# 11/14/2013PHY 113 C Fall 2013 -- Lecture 221 PHY 113 C General Physics I 11 AM – 12:15 PM MWF Olin 101 Plan for Lecture 22: Chapter 21: Ideal gas equations.

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11/14/2013PHY 113 C Fall 2013 -- Lecture 221 PHY 113 C General Physics I 11 AM – 12:15 PM MWF Olin 101 Plan for Lecture 22: Chapter 21: Ideal gas equations 1.Molecular view of ideal gas 2.Internal energy of ideal gas 3.Distribution of molecular speeds in ideal gas 4.Adiabatic processes

11/14/2013 PHY 113 C Fall 2013 -- Lecture 222

11/14/2013PHY 113 C Fall 2013 -- Lecture 223 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. 387 J/(kg* o C) 4186 J/(kg* o C) 900 J/(kg* o C) (From Table 20.1)

11/14/2013PHY 113 C Fall 2013 -- Lecture 224 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 T F versus T I for the system? A.T F >T I B.T F { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/7/1710597/slides/slide_4.jpg", "name": "11/14/2013PHY 113 C Fall 2013 -- Lecture 224 From Webassign (Assignment #19) A thermodynamic system undergoes a process in which its internal energy decreases by 465 J.", "description": "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 T F versus T I for the system. A.T F >T I B.T F

11/14/2013PHY 113 C Fall 2013 -- Lecture 225 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.

11/14/2013PHY 113 C Fall 2013 -- Lecture 226 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.

11/14/2013PHY 113 C Fall 2013 -- Lecture 227 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.

11/14/2013PHY 113 C Fall 2013 -- Lecture 228 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.

11/14/2013PHY 113 C Fall 2013 -- Lecture 229 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.

11/14/2013PHY 113 C Fall 2013 -- Lecture 2210 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?

11/14/2013PHY 113 C Fall 2013 -- Lecture 2211 Review: Consider the process described by A B C A iclicker exercise: What is the net work done on the system in this cycle? A.-12000 J B.12000 J C.0

11/14/2013PHY 113 C Fall 2013 -- Lecture 2212 Equation of state for ideal gas (from experiment) pressure in Pascals volume in m 3 # of moles temperature in K 8.314 J/(mol K)

11/14/2013PHY 113 C Fall 2013 -- Lecture 2213 Ideal gas -- continued Note that at this point, the above equation for E int is completely unjustified…

11/14/2013PHY 113 C Fall 2013 -- Lecture 2214 From The New Yorker Magazine, November 2003

11/14/2013PHY 113 C Fall 2013 -- Lecture 2215 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.

11/14/2013PHY 113 C Fall 2013 -- Lecture 2216 Proof: Force exerted on wall perpendicular to x-axis by an atom which collides with it: average over atoms What we can show is the pressure exerted by the atoms by their collisions with the walls of the container is given by: d x v ix -v ix number of atoms volume

11/14/2013PHY 113 C Fall 2013 -- Lecture 2217

11/14/2013PHY 113 C Fall 2013 -- Lecture 2218 iclicker question: What should we call ? A.Average kinetic energy of atom. B.We cannot use our macroscopic equations at the atomic scale -- so this quantity will go unnamed. C.We made too many approximations, so it is not worth naming/discussion. D.Very boring.

11/14/2013PHY 113 C Fall 2013 -- Lecture 2219 for mono atomic ideal gas

11/14/2013PHY 113 C Fall 2013 -- Lecture 2220 Average atomic velocities: (note =0) Relationship between average atomic velocities with T

11/14/2013PHY 113 C Fall 2013 -- Lecture 2221 Periodic table: http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg

11/14/2013PHY 113 C Fall 2013 -- Lecture 2222 Periodic table: http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg Molecular mass

11/14/2013PHY 113 C Fall 2013 -- Lecture 2223 Periodic table: http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg Molecular mass

11/14/2013PHY 113 C Fall 2013 -- Lecture 2224 For monoatomic ideal gas: General form for ideal gas (including mono-, di-, poly- atomic ideal gases):

11/14/2013PHY 113 C Fall 2013 -- Lecture 2225 Macroscopic Microscopic 8.314 J/mole o K1.38 x 10 -23 J/molecule o K

11/14/2013PHY 113 C Fall 2013 -- Lecture 2226 Internal energy of an ideal gas: derived for monoatomic ideal gas more general relation for polyatomic ideal gas Gas (theory) exp) He5/31.67 N2N2 7/51.41 H2OH2O4/31.30 Big leap!

11/14/2013PHY 113 C Fall 2013 -- Lecture 2227 Comment on big leap – case of diatomic molecule v CM Note: We are assuming that molecular vibrations are not taking much energy

11/14/2013PHY 113 C Fall 2013 -- Lecture 2228 Comment on big leap – continued Internal energy of an ideal gas: derived for monoatomic ideal gas more general relation for polyatomic ideal gas Big leap! can be measured for each gaseous system Note: = C P /C V

11/14/2013PHY 113 C Fall 2013 -- Lecture 2229 Determination of Q for various processes in an ideal gas: Example: Isovolumetric process – (V=constant W=0) In terms of heat capacity:

11/14/2013PHY 113 C Fall 2013 -- Lecture 2230 Example: Isobaric process (P=constant): In terms of heat capacity: Note: = C P /C V

11/14/2013PHY 113 C Fall 2013 -- Lecture 2231 Summary

11/14/2013PHY 113 C Fall 2013 -- Lecture 2232 iclicker question: The previous discussion A.Made me appreciate the factor in thermo analyses B.Made me want to scream C.Put me to sleep D.No problem – as long as this is not on the test

11/14/2013PHY 113 C Fall 2013 -- Lecture 2233 More examples: Isothermal process (T=0) T=0 E int = 0 Q=-W

11/14/2013PHY 113 C Fall 2013 -- Lecture 2234 Even more examples: Adiabatic process (Q=0)

11/14/2013PHY 113 C Fall 2013 -- Lecture 2235

11/14/2013PHY 113 C Fall 2013 -- Lecture 2236 iclicker question: Suppose that an ideal gas expands adiabatically. Does the temperature (A) Increase (B) Decrease (C) Remain the same

11/14/2013PHY 113 C Fall 2013 -- Lecture 2237 Review of results from ideal gas analysis in terms of the specific heat ratio C P /C V : For an isothermal process, E int = 0 Q=-W For an adiabatic process, Q = 0

11/14/2013PHY 113 C Fall 2013 -- Lecture 2238 Note: It can be shown that the work done by an ideal gas which has an initial pressure P i and initial volume V i when it expands adiabatically to a volume V f is given by:

11/14/2013PHY 113 C Fall 2013 -- Lecture 2239 P (1.013 x 10 5 ) Pa ViVi VfVf PiPi PfPf A BC D Examples process by an ideal gas: A BB CC DD A Q W0-P f (V f -V i )0P i (V f -V i ) E int Efficiency as an engine: e = |W net/ |/Q input

11/14/2013PHY 113 C Fall 2013 -- Lecture 2240 From Webassign (#19) An ideal gas initially at P i, V i, and T i 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?

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