Chemical Thermodynamics
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CHEMICAL THERMODYNAMICS Handbook of Exercises Valentim Maria Brunheta Nunes 2013 Chemical Thermodynamics _____________________________________________________________________________________ "A theory is the more impressive the greater the simplicity of its premises, the more varied the kinds of things that it relates and the more extended the area of its applicability. Therefore classical thermodynamics has made a deep impression on me. It is the only physical theory of universal content which I am convinced, within the areas of the applicability of its basic concepts, will never be overthrown." -- Einstein (1949) INTRODUCTION Thermodynamics it’s a discipline that is very important for many engineering degree programs like Chemical and Biochemical Engineering, Environment Engineering or Mechanical Engineering. With the Chemical Thermodynamics course we intend to introduce the principles of thermodynamics, and apply them to systems, that are solids, liquids or gases, with an interest in chemical engineering, don’t forgetting environmental issues. This course is also fundamental in the development of important calculation techniques in engineering. This exercise book will serve to accomplish the lectures, and the problems presented seek to encompass the entire program taught to prepare students for the final evaluations. The resolution of examinations of previous academic years may be also quite helpful for the students. Handbook of Exercises 2 _________________________________________________________________________________ Chemical Thermodynamics _____________________________________________________________________________________ 1st Series of Exercises - Gaseous State 1. Calculate the volume occupied by 3 moles of a perfect gas at 2 bar and 350 K. 2. Calculate the final pressure when 1 mole of nitrogen at 300 K and 100 atm is heated, at constant volume, until attaining 500 K. 3. The mass percentage of dry air at sea level is approximately: N2, 75%; O2, 23.2% and Ar, 1.3%. What are the partial pressures of each component when the total pressure is 1 atm? 4. Calculate the pressure exerted by 1 mole of CO 2 behaving as (a) perfect gas and (b) van der Waals gas, when it is confined in the following conditions: T = 273.15 K and V = 22.414 L (constants of the van der Waals equation: a = 3.592 atm L 2mol -2 and b = 4.267 ×10 -2 Lmol -1) 5. Calculate the molar volume of nitrogen at 100 °C and 30.5 atm using: 5.1. The perfect gas equation. 5.2. The van der Waals equation (a = 1.35 dm 6.atm.mol -2; b = 38.6 ×10 -3 dm 3.mol -1) 5.3. The virial equation: Z = 1 – 5.3 ×10 -4 p + 4.8 ×10 -6 p2, with p in atm. 6. At 300 K and 20 atm, the compressibility factor of a given gas is 0.86. Calculate (a) the volume occupied by 8.2 mmol of gas (b) the approximate value of the second virial coefficient at 300 K. Handbook of Exercises 3 _________________________________________________________________________________ Chemical Thermodynamics _____________________________________________________________________________________ 7. Calculate the pressure exerted by 1 mole of ethane (C 2H6) behaving as: (a) perfect gas; (b) van der Waals gas, when confined in the following conditions: i) 273.15 K and V = 22.414 L; ii) 1000 K and 100 cm3. Data: a = 5.489 L 2.atm.mol -2 and b = 6.380 ×10 -2 L.mol -1. 8. Suggest the pressure and temperature for 1 mole of He to be in corresponding states with 1 mole of H2 at 1 atm and 25 °C. Data: Pc(H 2) = 12.8 atm; Tc(H 2) = 33.23 K; Pc(He) = 2.26 atm; Tc(He) = 5.2 K. 3 -1 9. The critical constants of methane are pc = 45.6 atm, V c = 98.7 cm .mol and Tc = 190.6 K. Calculate the parameters of the van der Waals equation for this gas and make an estimative of the molecules radius (considered as spheres). 10. A sample of 87 mg of an ideal gas at 0.600 bar duplicates its volume and triplicates its temperature. What will be the final pressure? 11. Two balloons of equal volume and in vacuum are connected by a tube with negligible volume. The first one it’s in a thermostatic bath at 200 K and the other in a bath at 300 K; then it is injected in the system 1 mole of an ideal gas. What will be the final number of moles in each balloon? 12. One compound made of nitrogen and oxygen as the following variation of density with pressure at 0 °C: p / atm 0.333 0.500 0.667 1.000 (ρ/p)/g.L-1 .atm -1 1.9694 1.9722 1.9746 1.9804 Handbook of Exercises 4 _________________________________________________________________________________ Chemical Thermodynamics _____________________________________________________________________________________ 12.1. Calculate the molar mass of the compound and suggest the correspondent molecular formula. 