Liquid gas coexistence

Coexistence region

The van der Waals equation of state led to two phases below some critical temperature $T_c$.

Conclusions for a vdW gas cooled below $T_C$:

  • Phase separation happens when the sum of the Helmholz energies of two phases is lower than the Helmholz energy of a single, homogeneous phase.
  • The two phases are in equilibrium (coexist) at the same pressure (mechanical equilibrium).
  • At a given temperature, the pressure at which the two phases are in equilibrium is unique. (Given by the Maxwell construction.)

This last condition amounts to an additional(*) constraint on a system in the coexistence region:

  • *If* two phases are in equilibrium (coexistence region),
  • *Then* $P$ is determined by $T$ (or vice versa) for any mixture of phases.

(*) ...in addition to the state equation.

This constraint means that on on a $P$-$T$ diagram, the coexistence region has one less dimension than the area for a homogeneous phase.

That is, only a single thermodynamic variable--either $P$ or $T$--is needed to uniquely specify the thermodynamic state. The coexistence region on the $P$-$T$ diagram is the blue line.

How does the $P$-$T$ diagram relate to the $P$-$V$ diagrams we've been using? See this rotation of a typical $P-v-T$ surface.

Liquid/gas equilibrium - atomic view

Atomic picture. Equilibrium means:

# of atoms going liquid$\to$gas = # of atoms going gas$\to$liquid.

In a container of fixed volume, the vapor pressure (or saturation pressure) is the pressure exerted by the gas phase when liquid and gas of a pure substance coexist in equilibrium.

The vapor pressure (closed container, pure substance) is an indication of the evaporation rate of the liquid under any circumstances (open or closed container).

What do you notice about the saturation pressure (closed container) of water at 100 C?

Apparently the boiling temperature in an open container occurs when the pressure of the environment (open container) is equal to the vapor pressure (closed container) at that temperature!

How will the temperature of boiling water change if you go camping high up in the Rocky Mountains?

Partial pressure

Consider different pure gases, each in their own tank (all of identical size), with a pressure $P_i$ in the $i$-th tank.

John Dalton found ("Dalton's Law") that if you put the contents of all the tanks together in one of the same volume, that the pressure $P$ of the gas mixture in this tank is: $$\sum_i P_i=P.$$

In the limit that the gases are ideal, and do not react with each other, the ratios of the partial pressures, $P_i$, to the mole fractions, $\chi_i$ of each gas in the final mixed state are: $$\frac{P_i}{P}=\frac{n_i}{n}\equiv \chi_i.$$

See also:

Relative humidity

If $P_w$ is the partial pressure of water vapor in the atmosphere, and $P_s$ is the saturation pressure, then the relative humidity is: $$\text{relative humidity}=\frac{P_w}{P_s}.$$

Saturation curve near room temperature, $P_s(T)$:

What does this say about cloud formation? dew on the grass in the morning?

[You will also see the vertical axis in saturation curves with units of grams of water per volume of air.]

Image credits

HellTchi, Max Dodge, Ytrottier