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PHY 113 C General Physics I 11 AM – 12:15 P M MWF Olin 101 Plan for Lecture 22: Chapter 21: Ideal gas equations Mol

PHY 113 C General Physics I 11 AM – 12:15 P M MWF Olin 101 Plan for Lecture 22: Chapter 21: Ideal gas equations Molecular view of ideal gas Internal energy of ideal gas Distribution of molecular speeds in ideal gas Adiabatic processes. From Webassign (Assignment #19)

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PHY 113 C General Physics I 11 AM – 12:15 P M MWF Olin 101 Plan for Lecture 22: Chapter 21: Ideal gas equations Mol

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  1. PHY 113 C General Physics I 11 AM – 12:15 PM MWF Olin 101 Plan for Lecture 22: Chapter 21: Ideal gas equations Molecular view of ideal gas Internal energy of ideal gas Distribution of molecular speeds in ideal gas Adiabatic processes PHY 113 C Fall 2013 -- Lecture 22

  2. PHY 113 C Fall 2013 -- Lecture 22

  3. From Webassign (Assignment #19) A combination of 0.250 kg of water at 20.0°C, 0.400 kg of aluminum at 26.0°C, and 0.100 kg of copper at 100°C is mixed in an insulated container and allowed to come to thermal equilibrium. Ignore any energy transfer to or from the container and determine the final temperature of the mixture. 4186 J/(kg*oC) 900 J/(kg*oC) 387 J/(kg*oC) (From Table 20.1) PHY 113 C Fall 2013 -- Lecture 22

  4. From Webassign (Assignment #19) A thermodynamic system undergoes a process in which its internal energy decreases by 465 J. Over the same time interval, 236 J of work is done on the system. Find the energy transferred from it by heat. Note: Sign convention for Q : Q>0  system gains heat from environment • iclicker question: • Assuming the system does not change phase, what can you say about TF versus TI for the system? • TF>TI • TF<TI PHY 113 C Fall 2013 -- Lecture 22

  5. From Webassign (Assignment #19) A 2.20-mol sample of helium gas initially at 300 K, and 0.400 atm is compressed isothermally to 1.80 atm. Note that the helium behaves as an ideal gas. (a) Find the final volume of the gas.(b) Find the work done on the gas.(c) Find the energy transferred by heat. PHY 113 C Fall 2013 -- Lecture 22

  6. From Webassign (Assignment #19) A 2.20-mol sample of helium gas initially at 300 K, and 0.400 atm is compressed isothermally to 1.80 atm. Note that the helium behaves as an ideal gas. (a) Find the final volume of the gas. PHY 113 C Fall 2013 -- Lecture 22

  7. From Webassign (Assignment #19) A 2.20-mol sample of helium gas initially at 300 K, and 0.400 atm is compressed isothermally to 1.80 atm. Note that the helium behaves as an ideal gas. (b) Find the work done on the gas.(c) Find the energy transferred by heat. PHY 113 C Fall 2013 -- Lecture 22

  8. From Webassign (Assignment #19) One mole of an ideal gas does 2 900 J of work on its surroundings as it expands isothermally to a final pressure of 1.00 atm and volume of 28.0 L. (a) Determine the initial volume of the gas.(b) Determine the temperature of the gas. PHY 113 C Fall 2013 -- Lecture 22

  9. From Webassign (Assignment #19) One mole of an ideal gas does 2 900 J of work on its surroundings as it expands isothermally to a final pressure of 1.00 atm and volume of 28.0 L. Determine the initial volume of the gas. Determine the temperature of the gas. PHY 113 C Fall 2013 -- Lecture 22

  10. From Webassign (Assignment #19) • In the figure, the change in internal energy of a gas that is taken from A to C along the blue path is +795 J. The work done on the gas along the red path ABC is -530 J. • (a) How much energy must be added to the system by heat as it goes from A through B to C?(b) If the pressure at point A is five times that of point C, what is the work done on the system in going from C to D?(c) What is the energy exchanged with the surroundings by heat as the gas goes from C to A along the green path?(d) If the change in internal energy in going from point D to point A is +495 J, how much energy must be added to the system by heat as it goes from point C to point D? PHY 113 C Fall 2013 -- Lecture 22

  11. Review: Consider the process described by ABCA • iclicker exercise: • What is the net work done on the system in this cycle? • -12000 J • 12000 J • 0 PHY 113 C Fall 2013 -- Lecture 22

  12. Equation of “state” for ideal gas (from experiment) 8.314 J/(mol K) temperature in K volume in m3 # of moles pressure in Pascals PHY 113 C Fall 2013 -- Lecture 22

  13. Ideal gas -- continued Note that at this point, the above equation for Eint is completely unjustified… PHY 113 C Fall 2013 -- Lecture 22

