“Coexistence of Phases”#

Coexistence of Phases#

Consider a substance in isothermal such that we are able to vary its volume and pressure. We may graph this in a curve called the isotherm.

Let’s focus on liquid and gas phase (all of this applies between solid-gas and solid-liquid as well). At a certain isotherm the two liquid and gas phase may coexist if:

  • the isotherm lies below a critical temperature \(\tau_c\) such that

  • the two phases are in thermal, diffusive, and pressure equilibrium

    \[\tau_l=\tau_g, \quad \mu_l=\mu_g, \quad P_l = P_g\]

  • note that the the substance will in a certain phase if that phase’s chemical potential is the lowest:

    • Gas: \(\mu_g < \mu_l\)

    • Liquid: \(\mu_l < \mu_g\) and \(\mu_l < \mu_s\)

    • Solid: \(\mu_s < \mu_l\)

Coexistence Curve and Clausius-Claperyon Relation#

Alias: Vapor Pressure Curve, Vapor Pressure Equation

The coexistence curve is a pressure-temperature curve along which two phases may exist (coexist). For the case of liquid and gas, the coexistence curve has the slope given by the Clausius-Claperyon Relation:

\[\boxed{\frac{dP}{d\tau} = \frac{L}{\tau \Delta \nu}}\]

  • \(L\) : Latent heat of vaporization

  • \(\Delta \nu\) : The change in volume if a single particle would change from liquid to gas

Derivation#

The coexistence curve follows the following differential equation:

\[ \frac{dP}{d\tau} = \frac{\frac{\partial \mu_l}{\partial \tau} - \frac{\partial \mu_g}{\partial \tau}}{\frac{\partial \mu_g}{\partial P} - \frac{\partial \mu_l}{\partial P}} \]

We can represent this better using the definition of the differentials of the Gibbs free energy

\[\begin{split} \begin{gathered} \frac{1}{N}\left(\frac{dG}{dP}\right)_{N,\tau} = \left(\frac{d\mu}{dP}\right)_{\tau} = \frac{V}{N} \equiv \nu\\ \frac{1}{N}\left(\frac{dG}{d\tau}\right)_{N,P} = \left(\frac{d\mu}{d\tau}\right)_{P} = \frac{\sigma}{N} \equiv s\\ \boxed{\frac{dP}{d\tau} = \frac{s_g-s_l}{\nu_g-\nu_l} = \frac{\Delta s}{\Delta\nu}} \end{gathered} \end{split}\]

While confusing we’ll use:

  • \(\nu\) which relates to the volume per particle. \(\Delta\nu\) is the change in volume if we were to transfer one particle from one phase to another

  • \(s\) the entropy per particle. This changes the differential equation to a state equation. \(\Delta s\) is the change in entropy if we were to transfer one particle from one phase to another

We may use the latent heat of vaporization to simplify the vapor pressure equation.

\[ L \equiv \frac{dQ_{l\rightarrow g}}{N} = \frac{\tau(\sigma_g - \sigma_l)}{N}\]
\[L \equiv \tau(s_g - s_l)\]

Replacing this definition to \(dP/d\tau\) above gives the Clausius-Claperyon equation,

\[ \boxed{\frac{dP}{d\tau} = \frac{L}{\tau \Delta \nu}} \]

Clausius-Claperyon of the Ideal Gas#

The Clausius-Claperyon relation has a useful form in the ideal gas such that

  • \(\Delta \nu \approx v_g = V_g/N_g\) since gas takes up more volume by many orders \(v_g / v_l \approx 10^3\)

  • Ideal gas law is followed so that \(\Delta v \approx V_g/N_g = \tau/P\)

The vapor pressure equation them becomes,

\[\begin{split} \begin{gathered} \frac{dP}{d\tau} = \frac{L}{\tau^2}P \\ \boxed{\frac{d}{d\tau}\log{P} = \frac{L}{\tau^2}} \end{gathered} \end{split}\]

Coexistence Curve of the Ideal Gas#

If we assume that \(L=L_0\) does not depend on temperature then we can easily integrate the Clausius-Claperyon equation to the coexistence curve.

\[\begin{split} \begin{gathered} \int{\frac{dP}{P}} = L_0 \int{\frac{1}{\tau^2} d\tau} \\ \boxed{P(\tau) = P_0e^{-L_0/\tau}} \end{gathered} \end{split}\]

Latent Heat of Vaporization#

The amount of heat transfered a single particle of liquid to gas is called the latnet heat of vaporization.

\[L \equiv \frac{dQ}{N}\]