12.2. Calculate the 2 nd virial coefficient at 0 °C. 13. Deduce the relation between pressure and density for a perfect gas with molar mass M. Confirm graphically that dimethyl ether at 25 °C behaves like a perfect gas at low pressures and calculate the correspondent molar mass of gas. p/torr 91.74 188.98 273 .3 452.8 639.3 760 ρ/ g.L -1 0.225 0.456 0.664 1.062 1.468 1.734 2nd Series of Exercises – The First Law of Thermodynamics 14. Two moles of an ideal gas at 500 K are isothermally and reversibly compressed until a final volume that is 1/10 of the initial volume. Calculate: (a) ∆U (b) ∆H (c) work done by the gas (d) heat absorbed by the gas. -1 -1 15. Two moles of an ideal gas for which Cp = 40 JK mol are heated from 300 to 400 K, (a) at constant pressure and (b) at constant volume. Calculate in each case (i) ∆U, (ii) ∆H, (iii) work done by the gas and (iv) heat absorbed by the gas. 16. One mole of an ideal gas expands from 10 L and 0 °C to 20 L and 100 °C. -1 -1 Taking CV = 20 JK mol , calculate ∆U, W and Q for each of the Handbook of Exercises 5 _________________________________________________________________________________ Chemical Thermodynamics _____________________________________________________________________________________ following alternative steps for the whole process: (a) Isothermic and reversible expansion at 0 °C from 10 L to 20 L, followed by a constant volume heating until 100 °C and (b) Heating of 10 L at constant volume until 100 °C, followed by an isothermic and reversible expansion until 20 L. 17. A given quantity of an ideal gas occupies 1 dm 3 at 1 atm and 300 K. It is subject to the following transformations: (a) Heated at constant pressure until attaining the volume of 2 dm 3. (b) Heated at constant volume of 2 dm 3 until the pressure attains 2 atm. (c) Cooled at constant pressure until the volume decreases to 1 dm 3. (d) Cooled at constant volume until the pressure attains 1 atm. Calculate the sum for the 4 steps for (i) ∆U, (ii) ∆H, (iii) Q e (iv) W 18. A sample of 4.5 g of methane occupies 12.7 L at 310 K. Calculate the work done by the gas when it expands isothermally against an external constant pressure of 200 torr until the volume increases 3.3 L. Calculate the work if the expansion was reversible. 19. Two moles of a gas suffers a reversible and isothermic expansion at 300 K from 1 dm 3 to 10 dm 3. Calculate the work considering (a) a perfect gas (b) the gas obeys the van der Waals equation, with a = 1.36 atm dm 6 mol -2 e b = 3.183 ×10 -2 dm 3mol -1. 20. Write an expression for d V, considering V as a function of p and T. Deduce an expression for d lnV in terms of the isobaric thermal expansion coefficient and the isothermic compressibility coefficient. Handbook of Exercises 6 _________________________________________________________________________________ Chemical Thermodynamics _____________________________________________________________________________________ 21. In order to project a given refrigerator we need to know the decrease of temperature due to the adiabatic expansion of a refrigerant gas. -1 For a given Freon, µµµJT = 1.2 K atm . What is the decrease of pressure necessary to produce a decrease of temperature of 5.0 K? 22. Calculate the amount of heat we have to furnish to 500 g of O2 in order to increase the room temperature from 298 K to 500 K, at constant pressure, knowing that: -3 5 2 -1 -1 Cp (O 2) = 7.16 + 1 ×10 T – 0.40 ×10 /T cal K mol 23. Consider one mole of nitrogen (Cp = 29.4 JK -1mol -1) at 300 K and 1 atm that suffers a reversible adiabatic compression until 400 K. Calculate: (a) The final pressure (b) Work, W, heat, Q and the internal energy change of the gas. 24. The following figure shows two reversible processes 1-2-3 and 1-4-3 in which a mole of Freon changes from state T1, p 1 to the state T2,p 3. The p 1 2 T1 4 T2 3 V curves T1 and T2 are isotherms, and the vertical lines are isochoric (V constant). Show that the heat involved in the two processes is deferent, assuming that the Freon is a perfect gas. Handbook of Exercises 7 _________________________________________________________________________________ Chemical Thermodynamics _____________________________________________________________________________________ 25. Consider a monoatomic perfect gas (CV = 3/2 R) that undergoes the following transformation: P/bar B 2 A C 1 273 546 T/K 25.1. From the p,T diagram obtain the correspondent p,V diagram 25.2. Complete the next table: A B C cycle ∆U Q W 26. One kmol of perfect gas undergoes a cyclic process with four steps as shown in the p,V diagram.