Critical Phenomena

All real gases can be liquefied. Depending on the gas this might require compression and/or cooling. However, there exists for each gas a temperature above which the gas cannot be liquefied. This temperature, above which the gas cannot be liquefied, is called the critical temperature and it is usually symbolized by, TC . In order to liquefy a real gas the temperature must be at, or below, its critical temperature.

There are gases, sometimes called the "permanent gases" which have critical temperatures below room temperature. These gases must be cooled to a temperature below their critical point, which means below room temperture, before they can be liquefied. Examples of "permanent gases" include, He, H2, N2, O2, Ne, Ar, and so on. Many substances have critical temperatures above room temperature. These substances exist as liquids (or even solids) at room temperature. Water, for example, has a critical temperature of 647.1 K, much higher than the 298.15 K standard room temperature. Water can be liquefied at any temperature below 647.1 K (although above 398.15 - the normal boiling point of water - you would have to apply a pressure higher than atmospheric temperature in order to keep it liquid.

It is convenient to think about liquefying substances and critical phenomena using a p-V diagram. This is a graph with pressure, p, plotted on the vertical axis and the volume, V, plotted along the horizontal axis. If we plot the pressure of a substance as a function of volume, holding temperature constant we get a series of curves, called isotherms. There is an example of such a plot in most physical chemistry texts. We provide here an Excel file1 which contains six isotherms for the van der Waals equation of state. (Temperatures are given in the top row of numbers and the volumes are given in the left two columns. Temperatures are relative to the critical temperature so that a temperature of 1.0 is the critical temperature, a temperature of 1.1 is above the critical temperature, and so on. The isotherms below the critical temperature, for example, temperature equals 0.9, are peculiar to the van der Waals equation of state and are not physically realistic. Since you have the entire Excel spreadsheet you can change the temperatures yourself and watch the isotherms change.)

Notice that when a substance is liquefied the isotherm becomes "flat," that is, the slope becomes zero. On the critical isotherm the slope "just barely" becomes flat at one point on the graph. A point where a decreasing function becomes flat before continuing to decrease is called a point of inflection. The mathematical characteristic of an inflection point is that the first and second derivatives are zero at that point. For our critical isotherm on a p-V diagram we would write,

(1)        ,
and
(2)         .

Equations (1) and (2) constitute a set of two equation in two unknowns, V, and T. One can test to see if an approximate equation of state gives a critical point by calculating these two derivatives for the equation of state and trying to solve the pair of equations. If a solution exists (and p and V are neither zero or infinity) then we say that the equation of state has a critical point.

Let's use this test to see if the ideal gas has a critical point. First we have to solve the ideal gas equation of state. PV = nRT, for pressure, p.

(3)         .

Now we can take the derivatives in Equations 1 and 2 and set them (independently2) equal to zero.

(4)

(5)         .

It is easy to see that the only way these two equations can be satisfied is if T = 0, or V = ∞ . Neither of these solutions is physically reasonable so we conclude that the ideal gas does not have a critical point.

Good exercises would be for you to see if the approximate equation of state,

,

has a critical point, or to verify for yourself that the van der Waals equation of state does have a critical point and to find the critical constants, VC , TC ,and pC.

1. The Excel file is an Excel 97 file. If you have Excel 97 or higher your browser should launch Excel and load the file automatically. If you want to down-load the file place your mouse arrow on the link and click the right button and then save the link. (This is on a PC. The file can be saved on a Mac, but you need to check with a Mac user if you don't know how to do it.) Earlier versions of Excel may not be able to read this file.

2. Sometimes people are tempted to set these two derivatives equal to each other. There is nothing wrong with that, but you now have one equation in two unknowns. There is more information in both derivatives equaling zero than there is in the two derivatives equaling each other.

From here you can: