CBE 141- Chemical Engineering Thermodynamics

Spring 2015

Homework Set # 1

Deadline: Friday 6th February, 2015 (In class by 1:10 pm)

Strict adherence to these rules:

1) Solve each question on a new page. 2) DO NOT staple your problems into one document; instead, make sure your name is on

each page. 3) No homework regrades. 4) Show all work and cite all sources/references (even if from the textbook, name the table).

Problem 1.

Two kmol of nitrogen gas are compressed isothermally and quasi-statically from an initial pressure Pi to a final pressure Pf (Pi < Pf). Derive an expression for, and calculate, the work done on the gas during the compression if it follows the equation of state

(a) PV RT

(b)

a TRT

PV b V V b b V b

Note that V is the molar volume in both cases. T =298K, Pi = 100 kPa and Pf = 500 kPa. Make a sketch of the system and label it. For (b), you are dealing with the Peng-Robinson equation of state, so keep in mind that you will have to solve a cubic equation and find its roots in order to calculate the work done. (Hint: Convert the equation to its Z form before proceeding to solve it.) Compare your answers for part (a) and (b) – are they similar or quite different? Why?

For this homework set, show all steps involved in this calculation even if you choose to use a program to check your answer.

Problem 2.

(a) Derive an expression for the isothermal compressibility of gas using the van der Waals (vdW) equation of state.

(b) What happens to this expression in the limit of infinite volume?

(c) You are trying to explain a PV diagram that depicts vdW isotherm for a particular chemical to another student in the class. You notice that the behavior within the vapor-liquid phase envelope is not typical and in fact, it may not make intuitive sense to you. Sketch (qualitatively) a typical vdW isotherm. Then show on the sketch, for a given pressure under the critical pressure, the roots of the equation at a given isotherm temperature. Which of these roots make physical sense and which do not? Explain why.

Problem 3.

CO2 can be captured from the flue gas of coal-fired power plants and stored in underground formations. This “carbon capture and sequestration” technology is a strategy to mitigate the emission of greenhouse gases to the atmosphere. Consider CO2 transported in a pipeline at 150 bar and 20 ̊C, and injected into a formation at 30 bar.

(a) When the CO2 is depressurized, its temperature falls to -15 ̊C. What is the percentage change in the molar volume before and after depressurization? State which correlation you will be using and provide justification.

(b) Draw the depressurization process on a PV diagram. Draw the initial and final isotherms and label the molar volume, pressure, and phase before and after depressurization. Diagram need not be to scale. (Hint: You may want to look up VC for CO2.)

(c) A concern is that the CO2 cools upon depressurization, leading to the formation of hydrates (also called clathrates) that could block the borehole. Is there a risk of hydrate formation? If so, what strategy would you suggest to avoid this undesirable reaction?

Remember to state where any additional information comes from for parts (b) and (c).

Problem 4. Use the van der Waals, the Soave-Redlich-Kwong, and the Peng-Robinson equations to determine the molar volumes of the saturated phases of ethylene at 260 K (the saturation pressure is 30.35 bar). Compare with the tabulated values in the literature (below) by calculating the error for each phase for each equation of state. VL = 0.0713 L/mol VV = 0.456 L/mol