1.7.1: Compressions and Expansions- Liquids

The isothermal compressions of solutions and liquids have been extensively studied and the subject has a remarkable history. The term compression, symbol \(\mathrm{K}\) describes the sensitivity of the volume of a system to an isothermal change in pressure, \((\partial V / \partial p)\). Reference is usually made to the voyage made by HMS Challenger and the report of experiments undertaken by Tait into the compression of water [1-3]. Kell summarises various equations which have been proposed describing the isothermal dependence of the molar volume of water on pressure [4]; see also references [5,6].

The dependence of the volume of water(\(\ell\)) at low pressures and at a given temperature on pressure can be represented by equation (a) where \(\mathrm{A}\) and \(\mathrm{B}\) are constants.

\[[\mathrm{V}(\text { ref })-\mathrm{V}] / \mathrm{V}(\text { ref }) \, \mathrm{p}=\mathrm{A} /(\mathrm{B}+\mathrm{p}) \nonumber \]

Here \(\mathrm{V}(\text{ref})\) is the volume ‘at zero pressure’, usually ambient pressure (i.e. approx \(105 \mathrm{~N m}^{-2}\)). This equation often called the Tait equation [4] has the form shown in equation (b).

\[-\left(1 / \mathrm{V}^{0}\right) \,(\partial \mathrm{V} / \partial \mathrm{p})=\mathrm{A} /(\mathrm{B}+\mathrm{p}) \nonumber \]

\[\text { Alternatively [4] } \mathrm{V}=\mathrm{V}^{0}\{1-\mathrm{A} \, \ln [(\mathrm{B}+\mathrm{p}) / \mathrm{B}]\} \nonumber \]

The challenge of measuring the isothermal compression of liquids has been taken up by many investigators; e.g. references [7-12]. The isothermal compressions of a liquid \(\mathrm{K}_{\mathrm{T}}\) is defined by equation (d) [13].

\[\mathrm{K}_{\mathrm{T}}=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} \nonumber \]

The isothermal compressibility is given by equation (e) [14].

\[\kappa_{\mathrm{T}}=-\mathrm{V}^{-1} \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} \nonumber \]

For all thermodynamic equilibrium states, both \(\mathrm{K}_{\mathrm{T}}\) and \(\kappa_{\mathrm{T}}\) are positive variables. A related variable is the isochoric thermal pressure coefficient, \((\partial p / \partial T)_{v}\).[15]

We develop the story in the context of systems containing two liquid components. For a closed system containing \(\mathrm{n}_{1}\) and \(\mathrm{n}_{2}\) moles of chemical substances 1 and 2, the Gibbs energy is a dependent variable and the variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]\) are the independent variables. Temperature \(\mathrm{T}\) is the thermal potential; pressure \(\mathrm{p}\) is the mechanical variable. The number of thermodynamic variables necessary to define the system is established using the Gibbs Phase Rule [16]. For a closed system (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) at thermodynamic equilibrium the composition/organisation is represented by \(\xi^{e q}\). The affinity for spontaneous change is zero consistent with the Gibbs energy being a minimum; equation (f).

\[\mathrm{A}=-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=0 \nonumber \]

The Gibbs energy, volume and entropy of a solution at equilibrium are state variables. We contrast these properties with those properties which are associated with a process (pathway). Thus we contrast the state variable V with an unspecified compression of a solution. We need to define the path followed by the system when the pressure is changed. The Gibbs energy of a closed system at thermodynamic equilibrium (where the affinity for spontaneous change is zero and where the molecular composition/organisation is characterised by \(\xi^{e q}\)) is described by equation (g).

\[\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right] \nonumber \]

The same state is characterised by the equilibrium volume and equilibrium entropy by equations (h) and (i) respectively.

\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right] \nonumber \]

\[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right] \nonumber \]

We use two intensive variables, \(\mathrm{T}\) and \(\mathrm{p}\), in the definition of extensive variables \(\mathrm{G}\), \(\mathrm{V}\) and \(\mathrm{S}\). When the pressure is increased by finite increments from \(\mathrm{p}\) to (\(\mathrm{p} + \Delta \mathrm{p}\)), the volume changes in finite increments from \(\mathrm{V}\) to (\(\mathrm{V} + \Delta \mathrm{V}\)). For an important pathway, the temperature is constant. However to satisfy the condition that the affinity for spontaneous change \(\mathrm{A}\) is zero, the molecular organisation/composition \(\xi\) changes. The volume at pressure (\(\mathrm{p} + \Delta \mathrm{p}\)) is defined using equation (j).

\[\mathrm{V}=\mathrm{V}\left[\mathrm{T},(\mathrm{p}+\Delta \mathrm{p}), \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right] \nonumber \]

In principle we plot the volume as a function of pressure at constant temperature, \(\mathrm{n}_{1}\), \(\mathrm{n}_{2}\), and at ‘\(\mathrm{A} = 0\)’. The gradient of the plot defined by equation (h) yields the equilibrium isothermal compression, \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)\); equation (k)

\[\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}=0} \nonumber \]

\(\mathrm{K}_{\mathrm{T}}(\mathrm{A} = 0)\) characterises the state defined by the set of variables, \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]\).

