Thermodynamic critical pressure gases

Clusius and Frank 61) find 83.78 K. for the melting point with 280.8 cal./gram atom for the heat of melting as well as 87.29 K. for the normal boiling point and 1558 cal./gram atom for the associated heat of vaporization. These vapor pressure data are substantiated by the more recent work of Clark, Din, Robb, Michels, Wassenaar, and Zwietering 57). Thermodynamic properties of the ideal gas have been calculated at the National Bureau of Standards 296). Kobe and Lynn 193) select 151 K. for the critical temperature and 48.0 atmospheres for the critical pressure. 

The solid is not stable at one atmosphere, and can only be obtained at elevated pressures. In the range from 0° to 1° K., the required pressure is reported by Simon and Swenson (304) as 25 atmospheres. At a pressure of 103 atmospheres, Keesom (174) reports the melting point to be 3.5 K., with an associated heat of 5 cal./gram atom. Keesom also reports the second order transition (lambda point) at 2.186 K., and the normal boiling point at 4.216 K. with the associated heat of vaporization of 20 cal./gram atom. Thermodynamic prop>erties for the ideal monatomic gas have been calculated at the National Bureau of Standards (395). Kobe and Lynn (193) report the critical temperature as 5.3 K. and the critical pressure as 2.26 atmospheres. 

In general, any substance that is above the temperature and pressure of its thermodynamic critical point is called a supercritical fluid. A critical point represents a limit of both equilibrium and stability conditions, and is formally delincd as a point where the first, second, and third derivatives of the energy basis function for a system equal zero (or, more precisely, where 9P/9V r = d P/dV T = 0 for a pure compound). In practical terms, a critical point is identifled as a point where two or more coexisting fluid phases become indistinguishable. For a pure compound, the critical point occurs at the limit of vapor-Uquid equilibrium where the densities of the two phases approach each other (Figures la and lb). Above this critical point, no phase transformation is possible and the substance is considered neither a Uquid nor a gas, but a homogeneous, supercritical fluid. The particular conditions (such as pressure and temperature) at which the critical point of a substance is achieved are unique for every substance and are referred to as its critical constants (Table 1). 

A supercritical fluid (SCF) is any substance at a temperature and pressure above its thermodynamic critical point. Such a fluid can diffuse through soHds such as a gas and dissolve materials such as a liquid. Carbon dioxide and water are the most commonly used SCFs. 

SFC is the application of a supercritical fluid, any substance at a temperature and pressure above its thermodynamic critical point (Figure 9.2) with both gas- and liquid-like abilities to diffuse through solids, and dissolve materials, respectively, as the mobile phase in the chromatographic process. The most widely used mobile phase for SFC is carbon dioxide because of its low critical pressure (73 atm), low critical temperature (31°C), inertness, low toxicity, and high purity at low cost [12,13], Historically there were two approaches in developing modern SFC the use of either the packed and microbore columns designed for HPLC application or the open-tubular capillary GC type columns [13,14], The conventional packed HPLC... 

A supercritical fluid is defined as one that is above its thermodynamic critical point, as identified by the critical pressure (p ) and critical temperature (Tc). Supercritical fluid behavior can be peculiar because of the variation of theimophysical properties such as density and specific heat near and at the critical point. Supercritical fluids have some properties similar to liquids (e.g., density), and some properties that are comparable to those of gases (e.g., viscosity). Thus, they cannot be considered either a liquid or a gas. 

If the system is cooled isobarically along a path above the critical pressure Pc (Fig. 5b, path a), the state functions continuously change from the values characteristic of a high-temperature phase (gas) to those characteristic of a low-temperature phase (liquid). The thermodynamic response functions, which are the derivatives of the state functions with respect to temperature (e.g., C ), have maxima at temperatures denoted Pmax (P) Remarkably these maxima are still prominent far above the critical pressure [31], and the values of the response functions at Pmax(P) (e-g-, C max) diverge as the critical point is approached. The lines of the maxima for different response functions asymptotically approach one another as the critical point is approached, since all response functions become expressible in terms of the correlation length. This asymptotic line is sometimes called the Widom line, and is often regarded as an extension of the coexistence line into the one-phase regime. ... 

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. 

The results of the approximation equations are given in Figure 15.5 for ethylene at temperatures of 296.4 and 443.2 K and pressures up to 270 MPa (2700 bar). Depending on the calculation model, highly differing values for the isentropic exponents are obtained, calculation with the equation of state being the most accurate method. Inaccuracies should be expected there if the coefficients of the equation of state are determined only with the thermodynamic critical values of the gas and not adapted to the measured data. 

As an alternative, the analytic solution based on Eq. (15.17) may be used for sizing a valve. Arithmetic averages between the inlet and the throat of the nozzle for the isentropic exponent, the real gas factor, and its gradients can only be recommended if the change of the isentropic coefficient and the real gas factor is almost linear that is, pressure and temperature are far from the thermodynamic critical condition. 

In practice, often an ideal behavior of gases is assumed at moderate pressures when sizing a safety valve for gas service. Real gas behavior is only assumed at a very high pressure, for example, at a pressure of more than 100 bar. In general, the real gas behavior is rather determined from the proximity of the thermodynamic critical point. With the reduced thermodynamic pressure and the reduced thermodynamic temperature, the deviation from ideal behavior can be described much better than with the absolute values of pressure and temperature. If the reduced pressure and the reduced temperatures at the entrance of the nozzle exceed p/pc > 0.5 or T/Tc > 0.9, the deviations from the ideal behavior are usually no longer tolerable. 

A chart which correlates experimental P - V - T data for all gases is included as Figure 2.1 and this is known as the generalised compressibility-factor chart.(1) Use is made of reduced coordinates where the reduced temperature Tr, the reduced pressure Pr, and the reduced volume Vr are defined as the ratio of the actual temperature, pressure, and volume of the gas to the corresponding values of these properties at the critical state. It is found that, at a given value of Tr and Pr, nearly all gases have the same molar volume, compressibility factor, and other thermodynamic properties. This empirical relationship applies to within about 2 per cent for most gases the most important exception to the rule is ammonia. 

The thermodynamic properties of a number of compounds are shown in Appendix D as pressure-enthalpy diagrams with lines of constant temperature, entropy, and specific volume. The vapor, liquid, and two-phase regions are clearly evident on these plots. The conditions under which each compound may exhibit ideal gas properties are identified by the region on the plot where the enthalpy is independent of pressure at a given temperature (i.e., the lower the pressure and the higher the temperature relative to the critical conditions, the more nearly the properties can be described by the ideal gas law). 

---