The Melting Curve for Water; Vapor Pressure

The melting curve for water

We mentioned in our initial discussion of one-component phase diagrams that the slope of the melting curve for water was negative.

Water is one of just a few known substances with this feature (bismuth is another), the melting curve for most substances has a positive slope. Let us consider the thermodynamics consequences of this negative slope. Since we are talking about the melting curve we must use the Clapeyron equation,
(1)
where, of course, the ΔS and ΔV refer to the melting process or the freezing process. For the purposes of this discussion let us consider the melting process. That is, we will assume that the phase change is
H2O(s) → H2O(l).
A negative slope on the phase diagram means that the derivative in Equation 1 is negative. This in turn requires that,
(2)
We know that ΔS for the melting process is positive because the system is going from the highly ordered crystalline solid state to the more disordered liquid state (and we can calculate it from ΔS = ΔHfusion/Tmp). Therefore, the only way the derivative in Equation 2 can be negative is if ΔV is negative. This means that the molar volume of the liquid is smaller than the molar volume of the solid at the melting point,
(3a, b)
Taking the reciprocal of Equation 3b we find that,
(4)
and, since the density is proportional to the reciprocal of the volume, this means that the density of the liquid is greater than the density of the solid (at the melting point).

This result is counterintuitive. We expect that materials will expand on melting and most materials do expand on melting. However, water is unusual in this respect and contracts on melting. The visible consequence of this fact is that ice floats on liquid water. We are so accustomed to this that we do not even notice that it is unusual. However, if you have worked in the laboratory with other solid/liquid systems you will be aware that most often the solid sinks in the melt.

You can argue that the slope of the melting curve of water makes a crucial contribution to the ability of our planet, Earth, to support life. If the slope were positive then ice would sink in water. In cold climates ice would form in winter and sink to the bottom. This ice would be insulated from the summer warmth by the layer of water above it and would only partially melt. The next winter more ice would accumulate on the bottom. Eventually, all rivers and lakes and possibly a large portion of the oceans would be frozen from the bottom up with only a shallow layer of liquid water being formed on the surface during the summer. There would likely be no animal or plant life in these frozen lakes and rivers.

Vapor pressure - What is vapor pressure?

We mentioned in our discussion of one-component phase diagrams that the vaporization curve is sometimes referred to as the vapor pressure curve. This is because this curve - along with the sublimation curve - gives a plot of the vapor pressure of the substance versus temperature.

The vapor pressure is a measure of the ability of molecules to escape from the surface of a solid or liquid. The following discussion uses ideas that are outside the scope of thermodynamics, but they may be useful in calibrating our intuition. A solid or liquid is held together by intermolecular attractive forces. However, the molecules have kinetic energy (energy of motion) which we will later see is proportional, on the average, to the Kelvin temperature. There is a competition. The attractive forces are trying to keep the molecules in the solid or liquid and the kinetic energy is trying to take them out. The vapor pressure is the net result of this competition.

As more and more molecules build up in the space above the solid or liquid more of them will collide back with the surface and some of them will stick. At a fixed temperature the system will reach a state of "dynamic equilibrium," where just as many molecules escape from the surface as collide back and stick. At this point the pressure exerted by the molecules in the space above the solid or liquid is, by definition, the vapor pressure of the solid or liquid.

As the temperature is increased the average kinetic energy increases so more molecules are able to overcome the intermolecular forces and the vapor pressure increases.

Increasing the vapor pressure by the application of an external pressure

One can also increase the vapor pressure of a solid or liquid by applying an external pressure. This effect is not as easy to understand qualitatively as the effect of temperature, but it is easy to calculate thermodynamically. Recall that for solids and liquids, which are not very compressible over relatively small pressure ranges, the Gibbs free energy change is given by,

(5)
In words, Equation 5 says that G increases when we "squeeze" a solid or a liquid. (The Gibbs free energy of a gas also increases when you "squeeze it, but gases are quite compressible so that Equation 5 does not hold for gases.) We would expect that if the Gibbs free energy of a solid or liquid increases its vapor pressure would also increase. We shall see if this is true. Rewrite Equation 5 as follows,
(6)
We will rearrange Equation 6 and then divide it by the number of moles of substance, n.
(7a, b, c, d)
In Equation 7d we have used the fact that the Gibbs free energy per mole is called the chemical potential and is symbolized by a lower case Greek "mu."

Let us consider two systems, all at temperature, T. The first system is a closed container which contains only, say, liquid water and water vapor in equilibrium. Since the system is in equilibrium the chemical potential of the liquid water and water vapor must be equal. Using the approximation that the vapor is an ideal gas we can write,

(8a, b)
(In Equation 8b we have used the expression for the variation of the chemical potential of an ideal gas with pressure.)

Our second system consists of a similar container of liquid water and water vapor, but also contains an inert gas, say air, at a pressure, P. It is the air that is squeezing the liquid and providing the "external pressure." We will call this system the "prime" system. In this second system the chemical potentials of the liquid water and water vapor must still be equal, but they are not equal to the same values as in the first system. In the "prime" system we have,

(9)
but
(10)
and
(11)
So we have two equations equating chemical potentials in the gas and liquid phases, one for the first system and one for the second system.
(12)
and
(8b)
When we subtract Equation 8b from Equation 12 we eliminate all the standard state chemical potentials and are left with
(13a, b)
so that
(14)
Equation 14 tells us that the vapor pressure of the liquid, in this case water, increases when we apply an external pressure. For water the vapor pressure increases by 0.074 % at 1.00 atm and 7.6 % at 100 atm.

WRS

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