A PVT relationship is one of the forms of the equations of state (see Equations of State), which relates the pressure, molar volume V and the temperature T of physically homogeneous media in thermodynamic equilibrium.
Equation of state of a liquid for reasons of not sufficiently advanced qualitative theory is based either on experimental data immediately or on one of the variants of the law of corresponding states with a limited amount of information on a particular substance. In this case, while forming rational empirical equations the general ideas of the statistical theory of the equation of state or a character of density changes characteristic of liquids are usually considered. To describe the properties of liquids, the equations of state are used which define the PVT relationships both of liquid and gas in the form of a single analytical expression. The simplest are the so-called cubic equations of state corresponding to a cubic dependence of pressure on a specific volume (density) of a liquid and being a modification of van der Waals' equation. The intermolecular repulsive forces in them are included in the term similar to that in van der Waals' equation; the effect of the attractive forces in conveyed by the terms of different form. The most prominent in this group are:
The Redlich-Kwong equation of state is
where is the molar volume, (m^{3}/mol), the universal gas constant (8.314J/mol k), T the temperature (k) and p the pressure (pa) whose two parameters a and b can be determined either by fitting the experimental data or related to the properties of a substance at the critical point:
In the last case the values of the coefficients Ω_{a} and Ω_{b} for a particular substance can be refined from the experimental data, and, in equations generalized for a number of substances, they are often represented in the form of functions of a reduced temperature T_{r} = T/T_{c} and the acentric (Pitzer's) factor ω = (log p_{c}/10p_{s})T_{r} = 0.7, p_{c}, p_{s} are the critical and saturation pressures.
The Soave equation of state is
where
a = 0.42748R/p_{c};
b = 0.08664RT_{c}/p_{c};
α = [1 + (1 − )(0.480 + 1.574ω − 0.176ω^{2})]^{2};
and for highly polar substances α = 1 + (1 – T_{r})(m + nT_{r}) with m and n being determined from the experimental data.
The Peng-Robinson equation of state is
in which
a = 0.45724R^{2}/p_{c}
b = 0.07780RT_{c}/p_{c};
α = [1 + (1 − )(0.37464 + 1.54226ω − 0.26992ω^{2})]^{2},
is the one of two parameter cubic equations which most closely predicts the thermodynamic properties of a liquid. The compressibility of substances at the critical point according to this equation is z_{c} = 0.307. The error of this equation for individual substances is 2-3% except near the critical point and for highly polar substances. Cubic equations have found wide application for the description of properties of mixtures of substances when carrying out calculations in industry when a large number of iterative calculations must be performed, especially in defining the state of phase equilibrium of mixtures. For this purpose, procedures for determining the parameters of the equations of state of mixtures from the parameters of equations for pure substances have been developed. A considerable body of modifications of cubic equations are known which allow us to increase the accuracy of calculations both through various methods of determining the parameters of the equation (a and b) or by invoking more parameters.
Another trend in modifying the van der Waals equations of state starts from the assumption that the term which reflects the influence of intermolecular attractive forces is assumed to be the same as that in one of the cubic equations under study, and the term of the equation regarding for the repulsive forces is transformed. The development of this trend promises a greater accuracy in describing the properties of a substance in the liquid phase, where the intermolecular repulsive forces are the determining ones. Thus, the combination of the Carnahan-Starling equation approximating the interaction of particles with the potential of a rigid sphere and of the Redlich-Kwong equation brings about an equation of state of the form
where y = b_{0}/4, b_{0} = 2πσ^{3}/3 is the second virial coefficient of rigid spheres, is the molar density and σ is the diameter of the molecules.
The more exact description of the PVT relationship can be obtained when applying multiconstant equations. The Benedict-Webb-Rubin equation of state is most generally used in practical calculations. Various sets of eight constants for a wide range of substances have been published repeatedly in literature. Starling, by adding additional terms, has come up with a new modification of this equation applied over the range of sufficiently low temperatures (not lower than T_{r} = 0.3) and for densities up to 3ρ_{r}. The Benedict-Webb-Rubin-Starling equation of state has the form
Its eleven constants can be either specific to a particular substance or generalized and expressed in terms of critical parameters of the substance and its acentric factor ω. In the latter case the PTV relationship is described less accurately. The rules of combining these constants in applying the equations to binary mixtures are also known.
When employing the law of corresponding states for representing the PTV relationships in a wide range of state parameters of liquid and gas, the method based on the Pitzer and Curl relationship has enjoyed the widest application
where z^{(0)} is the compressibility of "simple" liquid, z^{(1)} is the correction for the deviation from the behavior of "simple" liquid, and ω is the Pitzer acentric factor. Lee and Kesler have presented the most precise values of z^{(0)} and z^{(1)} in table form for a temperature range of 0.3 ≤ T_{r} ≤ 4 and pressures of 0.01 ≤ p_{r} < ≤ 10, and have also suggested generalized equations for describing them. The error in describing the properties by this method does not exceed 2-3%, except for the near critical region (T_{r} = 0.93 − 1.0) and for highly polarized substances. In an effort to increase the reliability, modifications of the method recommended for particular classes of liquids have been developed.
