The PT Projection


On this page we will examine aspects of the pressure-temperature (or PT) diagram.  The PT diagram for a pure substance is shown to the right.  As before, click on the hyperlink of a particular aspect to see how it appears on this figure.

First, we will talk about the vapor pressure curve  which is shown in red.  Remember that there is a one-to-one relationship between temperature and pressure in a two phase region.  This curve provides that relationship for the liquid-vapor region.  We have talked about how boiling temperature is the temperature inside the vapor-liquid region at a particular pressure.  We will turn this idea around on occasion and refer to the pressure inside the vapor-liquid region at a particular temperature.  We will call this pressure the vapor pressure.  The diagram at the right illustrates this.  The purple values of P and T on the diagram refer to one particular state in the two phase vapor-liquid region.  If we consider temperature as the independent variable, we refer to P as the vapor pressure at temperature T.  If we consider pressure as the independent variable, we refer to T as the boiling temperature at pressure P.  Whether we talk about boiling temperature or vapor pressure simply depends on whether P or T is the independent variable.  Note though that this diagram does not give any information about where inside the two phase region this state lies - it could be saturated liquid, saturated vapor or any one of the infinite number of mixtures of liquid and vapor between the two.

The vapor pressure curve terminates at the critical point.  Remember that this point marks the highest temperature (T = Tc) and highest pressure (P = Pc) for which vapor and liquid can coexist. 

The other end of the vapor pressure curve is marked by the triple point.  On the TV diagram this was a line that connected a coexisting solid state, liquid state and vapor state.  Since these three states are all at the same temperature and pressure (otherwise they wouldn't be coexisting) they collapse to a single point on a PT diagram.

On the Tv diagram, there was a two phase region for solid-liquid coexistence.  That region collapses to a single curve on the PT diagram and is called the melting curve.  It provides the one to one relationship between the pressure and temperature of a solid-liquid mixture  (i.e. it gives melting temperature as a function of pressure or melting pressure as a function of temperature).

On this diagram, the melting curve as been drawn with a positive slope, which is typically the case.  Water is an exception and its melting curve slopes backwards from the triple point (negative slope).  Click here if you want to see this.  If you continue in the study of thermodynamics, you will be able to show that this is a result of the fact that ice floats on liquid water.  For most substances, the opposite is true and the solid phase would sink in the liquid. 

There also was a region on the Tv diagram for solid-vapor coexistence, and this also becomes a single curve on the PT diagram.  It is known as the sublimation curve and it provides the one to one relationship between temperature and pressure in the solid-vapor coexistence region.

Most people know that "dry ice" is frozen carbon dioxide.  When dry ice is placed on a table it is observed to sublime, which is to say it passes directly from solid to vapor without forming a pool of liquid.  What is it about carbon dioxide that makes this happen?   Click here to see.  For most substances, as shown in the top figure, atmospheric pressure is well above the triple point pressure Pt.  So it is possible at ambient conditions to observe phase transitions beween solid and liquid and between liquid and vapor, as in the case of water.   For carbon dioxide however, as shown in the lower figure, atmospheric pressure is below the triple point pressure.  As a result, it is only possible to observe solid and vapor phases of carbon dioxide at ambient conditions.  Thus dry ice sublimes directly to vapor.   Of course it is possible to observe liquid carbon dioxide - it just has to be in a pressure vessel.

The only thing left to do is to identify the single phase regions on the PT diagram.  The single phase liquid region is located between the melting curve and vapor pressure curve.  Note that this diagram is consistent with our earlier convention that the liquid region terminates at the critical temperature.  The single phase vapor region is located below the sublimation and vapor pressure curves and terminates, as mentioned earlier, at the critical pressure.  The supercritical fluid region, as defined earlier, is located for temperatures above Tc and pressures above Pc.  Finally, the single phase solid region is located to the left of the melting curve.

This concludes our study of phase diagrams for pure substances. However, there is one more variable (in addition to P,T and v) that is of importance in the two phase regions.  We will define this variable on the next (and last) page of this tutorial.

Important Points:

  • The vapor pressure curve, melting curve, and sublimation curve give the one-to-one relationship between pressure and temperature in the vapor-liquid, solid-liquid and solid-vapor regions respectively.

  • In the vapor-liquid region at a particular pressure, the temperature is called the boiling temperature.   At a particular temperature, the pressure is called the vapor pressure.


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