## Posts Tagged ‘triple point’

It was the American mathematical physicist Josiah Willard Gibbs who introduced the concepts of phase and chemical potential in his milestone monograph On the Equilibrium of Heterogeneous Substances (1876-1878) with which he almost single-handedly laid the theoretical foundations of chemical thermodynamics.

In a paragraph under the heading “On Coexistent Phases of Matter” Gibbs mentions – in passing – that for a system of coexistent phases in equilibrium at constant temperature and pressure, the chemical potential μ of any component must have the same value in every phase.

This simple statement turns out to have considerable practical value as we shall see. But first, let’s go through the formal proof of Gibbs’ assertion.

An important result

Consider a system of two phases, each containing the same components, in equilibrium at constant temperature and pressure. Suppose a small quantity dni moles of any component i is transferred from phase A in which its chemical potential is μ’i to phase B in which its chemical potential is μ”i. The Gibbs free energy of phase A changes by –μ’idni while that of phase B changes by +μ”idni. Since the system is in equilibrium at constant temperature and pressure, the net change in Gibbs free energy for this process is zero and we can write

hence

This result can be generalized for any number of phases: for a system in equilibrium at constant temperature and pressure, the chemical potential of any given component has the same value in every phase.

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Visualizing variance

The equality of pressure P, temperature T and component chemical potentials μn between coexistent phases in equilibrium provides a convenient way to visualize variance, or the number of degrees of freedom a system possesses. For example, the triple point of a single component system can be visualized as the array

where the solid, liquid and vapor phases are indicated by one, two and three primes respectively.

Each row represents a single variable, so the number of rows equates to the total number of variables. Each column lists the variables in a single phase. All but one of these may be independently varied; the last is determined by the Gibbs-Duhem relation

There are one of these for each phase, so the number of columns equates to the number of relations (=constraints) to which the system variables are subject. The variance, or number of degrees of freedom (f) of the system is defined

For arrays of the kind presented above, this transposes into

For the triple point of a single component system, there are three rows and three columns, so f =0. With zero degrees of freedom, the triple point is not subject to independent variation and is represented by a fixed point in the PT plane.

The above rule implies that a system of coexistent phases in equilibrium cannot have more phases than intensive system variables.

– – – –

Generating useful equations

For a component present in any pair of coexistent phases in equilibrium at constant temperature and pressure, the chemical potential of that component has the same value in both phases

From this general relation, equations may be deduced for computing various properties of thermodynamic systems such as ideal solutions, for example the elevation of boiling point, the depression of freezing point, and the variation of the solubility of a solute with temperature.

The key point to grasp is that μi is the chemical potential of component i in an arbitrary state, i.e. in a mixture of components. In order to compute this potential we need to know two things: the chemical potential of the pure substance μi0 at a pressure p (such as that of the atmosphere), and the mole fraction (xi) of the component in the mixture. Assuming an ideal solution, use can then be made of the textbook formula

… (1)

where for a given phase, μi is the arbitrary chemical potential of i in the mixture, μ°i is the chemical potential of the pure substance, and xi is the mole fraction of the component.

As an example, let us take the equilibrium relation

…(2)

where the chemical potential of the solid solvent is necessarily the standard potential because the mole fraction x is unity. The above relation will generate an equation for the depression of the solvent freezing point in a solution at a fixed pressure (p).

Substituting (1) for the liquid phase in (2) gives

where by convention the subscript 1 refers to the solvent. Differentiating with respect to T at constant pressure

using the quotient rule for ΔG/T gives

… (3)

Now since

equation (3) simplifies to

Integrating from the pure solvent state, where the mole fraction x1=1 and T0fus is the freezing point of the pure solvent, to the solution state where the mole fraction x1= x1 and Tfus is the freezing point of the solvent in the solution

yields the equation for the depression of the solvent freezing point in a solution at a fixed pressure (p)

Since x1<1 in a solution, the logarithm is negative and therefore the freezing point of the solvent in the solution must be lower than the freezing point of the pure solvent.

– – – –

Ok, so maybe that wasn’t the simplest procedure for generating a useful thermodynamic equation. But the point to be made here is that the same procedure applies in the other cases, so you only need to understand the principle once.

For example, the equation for elevation of solvent boiling point in solution with a non-volatile solute at a fixed pressure (p) is

The similarity to the previous equation is evident.

– – – –

P Mander February 2015

The above image from the northern latitudes of Mars, taken by the Viking 2 Lander in May  1979, revealed the presence of a white substance on and around the red rocks. We get this white stuff on Earth too. It’s called frost.

I can remember when I first saw this image confirming the presence of water frost on the Martian surface, I took an instant step closer to believing in the possibility of Little Green Men. During the Martian summer, the temperature can rise above freezing point, and lo! there would be liquid water, the swimming pool of life.

After revelling in this idea for a while, the thought struck me that I had directly transferred the phase change behavior of water on Earth to another planet. On reflection however, this appeared perfectly valid. There are no planet-dependent terms in the Clausius-Clapeyron equation, so the phase diagram of water must look exactly the same up there as it does down here.

Sound enough reasoning. The phase diagram is a PT plane, and according to what I was taught, those  thermodynamic variables, along with state functions such as enthalpy of fusion, were considered to apply universally. But then duh! the realization dawned on me – I had failed to consider differences in atmospheric pressure.

At the time, I knew nothing at all about the atmosphere on Mars, apart from the fact that it had one. But at least I could calculate the minimum surface pressure that the atmosphere would need to exert in order for water to undergo a solid-liquid phase change at 0°C. The Magnus-Tetens approximation, which generates saturated vapor pressure (hPa) as a function of temperature (°C)

made things simple since when T=0 the answer is immediately 6.1094 hectopascals, or if you prefer, 6.1094 millibars.

Back then, I never did find out what the atmospheric pressure on Mars was, and my inquiry stalled. Years later I came upon the information by happenstance. The atmospheric surface pressure on Mars is just below 6.1 hectopascals.

Now a glance at the phase diagram for water shows that the triple point temperature is a gnat’s whisker from 0°C, so for practical purposes the triple point pressure can be taken as 6.1 hPa. This is significant because at no point in the PT plane below the horizontal drawn through the triple point does the liquid phase exist. Since the atmospheric surface pressure on Mars lies below this horizontal, the only possible phase change is between solid and vapor. No liquid water can exist on the surface of Mars.

Except. Well, there are craters on Mars, so if at the bottom of a sufficiently deep one the atmospheric pressure were to nudge above 6.1 hPa, and if the temperature crept above 0°C, and if there was surface frost to start with, then liquid water would form. That’s a lot of ifs, but a possibility all the same. There might not be Little Green Men down there, but there might be something else that answers the description of ‘life form’.

Photo credits: (top) Mars / NASA, (below) Little Green Men / Wikimedia
Triple point diagram credit: National Metrology Institute of Japan
(I have added the horizontal line through the triple point)