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Elementary Thermodynamics - Thermodynamic Degrees of Freedom

If we look at the areas bounded by phase transition curves in a phase diagram (solid area, liquid area, and vapor area), we see that both pressure and temperature may change independent of one another. A vessel filled with liquid water, for instance, may be at 30 degrees Celsius and 2 atmospheres, or at 50 degrees Celsius and 2 atmospheres, or at 50 degrees Celsius and 1 atmosphere, all equally stable. A more technical way to state this is to say the liquid water has two degrees of freedom. Here, the word “degree” has a completely different meaning than it does when used to denote a unit of temperature or angle. In this context, “degree” may be thought of loosely as “dimension.” A cube has three physical dimensions, a square has two and a line has one. A point within a cube has three degrees of freedom (motion), while a point within a square only has two, and a point along a line only has one. Here, we use the word “degree” to denote the number of independent ways a system may change. For areas bounded by phase transition curves in a phase diagram, pressure and temperature are the two “free” variables, because within those bounded areas we may freely alter pressure without altering temperature, and visa-versa.

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:

So long as the water is all liquid (one phase), we may adjust its pressure and temperature independently. In this state, the system has two thermodynamic degrees of freedom. However, if the water becomes hot enough to boil, creating a system of two phases in direct contact with each other (equilibrium), we will find that pressure and temperature become linked: one cannot alter one without altering the other. For a steam boiler, operation at a given steam pressure thus defines the temperature of the water, and visa-versa. In a single-component, two-phase system, there is only one degree of thermodynamic freedom.

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:

nfreedom + nphase = nsubstance + 2

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 fixed1 so long as all three phases coexist in equilibrium.


1The 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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