\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\frac{\alpha_{\mathrm{p}}}{\kappa_{\mathrm{T}}} \nonumber \]
[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.
- W. Westwater, H. W. Frantz and J. H. Hildebrand, Phys.Rev.,1928, 31,135.
- J. H. Hildebrand, Phys.Rev.,1929,34,649 and 984.
- 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
- Aust. J. Chem.,1976, 29 ,771 and 779.
- 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
- K. Colter and M. L. Clemens, J.Phys.Chem.,1964, 68 ,651.
- 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.