Physics 141. Lecture 24.

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Physics 141. Lecture 24. 0.5 µm particles in water, 50/50 glycerol-water, 75/25 glycerol-water, glycerol http://www.physics.emory.

1 Physics 141. Lecture µm particles in water, 50/50 glycerol-water, 75/25 glycerol-water, glycerol Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 1

2 December 5 th. An important day in the Netherlands. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 2

3 Physics 141. Lecture 24. Course Information. Quiz Continue our discussion of Chapter 13: The ideal gas law. The energy distribution of an ideal gas and energy exchange with its environment. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 3

4 Physics 141. Course information. Homework 10 is due on Friday December 2 at noon. Homework set 11 is due on Tuesday December 13 at noon. Exam # 3 will take place on Tuesday December 6 at 8 am in Hoyt. This exam will cover the material discussed in Chapters 10, 11, and 12 and equilibrium. Let s look at the timeline for lab # 5. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 4

5 Analysis of experiment # 5. Timeline (more details during next lectures). 11/14: collisions in the May room 11/20: analysis files available. 11/21: each student determines his/her best estimate of the velocities before and after the collisions (analysis during regular lab periods). 11/24: complete discussion and comparison of results with colliding partners and submit final results (velocities and errors) to professor Wolfs. 11/27: professor Wolfs compiles results, determines momenta and kinetic energies, and distributes the results. 11/ /5: office hours by lab TA/TIs to help with analysis and conclusions. 12/9: students submit lab report # 5. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 5

6 The Personal Response System (PRS). Quiz Lecture 24. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 6

7 The kinetic theory of gases. Thermodynamic variables. The volume of a gas is defined by the size of the enclosure of the gas. During a change in the state of a gas, the volume may or may not remain constant (this depends on the procedure followed). The temperature of a gas has been defined in terms of the entropy of the system (see discussion in Chapter 11). The pressure of a gas is defined as the force per unit area. The SI unit is pressure is the Pascal: 1 Pa = 1 N/m 2. Another common unit is the atm (atmospheric pressure) which is the pressure exerted by the atmosphere on us (1 atm = x 10 5 N/m 2 ). Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 7

8 A quick review: The equation of state of a gas. In order to specify the state of a gas, we need to measure its temperature, its volume, and its pressure. The relation between these variables and the mass of the gas is called the equation of state. The equation of state of a gas was initially obtained on the basis of observations. Boyle s Law ( ): pv = constant for gases maintained at constant temperature. Charle s Law ( ): V/T = constant for gases maintained at constant pressure. Gay-Lussac s Law ( ): p/t = constant for gases maintained at constant volume Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 8

9 The equation of state of a gas. Combining the various gas laws we can obtain a single more general relation between pressure, temperature, and volume: pv = constant T. Another observation that needs to be included is the dependence on the amount of gas: if pressure and temperature are kept constant, the volume is proportional to the mass m: pv = constant mt. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 9

10 The equation of state of a gas. The equation of state of a gas can be written as where p = pressure (in Pa). V = volume (in m 3 ). pv = NkT N = number of molecules of gas (1 mole = 6.02 x molecules or atoms). Note the number of molecules in a mole is also known as Avogadro s number N A. T = temperature (in K). Note: the equation of state is the equation of state of an ideal gas. Gases at very high pressure and/or close to the freezing point show deviations from the ideal gas law. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 10

11 The molecular point of view of a gas. Consider a gas contained in a container. The molecules in the gas will continuously collide with the walls of the vessel. Each time a molecule collides with the wall, it will carry out an elastic collision. Since the linear momentum of the molecule is changed, the linear momentum of the wall will change too. Since force is equal to the change in linear momentum per unit time, the gas will exert a force on the walls. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 11

12 The molecular point of view of a gas. Consider the collision of a single molecule with the left wall. In this collision, the linear momentum of the molecule changes by mv x - (-mv x ) = 2mv x. The same molecule will collide with this wall again after a time 2l/v x. The force that this single molecule exerts on the left wall is thus equal to Δp/Δt = (2mv x )/(2l/v x ) = mv x2 /l Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 12

13 The molecular point of view of a gas. The force that this single molecule exerts on the left wall is thus equal to F left = mv x2 /l If the pressure exerted on the left wall by this molecule is equal to p left = F left /A = mv x2 /(la) where A is the area of the left wall. The volume of the gas is equal to la and we can thus rewrite the pressure on the left wall: p left = mv x2 /V Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 13

14 The molecular point of view of a gas. The pressure that many molecules exerts on the left wall is equal to p left = m(v 1x2 +v 2x2 +v 3x )/V This equation can be rewritten in terms of the average of the square of the x component of the molecular velocity and the number of molecules (N): p left = mn(v x2 ) average /V Assuming that there is no preferential direction, the average square of the x, y, and z components of the molecular velocity will be the same: (v x2 ) average = (v y2 ) average = (v z2 ) average Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 14

15 The molecular point of view of a gas. The force on the left wall can be rewritten in terms of the average squared velocity p left = mn(v 2 ) average /3V Assuming there is no preferential direction of motion of the molecules, the pressure on all walls will be the same and we thus conclude: pv = mn(v 2 ) average /3 Compare this to the ideal gas law: pv = NkT K average = (1/2)m(v 2 ) average = (3/2) kt Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 15

Assume elastic collisions between the gas molecules and the walls.