  14. From The New Yorker Magazine, November 2003 PHY 113 C Fall 2013 -- Lecture 22

  15. Microscopic model of ideal gas: Each atom is represented as a tiny hard sphere of mass m with velocity v. Collisions and forces between atoms are neglected. Collisions with the walls of the container are assumed to be elastic. PHY 113 C Fall 2013 -- Lecture 22

  16. x d What we can show is the pressure exerted by the atoms by their collisions with the walls of the container is given by: Proof: Force exerted on wall perpendicular to x-axis by an atom which collides with it: -vix vix number of atoms average over atoms volume PHY 113 C Fall 2013 -- Lecture 22

  17. PHY 113 C Fall 2013 -- Lecture 22

  18. iclicker question: • What should we call ? • Average kinetic energy of atom. • We cannot use our macroscopic equations at the atomic scale -- so this quantity will go unnamed. • We made too many approximations, so it is not worth naming/discussion. • Very boring. PHY 113 C Fall 2013 -- Lecture 22

  19. for mono atomic ideal gas PHY 113 C Fall 2013 -- Lecture 22

  20. Average atomic velocities: (note <vi>=0) Relationship between average atomic velocities with T PHY 113 C Fall 2013 -- Lecture 22

  21. Periodic table: http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg PHY 113 C Fall 2013 -- Lecture 22

  22. Periodic table: http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg Molecular mass PHY 113 C Fall 2013 -- Lecture 22

  23. Periodic table: http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg Molecular mass PHY 113 C Fall 2013 -- Lecture 22

  24. For monoatomic ideal gas: General form for ideal gas (including mono-, di-, poly-atomic ideal gases): PHY 113 C Fall 2013 -- Lecture 22

  25. Macroscopic Microscopic 8.314 J/mole oK 1.38 x 10-23 J/molecule oK PHY 113 C Fall 2013 -- Lecture 22

  26. Big leap! Internal energy of an ideal gas: derived for monoatomic ideal gas more general relation for polyatomic ideal gas PHY 113 C Fall 2013 -- Lecture 22

  27. Comment on “big leap” – case of diatomic molecule w vCM Note: We are assuming that molecular vibrations are not taking much energy PHY 113 C Fall 2013 -- Lecture 22

  28. Comment on “big leap” – continued Big leap! Internal energy of an ideal gas: derived for monoatomic ideal gas more general relation for polyatomic ideal gas • can be measured for each gaseous system • Note: g = CP/CV PHY 113 C Fall 2013 -- Lecture 22

  29. Determination of Q for various processes in an ideal gas: Example: Isovolumetric process – (V=constant  W=0) In terms of “heat capacity”: PHY 113 C Fall 2013 -- Lecture 22

  30. Example: Isobaric process (P=constant): In terms of “heat capacity”: Note: g = CP/CV PHY 113 C Fall 2013 -- Lecture 22

  31. Summary PHY 113 C Fall 2013 -- Lecture 22

  32. iclicker question: • The previous discussion • Made me appreciate the g factor in thermo analyses • Made me want to scream • Put me to sleep • No problem – as long as this is not on the test PHY 113 C Fall 2013 -- Lecture 22

  33. More examples: Isothermal process (T=0) DT=0  DEint = 0  Q=-W PHY 113 C Fall 2013 -- Lecture 22

  34. Even more examples: Adiabatic process (Q=0) PHY 113 C Fall 2013 -- Lecture 22

  35. PHY 113 C Fall 2013 -- Lecture 22

  36. iclicker question: Suppose that an ideal gas expands adiabatically. Does the temperature (A) Increase (B) Decrease (C) Remain the same PHY 113 C Fall 2013 -- Lecture 22

  37. Review of results from ideal gas analysis in terms of the specific heat ratio gº CP/CV: For an isothermal process, DEint = 0  Q=-W For an adiabatic process, Q = 0 PHY 113 C Fall 2013 -- Lecture 22

  38. Note: It can be shown that the work done by an ideal gas which has an initial pressure Pi and initial volume Vi when it expands adiabatically to a volume Vf is given by: PHY 113 C Fall 2013 -- Lecture 22

  39. Pf B C P (1.013 x 105) Pa A D Pi Vi Vf Examples process by an ideal gas: Efficiency as an engine: e = |Wnet/ |/Qinput PHY 113 C Fall 2013 -- Lecture 22

  40. From Webassign (#19) An ideal gas initially at Pi, Vi, and Ti is taken through a cycle as shown below. (Let the factor n = 2.6.) (a) Find the net work done on the gas per cycle for 2.60 mol of gas initially at 0°C.(b) What is the net energy added by heat to the system per cycle? PHY 113 C Fall 2013 -- Lecture 22

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