Equation Beattie-Bridgeman

Kinetic considerations always lead to an equation of state giving the pressure as a function of the concentration and temperature, but for many purposes it is more convenient to know the molal volume as a function of the pressure and temperature. Beattie1 has indicated how the constants of an equation, similar in form to the Beattie-Bridgeman equation,2 may be obtained from p,V,T data. Often, however, the labor of this determination is not compensated by the added convenience. 

Beattie1 has also shown how an approximation sufficient for many purposes may be obtained making use of the constants already obtained for the Beattie-Bridgeman equation. The treatment is the same for any equation which may be expressed in the virial form. So expressed the Beattie-Bridgeman equation has four terms ... 

J.A. Beattie O.C. Bridgeman equation of state was first described in ProcAmAcadSci 63, 229(1928), which we did not consult One of the modifications of Beattie Bridgeman equation is given under "Su ... 

Generalized Beattie-Bridgeman Equation of State for Real Gases. It is written by Su Chang as ... 

Su Chang stated that their equation falls into the general form of the Lorentz Equation of State (qv). It may also be regarded as a simplified, generalized form of Beattie-Bridgeman Equation of State (qv)... 

The Beattie-Bridgeman equation, like most equations of state, is explicit in pressure. Certain calculations require an equation that is explicit in volume. Therefore, Beattie rearranged the Beattie-Bridgeman equation and modified it to give the following form. 

The constants are the same as for the Beattie-Bridgeman equation and the same numerical values may be used. The equation is less accurate than the Beattie-Bridgeman equation, but the agreement with the observed data is satisfactory. 

Both the Beattie-Bridgeman equation and the Beattie modification have been used with good accuracy at densities up to 2/3 of the critical density of the gas. Unfortunately, this does not cover the entire range of interest of die petroleum engineer. 

EXAMPLE 4-2 Calculate the molar volume of the gas given below at 100°F and 250 psia. Use the Beattie modification of the Beattie-Bridgeman equation. 

Third, use Beattie modification of the Beattie-Bridgeman equation to compute molar volume. 

Repeat Exercise 4-2. Use the Beattie-Bridgeman equation of state. 

A laboratory cell with volume of 0.007769 cu ft contains 0.001944 lb moles of the mixture given in the table below. Temperature is to be raised to 80°F. Calculate the pressure to be expected. Use the Beattie-Bridgeman equation of state. Compare your answers with experimental results of 1200 psia 2%. 

The permselectivity for membrane separations can also be calculated by substituting fugacities calculated from an equation of state, here using the Beattie-Bridgeman equation, into Equation (3) for the partial pressure values (4). The values of the permselectivities in Table IV are relatively constant at a fixed feed composition in agreement with the approximately linear behavior noted in Figures 9-11. 

Other semiempirical equations of state can be used to predict Joule-Thomson coefficients. Perhaps the best of these is the Beattie-Bridgeman equation, which can be written (for 1 mol) as... 

Plot the Leimard-Jones potentials for each of the gases studied. Obtain ft from Eqs. (16)-(18) by numerioal integration and compare the values from this two-parameter potential with those from the van der Waals and Beattie-Bridgeman equations of state. Optional A simple square-well potential model can also be used to eradely represent the interaction of two molecules. In place of Eq. (18), use the square-well potential and parameters of Ref. 6 to ealeulate /t. Contrast with the results from the Lennard-Jones potential and comment on the sensitivity of the calculations to the form of the potential.]... 

If the constants in a satisfactory equation of state, e.g., the Beattie-Bridgeman equation, were known, d V/dT )p could be expressed analytically as a function of the pressure, and then integrated in accordance with equation (21.12). The treatment may be illustrated in a simple manner by utilizing the van der Waals... 

It is of interest to note that (d P/dT )v is zero for a van der Waals gas, as well as for an ideal gas hence, Cv should also be independent of the volume (or pressure) in the former case. In this event, the effect of pressure on Cp is equal to the variation of Cp — Cv with pressure. Comparison of equations (21.4) and (21.13), both of which are based on the van der Waals equation, shows this to be true. For a gas obeying the Berthelot equation or the Beattie-Bridgeman equation (d P/dT )v would not be zero, and hence some variation (f Cv with pressure is to be expected. It is probable, however, that this variation is small, and so for most purposes the heat capacity of any gas at constant volume may be regarded as being independent of the volume or pressure. The maximum in the ratio y of the heat capacities at constant pressure and volume, respectively, i.e., Cp/Cv, referred to earlier ( lOe), should thus occur at about the same pressure as that for Cp, at any temperature. 

Even though a and b are derived from actual P-V-T data, the fugacities obtained from equation (29.14) may not be too reliable over a range of pressures because of the approximate nature of the van der Waals equation. By using a more exact equation of state, such as the Beattie-Bridgeman equation, better values for the fugacities can be obtained, but since this equation involves five empirical constants, in addition to R, the treatment is somewhat more complicated than that given above. ... 

Lastly I would like to consider the Beattie-Bridgeman equation and its offspring the Benedict, Webb, and Rubin equation which start with the VDW equation and replace 1/(V — Nb) by (V + Nb)/V2. Several additional parameters are introduced. However, the above replacement is so meaningless at high densities that I cannot bring myself to comment further on this family of empirical equations. 

Using the Beattie-Bridgeman equation, calculate the molar volume of ammonia at 300 °C and 200 atm pressure. 

Compare the molar volume of carbon dioxide at 400 K and 100 atm calculated by the Beattie-Bridgeman equation with that calculated by the van der Waals equation. 

At 300 K, for what value of the molar volume will the contribution to the pV product of the term in 1/F in the Beattie-Bridgeman equation be equal to that of the term in 1/V (a) for oxygen (b) What value of pressure corresponds to this molar volume ... 

Using the same technique as that used to obtain Eq. (3.8), prove the relation given in Table 3.4 between y and y for the Beattie-Bridgeman equation. 

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