The perfect gas equation of state PV = nRT is manifestly incapable of describing actual gases at low temperatures, since they undergo a discontinuous change of volume and become liquids. Van der Waals equation comes up to recognize the molecules that interact with each other. The equation puts two parameters to mimic this interaction. The first, an attractive intermolecular force at long distances, helps draw the gas together and therefore reduces the necessary outside pressure to contain the gas in a given volume (the gas is a little thinner near the walls). The second is to take account of the finite molecular volume. A real gas cannot be compressed indefinitely (it becomes a liquid, for all practical purposes incompressible).
Where p is the pressure, V is the volume, T is the temperature, n is the number of moles of gas present, R is the gas constant, and a and b are constants that depend upon the gas. In the usual situation, a and b are known and can be found in tables. One must watch the units used though. This rather crude approximation does in fact give sets of isotherms representing the basic physics of a phase transition quite well.
Van der Waals’s equation is approximate and quantitatively describes the properties of real gases only in regions of high temperatures and low pressures. However, it permits the qualitative description of the behavior of a gas at high pressures, the condensation of a gas into a liquid, and the critical state. Based on above picture, at low temperatures all three roots are real, but above a specific temperature Tc, called the critical temperature. Physically, this means that for T > Tc the substance can exist in only one state (the gaseous), and below Tc, in three states (two stable liquid ⊽1 and gaseous ⊽g and one unstable). This is expressed graphically in the following way: for T < Tc the isotherm has three points of intersection with the straight line ac, parallel to the volume axis. The points of the straight line ac correspond to the equilibrium of liquid and its saturated vapor.
Under the equilibrium conditions, for example, in the state corresponding to point b, the relative amounts of liquid and vapor are determined by the ratio of the segments bc/ba (the “rule of moments”). The saturated vapor pressure psy and the volume interval from ⊽1 to ⊽g correspond to the phase equilibrium at a specific temperature. At lower pressures (to the right of ⊽g), the isotherm characterizes the properties of the gas. To the left, the almost vertical part of the isotherm indicates a very low compressibility of the liquid. Segments ad and ec belong to superheated liquid and super-cooled vapor respectively (meta-stable states). Segment de is physically unrealized because here there is a volume increase with an increase in pressure. The set of points a, a’, a’’, … and c, c’, c’’, … determines a curve, called a bi-nodal, which outlines the region of simultaneous existence of gas and liquid. At the critical point-C the temperature, pressure, and volume (Tc, pc, and ⊽c) have values that are characteristic of each substance. However, if the relative values T/Tc, p/pc, and ⊽/⊽c are introduced into Van der Waals’s equation, then it is possible to obtain the so-called reduced Van der Waals’s equation, it does not depend on the individual properties of the substance.
With reduced quantities:
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