Week 2: Phase Equilibria

Chemical potential

A useful definition of the Chemical Potential is \(\left(\frac{\partial{G}}{\partial{n_i}}\right)_{T,P,n_{j\neq{i}}}\).

The change in chemical potential can be obtained rearranging the Gibbs Duhem equation for a single component system:

\[ d\mu=-sdT+vdP \]

This expression is useful in different contexts, from computing the chemical potential of an ideal gas, to deriving an equation for phase coexistence conditions.

Equilibrium conditions

As discussed in Week 1, two phases \(\alpha\) and \(\beta\) are in equilibrium when their temperatures are equal, their pressures are equal, and all the chemical potentials of all the species that are present in both phases are equal:

\[ T^\alpha=T^\beta \]

\[ P^\alpha=P^\beta \]

\[ \mu_i^\alpha=\mu_i^\beta \]

In conditions of constant temperature and pressure, the equality between chemical potentials defines the equilibrium conditions across phases: \(\mu_i^\alpha(T,P)=\mu_i^\beta(T,P)\)

Residual Chemical Potential and Fugacity

The chemical potential of an ideal gas at pressure \(P\) and temperature \(T\) can be computed by integrating the Gibbs-Duhem equation along a path where the temperature is raised from 0 to \(P\) at constant \(T\):

\[ \int_{0,T}^{P,T}d\mu=\int_T^T-sdT^\prime+\int_0^PvdP^\prime = RT\int_0^P\frac{1}{P^{\prime}}dP^\prime \]

which gives the chemical potential for an ideal gas at temperature \(T\) and pressure \(P\):

\[ \mu^{id}=RT\ln{P} \]

The chemical potential of a real gas can be expressed as the chemical potential of an ideal gas, plus a residual chemical potential capturing the effect of intermolecular interactions: \(\mu=\mu^{id}+\mu^{res}\).
Using the definition of \(\mu^{id}\) above as a blueprint we can define the residual chemical potential \(\mu^r=RT\ln{\phi}\), and thus the chemical potential of a real fluid at pressure \(P\) and temperature \(T\) becomes:

\[ \mu^{real}=RT\ln{P\phi} \]

where \(\phi\) is the so-called fugacity coefficient and the product \(P\phi\) defines the fugacity of a real gas. The fugacity coefficient of a pure fluid depends on \(T\) and \(P\), and it is indicated as \(\phi(T,P)\).

Excess chemical potential and activity

A mixture (an homogeneous fluid phase with more than one component) is considered ideal when its properties are a linear combination of the properties of the pure components. In particular the fugacity of component \(i\) in an ideal mixture, in which its molar fraction is \(x_i\) is:

\[ f_i^\circ(T,P,x_i)=x_if_i(T,P) \]

The activity of component \(i\) is defined as the ratio between the fugacity of component \(i\) in a real mixture (\(\hat{f}_i(T,P,x)\)) and \(f_i^\circ(T,P,x_i)\)

\[ \gamma_i(T,P,x)=\frac{\hat{f}_i(T,P,x)}{f_i^\circ(T,P,x_i)} \]

The activity of component \(i\) is then defined as:

\[ a_i(T,P,x)=x_i\gamma_i(T,P,x) \]

Equilibria in monocomponent systems: the Clapeyron Equation

Let’s consider varying the temperature and pressure in such a way that remain on the coexistence line between the liquid (\(L\)) and vapour phases (\(V\)), where \(\mu^L=\mu^V\). In this process \(d\mu^L=d\mu^V\), by using the Gibbs Duhem equation this can be restated as:

\[ v^LdP-s^LdT=v^VdP-sVdT \]

which can be rearranged as:

\[ \frac{dP}{dT}=\frac{s^V-s^L}{v^V-v^L} \]

Considering now that \(\mu^L=\mu^V\), which can be rewritten as \(h^L-Ts^L=h^V-Ts^V\) the expression above becomes:

\[ \frac{dP}{dT}=\frac{h^V-h^L}{T(v^V-v^L)}=\frac{\Delta{h}_{ev}}{T(v^V-v^L)} \]

x which takes the name of Clapeyron equation.

