Projections of the PvT Surface


We have examined the Tv diagram (at least the liquid-vapor part of it) and the PvT surface (shown right) of a pure substance in some detail.  Although the PvT surface provides a visualization of the relation between pressure, temperature, and specific volume, most people find it easier to work with two dimensional projections of a three dimensional surface.   Note that there are three different possible projections of the PvT surface, each obtained by looking down a different axis (as shown by the purple arrows).

One of these projections is the Tv diagram.  The other two are the PT diagram and the Pv diagram.  It is difficult to visualize these projections by looking at a two-dimensional rendering of the PvT surface.   However, if you click on the different projections below, the surface will rotate to the appropriate projection.


Oblique View   

Clicking here will simply restore the surface to its original view, in the event you wish to do so between examining different projections.


TV Projection

Note that the isobars remain horizontal when they cut through the solid-liquid region.  Thus, like the vapor-liquid region, there is a one-to-one relationship between temperature and pressure.  At a given pressure in the solid-liquid region, the temperature is called the melting temperature.   Isobars below the triple line would cut across the two-phase solid-vapor region in the same way.  In the solid-vapor region, the temperature at a particular pressure is called the sublimation temperature.

Note on this projection that the isotherms become horizontal.  This is as we would expect since they are lines of constant temperature on a diagram of temperature versus specific volume.

PT Projection

We will have more to say about this projection on the next page.  For now, note that all three of the two-phase regions collapse to single lines or curves in this view.  This is because coexisting phases always exist at the same temperature and pressure.  So on a PT diagram the infinite number of mixtures of liquid and vapor that could coexist at a particular pressure and temperature will all collapse onto a single PT point along a curve.

Note also in this diagram that the isotherms become vertical and the isobars horizontal, as one would expect.

Pv Projection

This diagram shows the relation between pressure and specific volume along paths of constant temperature (isotherms).  Note that isotherms also become horizontal as they cut across two phase regions.  This should make sense to you:   If in a two phase region, specifying the pressure fixes the temperature then it must follow that specifying the temperature must fix the pressure.  (Remember what the phrase one-to-one relationship means).

Take a minute to note that, in the single phase regions, isobars on a Tv diagram have positive slope while isotherms on a Pv diagram have negative slope.   Here's how you can remember this:  Imagine a constant pressure process (say in a piston-cylinder).  As you increase temperature, you expect the volume to increase.   Thus an increase in T corresponds to an increase in v and the isobar would have a positive slope.  Now consider an experiment in a piston-cylinder in which the piston is pushed down (decreasing the volume) at constant temperature.  What would you expect to happen to the pressure?  It would increase right?  Thus a decrease in v corresponds to an increase in P along an isotherm and isotherms have a negative slope.

The Tv and Pv projections provide roughly equivalent information.   However, the PT projection provides some new ways to interpret PvT behavior.   Therefore we will discuss the PT projection in detail on the next page.

Important Points:

  • Three different projections (PT, Pv and Tv) can be obtained from the PvT surface.
  • On the PT projection, two-phase regions collapse to curves.
  • There is a one-to-one relationship between pressure and temperature in two-phase regions.  In the solid-vapor region, the temperature at a given pressure is called the sublimation temperature.  In the solid-liquid region it is called the melting temperature.


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