NEED #2 SOLVED. THE REDLICH-KWONG EQUATION OF STATE EQUATION IS AT THE VERY TOP OF THE IMAGE.

P=VbRTV(V+b)a/TZ=1+ZBBZ+BA The second form (in terms of ZPV/RT ) is easier to use numerically for obtaining a solution when (T,P) are specified. This equation of state is capable of predicting a liquid phase (small V ) as well as a vapor phase (higher V ) for temperatures and pressures that are below the critical point. The parameters are determined as b=0.08664RTc/Pca=0.42747R2Tc2.5/PcB=bP/RTA=aP/R2T2.5 The parameters A and B combine a and b with the desired (T,P) in order to make the equation in Z easier to use. The labels Tc and Pc are the temperature and pressure of the critical point for the substance being considered. These are tabulated in many sources, including the books by Felder and Rousseau and by Dahm and Visco. The critical properties of isobutane are Tc=407.85K and Pc=36.40 bar. The gas constant in these units equals 83.14cm3bar/molK. 1. Use the Newton-Raphson method to solve for the molar volumes (in cm3/mol ) of isobutane vapor at the condition (T=350K,P=5 bar). NOTE: I encourage you to use the form of the equation of state in terms of Z, rather than P. You can use P, T, and the critical properties to determine A and B. Then solve for Z by root finding. Then determine V from V=ZRT/P. I know this seems like more work, but it is easier in the long run. Really. SUGGESTION: for an ideal gas, Z=1. This can be a useful first guess for determining the molar volume of a real gas. 2. This problem continues with use of the Redlich-Kwong equation of state. Use the Newton-Raphson method to solve for the molar volumes (in cm3/mol ) of isobutane liquid at the condition ( T=300K,P=30bar). SUGGESTION: I especially encourage you to use the Z approach for a liquid. Note that the minimum physical value of Z is slightly greater than B. Thus Z=(B+ a little) is a good initial guess for determining a liquid phase molar volume. The size of nanovoids in a material (such as those between neighboring atoms) can be measured using a technique called positron annihilation. A positron - an antimatter particle that resembles an electron but has a positive charge is emitted during radioactive decay of some atoms, such as 22Na. An experiment can be designed such that some positrons then extract an electron from the material of interest, forming a hydrogen atom-type complex called positronium. This pseudo-atom exists for a short time, 3 nanoseconds (1ns109s), before it collides with an electron of a neighboring atom and is annihilated in a photon of gamma radiation. With good electronics and appropriate data analysis, both the positron and subsequent gamma ray emissions can be measured in time and thus 3 can be determined. S. J. Tao, in the paper "Positronium Annihilation in Molecular Substances" (Journal of Chemical Physics, 1972, 56, 5499-5510), performed a quantum mechanical analysis and obtained an equation 3=[1R+1.656R+21sin(R+1.6562R)]1/2 that relates 3 (in ns) to the radius R (in ngstroms, 1A1010m=0.1nm ) of the cavity in which the positronium resides. (He assumed that the cavity is a sphere, so R is the radius of a sphere that is equivalent in some sense to the original volume.) It is not possible to solve for R analytically, given a measured value of 3. An experiment (Lind et al., J. Polym. Sci. A: Polymer Chemistry Edition, 1986, 24, 3033-3047) found the following positronium lifetimes in annealed isotactic polypropylene