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## Elementary Thermodynamics - Thermodynamic Degrees of Freedom

Such is not the case at any point lying on one of the phase transition curves. Any point along a curve is geometrically defined by a pair of coordinates, which means that for a two-phase mixture in equilibrium there will be exactly one temperature value valid for each unique pressure value. At any point along a phase transition curve, pressure and temperature are not independent variable, but rather are related. For any single substance, there is only one degree of freedom along any point of a phase transition curve.

To illustrate this concept, suppose we equip a closed vessel containing water with both a thermometer and a pressure gauge. The thermometer measures the temperature of this water, while the pressure gauge measures the pressure of the water. A burner beneath the vessel adds heat to alter the water’s temperature, and a pump adds water to the vessel to alter the pressure inside:

Our freedom to alter pressure and temperature becomes even more restricted if we ever reach the triple point of the substance. For water, this occurs (only) at a pressure of -14.61 PSIG (0.006 atmospheres) and a temperature of 0.01 degrees Celsius: the coordinates where all three phase transition curves intersect on the phase diagram. In this state, where solid (ice), liquid (water), and vapor (steam) coexist, there are zero degrees of thermodynamic freedom. Both the temperature and pressure are locked at these values until one or more of the phases disappears.

The relationship between degrees of freedom and phases is expressed neatly by Gibbs’ Phase Rule – the sum of phases and degrees of freedom equals the number of substances (“components”) plus two:

We may simplify Gibbs’ rule for systems of just one substance (1 “component”) by saying the number of degrees of freedom plus phases in direct contact with each other is always equal to three. So, a vessel filled with nothing but liquid water (one component, one phase) will have two thermodynamic degrees of freedom: we may change pressure or temperature independently of one another. A vessel containing nothing but boiling water (two phases – water and steam, but still only one component) has just one thermodynamic degree of freedom: we may change pressure and temperature, but just not independently of one another. A vessel containing water at its triple point (three phases, one component) has no thermodynamic freedom at all: both temperature and pressure are fixed_{1} so long as all three phases coexist in equilibrium.

_{1}The non-freedom of both pressure and temperature for a pure substance at its triple point means we may exploit different substances’ triple points as calibration standards for both pressure and temperature. Using suitable laboratory equipment and samples of sufficient purity, anyone in the world may force a substance to its triple point and calibrate pressure and/or temperature instruments against that sample.

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