1.14.34: Internal Pressure
- Page ID
- 386985
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.