EMPIRICAL EQUATIONS
2.4 THE CRITICAL TEMPERATURE
Vm (dm3)
2 1
0
p (bar)
73.8
FIGURE 2.4 Three isotherms of a van der Waals gas. The top isotherm is above Tc, the middle isotherm is at the critical temperature Tcand the bottom curve is below Tc. The critical pressure is 73.8 bar.
The apparent paradox that there are only two degrees of freedom in the equation of state of a pure substance which may have an infinite number of terms in an equation of state is removed by noting that each term contains only the pressure, p, and an adjustable parameter (not a variable) that is a function of the temperature. Hence the only true variables in the equation are p and T.
THE CRITICAL TEMPERATURE 25
1 2
p 3
V
4 gas
liquid equilibrium vapor
FIGURE 2.5 Conversion of a liquid to its vapor without boiling (1–4).
gas, but it is usually applied to a gas in equilibrium with its liquid form. When a liquid is in equilibrium with its vapor, heat can be applied with no change in temperature but with conversion of some or all of the liquid to its vapor. When spheroids of vapor rise from the bottom of a heated liquid to the top, we say that the liquid boils. It is sometimes said that “no gas can be liquefied above the critical point.” This is true, but it is a little misleading because there is no distinction between liquid and gas above the critical isotherm. Above the critical isotherm, the system is a supercritical fluid.
The segment of the critical isotherm forming the upper boundary of the liquid region is particularly interesting. The system passes from liquid to gas (or back again) with no discontinuity. That is, it goes from liquid to gas but it does not boil. Points just below the isotherm represent liquids of low density. Those above it represent gases of high density. On the isotherm, the liquid and gaseous states become one and the same.
To get a better feeling for the meaning of the critical isotherm, let us heat a subcritical liquid (1) to one of its supercritical isotherms (2), expand it (3), and cool it to its original temperature (4) as in Fig. 2.5. At the end of the process, the liquid has been transformed to a state that is clearly in the gaseous region, but there is no discernible phase change (boiling or vaporization→gas) during this process.
2.4.1 Subcritical Fluids
A subcritical curve in Fig. 2.5 has three real roots, predicting three different vol- umes for the same fluid below Tc. This seems absurd. How can a gas have three different volumes at the same time? The answer is that the term “fluid,” meaning that which flows, is more general than “gas.” The term fluid includes both liquids and gases. The volume of a subcritical liquid is small and is given by the leftmost intersection (root) of the subcritical isotherm with the horizontal. The volume of the vapor is large and is given by the rightmost intersection (root). The middle
root of the subcritical isotherm is not observed experimentally but has theoretical significance.
The constant pressure horizontal that intersects the subcritical isotherm represents vaporization going from left to right or condensation going from right to left. It is called an isobar. In vaporization, the volume of the system gets much larger but the pressure stays the same. Along an isobar, liquid and vapor are in equilibrium. If the pressure indicated by the isobar is atmospheric pressure, the subcritical isotherm is at the normal boiling point Tb.
2.4.2 The Critical Density
Below the critical temperature, one can measure the densities of a gas and a liquid at equilibrium in a closed container. Upon raising the temperature slightly at constant pressure, the density of the gas increases because liquid vaporizes. The density of the liquid decreases due to expansion. These densities, one increasing and the other decreasing, must approach each other as in Fig. 2.6. At the point where they meet, the densities of the gas and liquid and all other properties become identical. This is the critical temperature. Determination of the critical temperature of O2 by density measurements is shown in Fig. 2.6.
T (K)
155 150
145 140 135
130 125
120 115
Density
0 1
.
153.5
FIGURE 2.6 Densityρcurves for Liquid and Gaseous Oxygen. The straight line represents the arithmetic mean density of the liquid and gas (ρl+ρg)/2. The three curves meet at T= 153.5 K. The tabulated value for Tc is 154.6 K (CRC Handbook of Chemistry and Physics 2008–2009, 89th ed.).
THE LAW OF CORRESPONDING STATES, ANOTHER VIEW 27 2.5 REDUCED VARIABLES
The degree of nonideality for real gases is determined by how near the temperature is to the critical temperature. We can express the “nearness” of the temperature of a gas to its critical temperature as the unitless ratio T/Tc. This ratio is called the reduced temperature
TR = T Tc
The other reduced variables are defined in the same way. The (unitless) reduced pressure and reduced volume are, for one mole,
pR = p pc
and VR = V Vc
These new variables are scaling factors by which we take an entire family of isotherms suggested by the three representative isotherms in Fig. 2.4 and move, stretch, or compress them until their critical isotherms coincide. Having done that, at the same TR, the pR and, VR behavior of all gases fall approximately on the same family of curves. The gas data have been manipulated into corresponding states (Section 2.3). Knowing the behavior of one gas in terms of its reduced variables pR, VR, and TR, we know the behavior of any real gas, provided only that we know its critical constants pc, Vc, and Tc. Needless to say, industrial chemists and chemical engineers are delighted by this, and they have devoted considerable effort to construct standard tables and Z-curves in terms of the reduced variables.