Real gases critical point

Unlike the pressure where p = 0 has physical meaning, the zero of free energy is arbitrary, so, instead of the ideal gas volume, we can use as a reference the molar volume of the real fluid at its critical point. A reduced Helmlioltz free energy in tenns of the reduced variables and F can be obtained by replacing a and b by their values m tenns of the critical constants... 

According to the van der Waals equation of state, the value of compressibility at the critical point should be 3/8 = 0.375. When does a real gas depart significantly from an ideal gas We can write equation (4.9) as the reduced equation of state, with the reduced temperatures, pressures, and volumes = TITc, Pi = P/Pc, Vr = V/Vc- Then, all gases would have the same equation of state in the form of reduced parameters ... 

Tests of this prediction against experimental critical-point data of Table 2.4 reveal large deviations (e.g., an approximately 20% error even in the most favorable case of He) that reflect serious quantitative defects of the Van der Waals description. This is but one of many indications that the Van der Waals equation, although a distinct improvement over the ideal gas equation, is still a significantly flawed representation of real fluid properties. 

For chemical and biological applications, one typically wants to compute the volume V of a real gas over a specified range of pressures and temperatures, such as for pressures and temperatures well below their critical points, but more often for pressures and temperatures above the critical points Pc and Tc of the gas. Notice that the condition of any gas above its critical point Pc and Tc is not really gaseous, nor is it a liquid. Such a gas is called a supercritical fluid. 

In Chapter 1, we remarked on the oscillatory behavior of two-parameter equations of state, such as those of van der Waals, Berthelot, and Redlich-Kwong in the two-phase region. Usually, the parameters for these equations are determined from the critical point therefore, the equations should have some validity slightly below the critical point in the two-phase region. Because these equations of states are cubic equations, with three real roots below the critical point, they give three values of Vm for any combination of P and T. Referring to Fig. 7, VmX and Vm3 correspond to the liquid and gas, respectively, whereas Vna, where... 

Our table refers to the real fluid existing at the standard-state pressure of 1 bar. Since this pressure is below the critical pressure of water [p = 217.6 bar at T = 647.14 K ( ) ], there is a first order transition at the normal boiling point. Therefore, this is a two-phase (liquid-real gas) table for the real fluid at p=l bar. 

The ideal gas equation of state cannot describe real fluids in most situations because the fluid molecules themselves occupy a finite volume and because they exert forces of attraction and repulsion on each other. As the gas is cooled, and assuming its pressure is below the critical point, a temperature is reached where the intermolecular interactions result in a transition from the gas phase to the liquid phase. The ultimate fluid model would be one that could describe this transition as well as the fluid behavior over the entire range of temperature and pressure. Such a model would also be capable of representing mixtures as well as pure components. 

The sharp intersection of the liquid Pi(x)- and gas Pg branches at the critical point shown in Fig. 1 implies that the molar internal energy e(v,s) is not a continuous differentiable function of v and s along the CXC for both real and WMG-model fluids. It is the direct confirmation of the singular concept introduced to represent the actual CXC-data of any real fluid by FEOS (1-5). Put in thermodynamic terms, any state-point Ps,T of CXC including the critical point F oT c is, simultaneously, the one phase ... 

IIL There are three real equal roots present. At and above the point where a = / y, there can only be one value of v for any assigned value of p. This point K (Fig. 143) is no other than the well-known critical point of a gas. Write pe, vc, Tc, for the critical pressure, volume, and temperature of a gas. From (2),... 

According to some researchers, the two states are in fact two distinct phases, with real coexistence boundaries. If this claim can ever be verified, then it follows that there should exist a second critical point for water, akin to the critical point at 374°C, where the liquid and gas phases coalesce. It may, however, be impossible to confirm the existence of such a point by experiment, because it would lie well below Thom where freezing cannot be avoided. 

Figure 2.1-27 shows the fugacity coefficient vs. the reduced pressure p, each line being parametrized by a value of the reduced temperature. The line for the saturation limit (vapor pressure curve) ends at the critical point. If the critical real gas constant Z deviates from 0.27 then the fugacity coefficient complies with the following empirical law ... 

This behavior occurs until a certain high temperature is reached denoted and called the critical temperature. At that temperature, the constant pressure plateau shrinks into a single point (point C) called the critical point The molar volume at that point is called critical molar volume and the pressure is the critical pressure P. A gas cannot be condensed to a liquid at temperatures above and there is no clear distinction between the liquid and gaseous phases because the two states cannot coexist with a sharp boundary between them. Experimentally, if a certain amount of gas and liquid is placed inside a pressurized container with transparent quartz windows and kept below T, two layers will be observed, separated by a sharp boundary. As the tube is warmed, the boundary becomes less distinct because the densities, and therefore the refractive indices, of the liquid and gas approach a common value. When the T is reached, the boundary becomes invisible and the iridescent aspect exhibited by the fluid is called critical opalescence. Hence the following definitions can be drawn for the critical constants of a real gas. 

We list in Table 9.1 the values of the critical-point exponents a, y, 8, V, T, and il. In the first column are the classical, mean-field-theory values, in the second column the values for the two-dimensional Ising model (lattice gas), which are also known exactly, and in the third column those for real, three-dimensional fluids as determined from present-day critical-point theory, " from experiment, or from both. As theoretical values the latter are probably correct to 0-01, but as experimental values their uncertainties may be two or three times that. 

Nearby the thermodynamic critical temperature (T, = 1.05), the real gas factor drops off, at first very strongly, reaches a minimum at a reduced pressure of somewhat over 1, and then increases again. The further away the temperature of the gas is from the thermodynamic critical point, the less strongly pronounced the minimum is. 

The real behavior of a gas essentially depends on how far away the actual pressure and temperature are from the thermodynamic critical point and not on the absolute values of pressure or temperature of the gas. The assumption that a gas behaves ideally (Z = 1) may lead to significant errors in the sizing of safety valves. Basically, the required cross-sectional area of the valve seat is rather underestimated if a too small real gas factor is assumed. 

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