The Bender Equation of State

Bender developed an equation of state with a modified Benedict-Webb-Rubin form given by  

The Bender equation was one of the first modifications of the Benedict-Webb-Rubin equation that was specifically intended to describe vapour-liquid phase equilibria as well as energetic properties in the liquid phase with results that represent the measured values with an uncertainty suitable for technical applications. 

In 1973, Jacobsen and Stewart developed what is termed an advanced form of a modified Benedict-Webb-Rubin equation of state that has been given the acronym mBWR and is given by 

5 Thermodynamic Properties from Pressure-Explicit Equations of State 

The values of entropy S, enthalpy J/m, internal energy C/m, and heat capacity (Cpm, Cj/.m. or Csat) at various state points are calculated with the pressure explicit equation of state and an ancillary equation that represents the ideal gas heat capacity. For a pure fluid, the equations that represent the vapour pressure and melting curve are used to identify the temperatures of the phase changes from liquid to vapour and solid to liquid, respectively. Properties are evaluated through 

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Figure 5 shows the plots of the solubility of a-tocopherol in the supercritical gas phase as a function of the density of pure carbon dioxide, which has been calculated with the Bender equation of state. 

A new approach based on a combination of the non-local density functional theory and the Bender equation of state was successfully applied to high-pressure (up to SO MPa) argon, nitrogen, methane and helium adsorption. The approach allows reliably determining pore size distribution and adsorbent density, and most importantly it does not require helium experiments to determine the skeletal density. 

More recently, Ustinov and coworkers [72, 73] developed a thermodynamic approach based on an equation of state to model the gas adsorption equilibrium over a wide range of pressure. Their model is based on the Bender equation of state, which is a virial-like equation with temperature dependent parameters based on the Benedict-Webb-Rubin equation of state [74]. They employed the model [75, 76] to describe supercritical gas adsorption on activated carbon (Norit Rl) at high temperature, and extended this treatment to subcritical fluid adsorption taking into account the phase transition in elements of the adsorption volume. They argued that parameters such as pore volume and skeleton density can be determined directly from adsorption measurements, while the conventional approach of He expansion at room temperature can lead to erroneous results due to the adsorption of He in narrow micropores of activated carbon. 

Polt, A., Platzer, B., and Maurer, G. (1992). The Bender equation of state for 14 polyatomic pure substances. Chem. Tech. (Leipzig), 44, 216-24. 

The Bender equation of state for several of the older refrigerants, including R14, R114, and RC318. 

Platzer, B. and Maurer, G. (1993). Application of a generalized Bender equation of state to the description of vapour-liquid equihbria in binary systems. Fluid Phase Equilibria, 84, 79—110. 

E. Bender, Equations of State Exactly Representing the Phase Behaviour of Pure Substances, Proc. 5th Symp. Thermophys. Prop., 1970, 5, 227-235. 

The method of analysis of the uncertainty in the amount of fluid or density in the continuous phase in manometric adsorption measurements, calculated via Peng-Robinson and Bender equations of state is presented. It is applied for the evaluation of the specific surface excess amount and its uncertainty during high-pressure nitrogen adsorption by a microporous activated carbon cloth at pressures up to 17 MPa at 252.40 K. Adsorption data were analysed via the use of bilinear-interpolated data from a P-p-T matrix developed fi-om the NIST Chemistry WebBook fluid physical properties database. Deviations of calculated specific surface excess amounts from those calculated using NIST density data approach 0.2 %, considerably superior to either Peng-Robinson or Bender EoS, ranging from 6.4 to 3.0 %. 

For the evaluation of these measurements the thermodynamic properties of the C02"air mixtures were described with the equation of state by E. Bender, applying the mixing and combination rules proposed by Sievers and Schulz (3. As has already been described for pure gases, the temperature T at the Wilson point was calculated with this equation of state for the pressure p at the Wilson point and the stagnation entropy [IJ. 

