of Department Chemical Engineering & Materials Science University of California, Davis
ECM6: Homework # 1
Problem 1
The van der Waals equation of state (VDW EOS) is derived from the ideal gas law by adding two additional terms to the ideal gas law. The first term accounts for intermolecular attraction and is of the form a V 2 , where a is an empirical constant for each substance. The second term accounts for the fact that molecules have a finite volume and therefore there is a volume below which a system can never be compressed. The result is RT a P = - (1) V-b V2 where the constant b is specific for each substance. The constants can be determined at the critical point where the density of the liquid and gas phase are the same. At the critical point T = TC , P = PC and V = VC the following conditions hold P i 2 P y J N = 0, j z = 0 j 2 z V TC k V { TC (2)
Use Mathematica to determine the constants a and b in terms of TC and PC at the critical point. Note that at the critical point (2) holds. Hint: Use the Mathematica function Solve to find a and b
Problem 2
(i) Calculate the molar volume and compressibility factor defined as PV Z (3) RT for gaseous ammonia at a pressure of P=56 atm and a temperature of T=450 K using the van der Waals equation of state (Problem 1) . The critical parameters for ammonia are The gas constant R = 0.08206 atm L mol K. (ii) Repeat the calculation in (i) for the following reduced pressures Pr = 2, 4, 6, 8, 10, the where reduced pressure is defined as (4) P Pr = PC (5) TC = 405.5 K, PC = 111.3 atm
Present your results in a table that has the following columns P(atm), Pr , V , Z for the pressures listed above.
2
ECM6HW1_07.nb
Problem 3
Consider the following function 1 g HxL = (7) x5 Ex2 - 1 (i) Plot this function over the range 0 x 2. Not that E is the exponential constant e (ii) Can you find the value of x where the function is underfined. Hint: Explore the use of the function FindRoot. (iii) Find a Taylor series expansion of the the function about x=0 and also about x=0.95. Take terms up to order x10 . (iv) If g10 Hx - x0 ) represents the Taylor series expansion about x = x0 found in (iii), determine the error in the series at x=0.2 and , x=1.02, were the error is defined as How many terms in the series is needed such that the error in the function represented by the series at x=1.02 satifies error H1.02L < 10-6 (9) g10 Hx - x0 L - g HxL error HxL = g HxL (8)
Problem 4
In a calculus text book the following function was defined
H-1L x f(p)= ,, n p n=0
m=1 n
where p is an integer. (i) Use Mathematica to write a function that evaluates the sum for arbitrary values of p. Display the result when p=10. (ii) Use Table to generate a list of partial sums for p=1, 2,...10. Then use Plot to display the partial sums for 0 x 10. Be sure to wrap the first argument of Plot with Evaluate. See what happens if you do not use evaluate.
H2 mL2
2n
(10)