16 Simulating an ideal gas. Ideal gas simulations: Assume elastic collisions between the gas molecules. Assume elastic collisions between the gas molecules and the walls. Results agree very well with measured values. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 16

Ask me a quick question. Enjoy the fantastic music.

17 2 Minute 48 Second Intermission. Since paying attention for 1 hour and 15 minutes is hard when the topic is physics, let s take a 2 minute 48 second intermission. You can: Stretch out. Talk to your neighbors. Ask me a quick question. Enjoy the fantastic music. Solve a WeBWorK problem. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 17

18 The real equation of state. Different points of view. Ideal gas law Critical point Note: 1. T > T c : gas 2. T < T c : liquid and/or vapor. Different temperatures Critical temperature Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 18

19 The real equation of state. Different points of view. Note the curvature of the solid-liquid line. Curvature to the left implies expansion on cooling. Direct change from solid to vapor. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 19

20 The real equation of state. Different points of view. Note the curvature of the solid-liquid line. Curvature to the left implies expansion on cooling. Water CO 2 Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 20

21 The first law of thermodynamics. Adding/removing heat from a system. Consider a closed system: Closed system No change in mass Change in energy allowed (exchange with environment) Isolated system: Closed system that does not allow an exchange of energy The internal energy of the system can change and will be equal to the heat added tot he system minus the work done by the system: ΔU = Q - W (note: this is the work-energy theorem). Note: keep track of the signs: Heat: Q > 0 J means heat added, Q < 0 J means heat lost Work: W > 0 J mean work done by the system, W < 0 J means work done on the system Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 21

The first law of thermodynamics. Isothermal processes. An isothermal process is a process in which the temperature of the system is kept constant.

22 The first law of thermodynamics. Isothermal processes. An isothermal process is a process in which the temperature of the system is kept constant. This can be done by keeping the system in contact with a large heat reservoir and making all changes slowly. Since the temperature of the system is constant, the internal energy of the system is constant: ΔU = 0 J. The first law of thermodynamics thus tells us that Q = W. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 22

23 The first law of thermodynamics. Adiabatic processes. An adiabatic process is a process in which there is no flow of heat (the system is an isolated system). Adiabatic processes can also occur in non-isolated systems, if the change in state is carried out rapidly. A rapid change in the state of the system does not allow sufficient time for heat flow. The expansion of gases differs greatly depending on the process that is followed (see Figure). Q = 0 J ΔU = 0 J Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 23

24 Work done during expansion/compression. Consider an ideal gas at pressure p. The gas exerts a force F on a moveable piston, and F = pa. If the piston moves a distance dl, the gas will do work: dw = Fdl Note: F and dl are parallel. The work done can be expressed in terms of the pressure and volume of the gas: dw = padl = pdv Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 24

25 Work done during expansion/compression. Isobaric and isochoric processes. Isobaric process: Processes in which the pressure is kept constant. W A->B = pdv = p A (V B - V A ) Isochoric process: Processes in which the volume is kept constant. W A->B = p A (V B - V A ) = 0 Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 25

26 Work done during expansion/compression. Isothermal process. Isothermal process: p = NkT V The work done during the change from state A to state B is V B W = pdv V A = NkT = NkT ln V B V A V B V A 1 V dv Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 26

27 Work done during expansion/compression. The work done during the expansion of a gas is equal to the area under the pv curve. Since the shape of the pv curve depends on the nature of the expansion, so does the work done: Isothermal: W = NkT ln(v B /V A ) Isochoric: W = 0 Isobaric: W = p B (V B - V A ) The work done to move state A to state B can take on any value! Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 27

Done for today! http://www.chinfo.navy.mil/navpalib/images/sndbarphoto.html Frank L. H.

28 Done for today! Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 28

Physics 141. Lecture 24. December 5 th. An important day in the Netherlands. Physics 141. Lecture 24. Course Information. Quiz

Physics 141. Lecture 24. December 5 th. An important day in the Netherlands. Physics 141. Lecture 24. Course Information. Quiz Physics 141. Lecture 24. 0.5 µm particles in water, 50/50 glycerol-water, 75/25 glycerol-water, glycerol http://www.physics.emory.edu/~weeks/squishy/brownianmotionlab.html Frank L. H. Wolfs Department