We turn our attention to another property starting with a system having a volume defined by equation (h). The system is perturbed by a change in pressure from \(\mathrm{p}\) to (\(\mathrm{p} + \Delta \mathrm{p}\)) in an equilibrium displacement. However on this occasion we require that the entropy of the system remains constant at a value defined by equation (i). In principle we plot the volume \(\mathrm{V}\) as a function of pressure at constant \(\mathrm{n}_{1}\), \(\mathrm{n}_{2}\), at ‘\(\mathrm{A}=0\)’ and at a constant entropy defined by equation (i). The gradient of the plot at the point where the volume is defined by equation (g) yields the equilibrium isentropic compression \(\mathrm{K}_{\mathrm{S}} (\mathrm{A}=0)\); equation (l) where isentropic = adiabatic and ‘at equilibrium’.

\[\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{S}, \mathrm{A}=0} \nonumber \]

The equilibrium state characterised by \(\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)\) is defined by the variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]\). In other words an isentropic volumetric property describes a solution defined in part by the intensive variables \(\mathrm{T}\) and \(\mathrm{p}\). Significantly the condition on the partial derivative in equation (l) is an extensive variable, entropy. For a stable phase \(\mathrm{K}_{\mathrm{S}}\) is positive.

The arguments outlined above are repeated with respect to both isobaric equilibrium expansions \(\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)\) and isentropic equilibrium expansions, \(\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)\); equations (m) and (n).

\[\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)=-(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}, \mathrm{A}=0} \nonumber \]

\[\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)=-(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{S}, \mathrm{A}=0} \nonumber \]

The (equilibrium) volume intensive isothermal \(\kappa_{\mathrm{T}}\) and isentropic \(\kappa_{\mathrm{S}}\) compressibilities are defined by equations (o) and (p) .

\[\kappa_{\mathrm{T}}=-(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}=\mathrm{K}_{\mathrm{T}} \, \mathrm{V}^{-1} \nonumber \]

\[\kappa_{\mathrm{s}}=-(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=\mathrm{K}_{\mathrm{s}} \, \mathrm{V}^{-1} \nonumber \]

In 1914 Tyrer reported isentropic and isothermal compressibilities for many liquids [9]. Equations (q) and (r) define two (equilibrium) expansibilities, isentropic and isobaric, volume intensive properties.

\[\alpha_{\mathrm{s}}=(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{s}}=\mathrm{E}_{\mathrm{S}} \, \mathrm{V}^{-1} \nonumber \]

\[\alpha_{p}=(1 / V) \,(\partial \mathrm{V} / \partial \mathrm{T})_{p}=\mathrm{E}_{\mathrm{p}} \, \mathrm{V}^{-1} \nonumber \]

Rowlinson and Swinton state that the property \(\alpha_{\mathrm{S}}\) is ‘of little importance’ [17]. The isobaric heat capacity per unit volume \(\sigma\) is the ratio \(\left[\mathrm{C}_{\mathrm{p}} / \mathrm{V}\right]\). A property of some importance is the difference between compressibilities, \(\delta\); equation (s).

\[\delta=\kappa_{\mathrm{T}}-\kappa_{\mathrm{S}}=\mathrm{T} \,\left[\alpha_{\mathrm{p}}\right]^{2} \, \mathrm{V} / \mathrm{C}_{\mathrm{p}}=\mathrm{T} \,\left[\alpha_{p}\right]^{2} / \sigma \nonumber \]

The property \(\sigma\) is given different symbols and names; e.g. volumetric specific heat. Here we identify \(\sigma\) as the thermal (or, heat) capacitance. The property \(\varepsilon\) is the difference between isobaric and isentropic expansibilities; equation (t).

\[\varepsilon=\alpha_{p}-\alpha_{s}=\kappa_{T} \, \sigma / T \, \alpha_{p} \nonumber \]

The Newton–Laplace equation is the starting point for the determination of isentropic compressibilities of liquids using sound speeds and densities; equation (u).

\[u^{2}=\left(\kappa_{\mathrm{s}} \, \rho\right)^{-1} \nonumber \]

The isentropic condition on \(\kappa_{\mathrm{S}}\) means that as a sound wave passes through a liquid the pressure and temperature fluctuate within each microscopic volume but the entropy remains constant.

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