Liquid-Vapour Equilibria in monocomponent systems: the Clausis-Clapeyron Equation

By introducing the approximation \((v^V-v^L)\simeq{v^V}\simeq{\frac{RT}{P}}\) in the Clapeyron equation, we obtain the so called Clausis-Clapeyron equation:

\[ \frac{dP}{dT}=\frac{P\Delta{h}_{ev}}{RT^2} \]

which integrated provide an expression for the equilibrium vapour pressure \(P^\circ(T)\) as a function of the temperature:

\[ \int\frac{dP}{P}=\int\frac{\Delta{h}_{ev}}{RT^2}dT\simeq{\frac{\Delta{h}_{ev}}{R}\int\frac{dT}{T^2}} \]

\[ \ln{P^\circ(T)}=-\frac{\Delta{h}_{ev}}{RT}+C \]

Solid-Vapour Equilibria in monocomponent systems

Solid-vapour equilibria are can be characterised by a Clausis-Clapeyron equation, analogous to the liquid-vapour equilibria discussed above. The Clapeyron equation is obtained without introducing any phase specific semplification, and for a solid-vapour equilibrium would look like:

\[ \frac{dP}{dT}=\frac{h^V-h^S}{T(v^V-v^S)}=\frac{\Delta{h}_{sub}}{T(v^V-v^S)} \]

where the superscript \(S\) refers to the solid phase, and \(\Delta{h}_{sub}\) is the sublimation enthalpy defined as \(h^V-h^S\).

The semplifications introduced above to obtain the Clausius-Clapeyron apply also in this case, \((v^V-v^S)\simeq{v^V}\simeq{\frac{RT}{P}}\), leading to the following integrated expression for the equilibrium vapour pressure with respect to a solid:

\[ \ln{P^s(T)}=-\frac{\Delta{h}_{sub}}{RT}+C \]

Liquid/Vapour Equilibrium in multicomponent systems: deriving the Raoult Equation

To express the partial pressure of component \(i\) in a vapour phase in equilibrium with a multicomponent liquid phase we shall begin by expressing the fundamental condition that has to be fulfilled by all species at thermodynamic equilibrium at constant \(T\) and \(P\):

(1)\[ \mu_i^L(T,P,\vec{x})=\mu_i^V(T,P,\vec{y}) \]

where \(\mu_i^L(T,P,\vec{x})\) and \(\mu_i^V(T,P,\vec{y})\) are the chemical potentials for specie \(i\) in the liquid and vapour phase, respectively.

For mixtures of ideal gases the chemical potential is conveniently expressed in differential form as a function of the partial pressure of component \(i\), \(p_i\) as \(d\mu_i=RTd\ln{p_i}\). This result is obtained integrating the Gibbs-Duhem equation (can you show it?).

In analogy with this expression in the case of real gases fugacity \(f_i\) is introduced, defined as the partial pressure multiplied by a fugacity coefficient \(\phi_i\), and leading to the differential expression \(d\mu_i=RTd\ln{f_i}\).

Integrating this expression between the phases in equilibrium yields:

\[ \int_{L,T,P,\vec{x}}^{V,T,P,\vec{y}}d\mu_i=\int_{L,T,P,\vec{x}}^{V,T,P,\vec{y}}RTd\ln{f_i} \]

(2)\[ \mu_i^V(T,P,\vec{y})-\mu_i^L(T,P,\vec{x})=RT\ln\frac{\hat{f}_i^V(T,P,\vec{y}}{\hat{f}_i^L(T,P,\vec{x})} \]

Hence, the equilibrium condition expressed by Eq. (1) can be conveniently stated as:

(3)\[ \hat{f}_i^V(T,P,\vec{y})=\hat{f}_i^L(T,P,\vec{x}) \]

The left-hand term can be written as:

(4)\[ \hat{f}_i^V(T,P,\vec{y})=\hat{\phi}(T,P,\vec{y})Py_i \]

where \(\hat{\phi}(T,P,\vec{y})\) is the fugacity coefficient, \(P\) the pressure and \(y_i\) the molar fraction in the vapour phase.

The right hand side term can instead be written as:

(5)\[ \hat{f}_i^L(T,P,\vec{x})=\gamma_i(T,P,\vec{x})x_i\phi(T,P^\circ(T))P^o(T)e^{\frac{v^L_i(P-P^o(T))}{RT}} \]

where \(x_i\) is the molar fraction of component \(i\) in the liquid phase, \(\gamma_i(T,P,\vec{x})\) is the activity coefficient for component i in the liquid mixture, \(P^o(T)\) is the equilibrium vapour pressure of the pure component \(i\), \(\phi(T,P^o(T))\) is the fugacity coefficient for the pure component \(i\) at \(T\) and \(P^o(T)\), and the term \(e^{\frac{v_i(P-P^o(T))}{RT}}\) is the Poynting correction, which captures the difference in fugacity of the pure liquid component associated with the difference in pressure between \(P\) and \(P^o(T)\).

Introducing Eq.(4) and Eq. (5) in Eq.(3), while considering negligible the Poynting correction, and ideal gas approximation applicable yields:

(6)\[ Py_i=\gamma_i(T,P,\vec{x})P^o(T)x_i \]

It should be noted that introducing the further hypothesis of ideal liquid mixture yields the Raoult law:

(7)\[ Py_i=P^o(T)x_i \]