Bender, E. The Calculation of Phase Equilibria from a Thermal Equation of State Applied to the Pure Fluids Argon, Nitrogen, Oxygen and their Mixtures. C.F. Mil 11 er-Ver 1 ag, Karlsruhe, 1973. 

Historically, the first experimental determinations of the vapor densities and pressures approaching the critical region of a metal were made for mercury. Bender (1915, 1918) carried out pioneering measurements of vapor densities up to about 1400 °C. The samples in these studies were enclosed in strong fused quartz capillaries. In 1932, Birch made the first measurements of the vapor pressure of mercury and obtained realistic values for the critical temperature and pressure. Birch found values = 1460 °C and = 1610 bar, results that are remarkably close to the most accurate values available today (Table 1.1). A number of groups in various countries have contributed subsequently to the pool of pVT data currently available (Hensel and Franck, 1966, 1968 Kikoin and Senchenkov, 1967 Postill et al., 1968 Schonherr et al., 1979 Yao and Endo, 1982 Hubbard and Ross, 1983 Gotzlaff, 1988). The result is that the density data for mercury are now the most extensive and detailed available for any liquid metal. Data have been obtained by means of isothermal, isobaric, or isochoric measurements, but as we have noted in Sec. 3.5, those obtained under constant volume (isochoric conditions) tend to be preferable. In Fig. 4.10 we present a selection of equation-of-state data that we believe to be the most reliable now available for fluid... 

Figure 6. Comparison of the residual isobaric heat capacities of propylene of Bier et al. with those predicted by the equation of state of Bender...

Bender, E. (1970). Equations of state exactly representing the phase behavior of pure... 

Using the formula for one-dimensional compressible flow presented in Section IV.B, we calculated the pressure, temperature, and velocity profiles describing the subsonic adiabatic expansion of pure carbon dioxide inside the orifice and the capillary up to the nozzle exit (i.e., point 2 in Figures 3 and 5). Both the Bender (38) and Camahan-Starling-van der Waals (39) equations of state were used to calculate the necessary PvT properties for CO2, and results using either of the two equations were essentially identical. Downstream of the nozzle exit, we calculated the pressures and temperatures on the upstream and downstream sides of the Mach disk by using the formulas of Ashkenas and Sherman (36) (see Section V). These formulas assume an ideal gas with y = 1.286, close to the value of CO2 at ambient conditions. We should remember, however. 

Acylation and deacylation in equation (13) proceed through similar transition states. If deacylation occurs through attack of an alcohol molecule R OH rather than water on the carbonyl carbon atom, then deacylation is the microscopic reverse of acylation. Bender and coworkers (Bender and Kezdy, 1965) have demonstrated the symmetry of the reaction about the acyl enzyme in reactions in which reversibility can be observed. 

After solving Eq. (3.45) we can obtain the eigenvalues and eigenvectors of the inversion—rotation states of ammonia in the ground vibrational state i.e. we can calculate - in the rigid bender approximation - the rotational dependence of the inversion splittings in the states of ammonia. Note that the / and k dependent terms in the Schrodinger equation [Eq. (3.45)] represent a modification of the double-minimum potential function fo(p) for each rotational state/, k (see further Sections 5.1 and 5.2). 

The cause of the scatter is the non-systematic influence of the substituent on the microscopic environment of the transition structure. The linear free energy relationship between product state XpyH (Equation 22) and the transition structure (Xpy. .. PO32 . . . isq) will be modulated by second-order non-systematic variation because the microscopic environment of the reaction centre in the standard (XpyH ) differs slightly from that (Xpy-PO ) in the reaction under investigation giving rise to specific substituent effects. These effects are mostly small. An unusually dramatic intervention of the microscopic medium effect may be found in Myron Bender s extremely scattered Hammett dependence of the reaction of cyclodextrins with substituted phenyl acetates.22 The cyclodextrin reagent complexes the substrate and interacts... 

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