Question

For any gas whose internal energy depends on temperature only, the relation between pressure p, molar volume v, and absolute temperature T can be approximated by using the Ideal Gas law: ??=?? where R is the universal gas constant. If p is measured in atmospheres, v in liters/mole, and T in Kelvin, then R=0.08205 ????? ?????? ?.
However, for certain physical conditions (notably high T and/or high p), gases may behave in a nonideal way. A more general description of their behavior is given by the following higher-order approximation, known as the Beattie-Bridgeman Equation: ?=???+??2+??3+??4,
where ?,?,and ? are temperature dependent quantities given as follows: ?=?0??−?0−???2 ; ?=−?0???+?0?−?0???2 ; ?=?0????2.
Constants ?0,?0,?,?,and ? have been determined by fitting curves to experimental data. For methane (CH4), the values of the constants (in units consistent with the ones above) are: ?0=2.2769; ?0=0.05587; ?=0.01855; ?=−0.01857; and ?−12.81?4.
a. (24 points) Write a program that, when given T in kelvin and p in atmospheres, uses Muller’s Method to solve the Beattie-Bridgeman Equation for the molar volume of CH4. You are not required use complex variables, but you should include a section in your program that checks to see whether ??+1 is complex, and, if it is, stops the method. Close initial guesses chosen in part b will allow you to find the root without venturing into the complex plane. Try to reduce the number of function evaluations.
Before the iteration begins, your program should print the starting estimates ?−2,?−1,?0 and their corresponding ?(?) values. At each iteration, your program should print out the iteration number (?),??,?(??), and the quantity |??−??−1||??|. Implement a convergence criterion of the form: |??−??−1||??|≤?,?ℎ??? ?=10−5
When your program finishes running, it should either print out the converged solution and the number of iterations required, or else it should print out a statement that a solution could not be found (perhaps because the method found a complex ??+1.
b. (8 points) Run your program for the following data (note that there are 10 possible combinations): ?=500 and 1500 ?,with ?=1,5,10,50 and 100 atm. In each case, choose your own starting points (perhaps using the Ideal EoS).

Answer 1