linguistic:paraphrasing
The van der Waals equation, named for its originator, the Dutch physicist Johannes Diderik van der Waals, is an equation of state that extends the ideal gas law to include the non-zero size of gas molecules and the interactions between them. As a result the equation is able to model the phase change, liquid ↔ vapor. It also produces simple analytic expressions for the properties of real substances that shed light on their behavior. One way to write this equation is: p = R T v − b − a v 2 where p is pressure, T is temperature, and v = V N A / N is molar volume, N A is the Avogadro constant, V is the volume, and N is the number of molecules. In addition R = N A k is the universal gas constant, k is the Boltzmann constant, and a and b are experimentally determinable, substance-specific constants. The constant a expresses the strength of the molecular interactions. It has dimension pressure times molar volume squared, [pv⁲] which is also molar energy times molar volume. The constant b denotes an excluded molar volume due to a finite molecule size, since the centers of two molecules cannot be closer than their diameter, σ. A theoretical calculation of these constants for spherical molecules with an interparticle potential characterized by, σ, and a minimum energy, − ε, as shown in the accompanying plot produces b = 4 [ 4 π 3 / 3 ] N A. Multiplying this by the number of moles, N / N A, gives the excluded volume as 4 times the volume of all the molecules. This theory also produces a = I ε N A b where I is a number that depends on the shape of the potential function, φ. In his book Boltzmann wrote equations using V / M in place of V N A / N used here, Gibbs did as well, so do most engineers. Also the property, V / N the reciprocal of number density, is used by physicists, but there is no essential difference between equations written with any of these properties. Equations of state written using molar volume contain R, those using specific volume contain R / m ¯, and those written with number density contain k. Once a and b are experimentally determined for a given substance, the van der Waals equation can be used to predict the boiling point at any given pressure, the critical point, and other attributes. These predictions are accurate for only a few substances. For most simple fluids they are only a valuable approximation. The equation also explains why superheated liquids can exist above their boiling point and subcooled vapors can exist below their condensation point. The graph on the right is a plot of T vs v calculated from the equation at four constant pressure values. On the red isobar, p = 2 p c, the slope is positive over the entire range, b ≤ v < ∞. This describes a fluid as a gas for all T, and is characteristic of all isobars p > p c. The green isobar, p = 0.2 p c, has a physically unreal negative slope, hence shown dotted gray, between its local minimum, T m i n, v m i n, and local maximum, T m a x, v m a x. This describes the fluid as two disconnected branches; a gas for v ≥ v m i n, and a denser liquid for v ≤ v m a x ≤ v m i n. The thermodynamic requirements of mechanical, thermal, and material equilibrium together with the equation specify two points on the curve,, and, shown as green circles that designate the coexisting boiling liquid and condensing gas respectively. Heating the fluid in this state increases the fraction of gas in the mixture; its v, an average of v f and v g weighted by this fraction, increases while T s remains the same. This is shown as the dotted gray line, because it does not represent a solution of the equation; however, it does describe the observed behavior. The points above T s, superheated liquid, and those below it, subcooled vapor, are metastable; a sufficiently strong disturbance causes them to transform to the stable alternative. Consequently they are shown dashed. Finally the points in the region of negative slope are unstable. All this describes a fluid as a stable gas for T > T s, a stable liquid for T < T s, and a mixture of liquid and gas at T = T s, that also supports metastable states of subcooled gas and superheated liquid. It is characteristic of all isobars 0 < p < p c, where T s is a function of p. The orange isobar is the critical one on which the minimum and maximum are equal. The black isobar is the limit of positive pressures, although drawn solid none of its points represent stable solutions, they are either metastable or unstable (negative slope. All this is a good explanation of the observed behavior of fluids.