The density (specific volume) of many liquids at atmospheric pressure or in a state of saturation has been measured experimentally and is given in handbooks. At the same time a number of methods of approximate determination of these properties for poorly studied liquids on the basis of a limited number of data have been developed.
A number of additive methods have been suggested for finding a molar volume of liquids. The idea behind these methods is that each chemical element or each type of chemical bonds is ascribed certain numerical values of the components whose sum allows us to calculate the molar volume. The error in calculated values for different liquids is 3-4% except for highly accociated liquids. A somewhat more accurate result (except for low boiling substances and polar nitrogen and fluorine containing compounds) for molar volume of boiling liquid can be determined by the Tyn and Calus method, if the value of the critical volume of the substance is known,
where and are expressed in cm^{3}/ mole.
Some other methods for describing the temperature dependence of a specific volume of a saturated liquid have also been suggested. The most exact among them are evidently the methods, which use various modifications of the Rackett equation:
where T_{r} = T/T_{c} is the reduced dimensionless temperature, z_{c} = is the compressibility of the substance at the critical point. This equation makes it possible to describe a wide range of substances with an error less than 1.5%; however, much worse results are obtained when the method applied to quantum liquids and to substances whose molecules contain cyclic groups, to alcohols, nitriles, etc. Its modifications consist either in replacing the quantity z_{c} by the empirical constant peculiar to each substance or in using as the reference state the state at a certain temperature instead of the critical point, for which the specific volume of saturated liquid is well known. The value of z_{c} in this case can be calculated from the relationship
where ω = (log p_{c}/10 p_{s})T_{r} = 0.7 is the Pitzer acentric factor.
In such a case, the accuracy of calculation of specific volumes of saturated liquid increases. Thus, for a majority of polar substances the error does not exceed 1%. In order to determine the pressure at which the saturated liquid is, we can make use of one of the generalized methods most of which are based on integration of the Clapeyron-Clausius Equation on condition that this or that additional assumption will be made. One of the most precise methods among them is the Riedel-Plank-Miller method to use which we must have an information on normal boiling temperature (expressed as Tb,r which is the reduced value of this temperature) and the critical point. The saturation pressure in this method is calculated as
where
G = 0.4835 + 0.4605h;
h = T_{b,r} ln p_{c}/(1 − T_{b,r} );
k = h/G − (1 + T_{r,b})/[(3 + T_{b,r})(1 − T_{b,r})^{2}].
The error in calculating p_{s,r} at pressures higher than 10 mm Hg for nonpolar liquids does not exceed 2-4% but is somewhat higher for polar liquid.
The density of a compressed liquid, i.e., of liquid at a pressure higher than the saturation pressure, within the temperature range from the triple point temperature up to temperatures somewhat exceeding the normal boiling temperature, changes only slightly with increase in the pressure. Its isothermal compressibility β_{T} in this case depends slightly on pressure. This fact is the basis for a whole family of empirical equations of state of a compressed liquid in which this or that form of dependence of isothermal compressibility or its reciprocal quantity k_{T} = , on pressure, is assumed as the basis. The most convenient among these is the secant bulk-modulus equation. Its linear variant, suggested by the Tait in 1888, is as follows
where the quantities with subscript "0" refer to the initial pressure, makes it possible to describe a specific volume of a large number of liquids at pressures of several tens of MPa. The addition of terms containing p^{2} and p^{3} enables the field of application of these equations, for instance, for water, water solutions and liquid metals to be extended up to the pressures of several hundreds MPa. Extensive application has also been made of an equation, which is erroneously called the Tait equation and approximates the linear secant bulk-modulus equation,
which corresponds to the implicit assumption that . Also known are the variants generalizing this equation, in which the specific volume of the compressed liquid is related to its specific volume in the state of saturation ( )
the parameter as a function of C in this case being represented as a function of the Pitzer acentric factor, and B by the reduced temperature and the Pitzer factor.
An equation of the form
is used for describing the properties of compressed liquids in a wider range of temperatures, for instance, for organic liquids up to T_{r} = 0.9 and works well.
The application of the above equations is limited to the conditions under which liquid is the existing phase.
REFERENCES
Reid, R., Prausnitz, J. M., and Sherwood, T. K. (1977) The Properties of Gases and Liquids, 3rd edn., McGraw-Hill.
Wales, S. M. (1985) Phase Equilibria in Chemical Engineering, Butterworth Publ.
References
- Reid, R., Prausnitz, J. M., and Sherwood, T. K. (1977) The Properties of Gases and Liquids, 3rd edn., McGraw-Hill.
- Wales, S. M. (1985) Phase Equilibria in Chemical Engineering, Butterworth Publ.
Heat & Mass Transfer, and Fluids Engineering