Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 1

http://www.physics.emory.edu/~weeks/squishy/BrownianMotionLab.html

0.5 µm particles in water, 50/50 glycerol-water, 75/25 glycerol-water, glycerol

Physics 141.Lecture 24.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 2

December 5th.An important day in the Netherlands.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 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 itsenvironment.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 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 inHoyt. This exam will cover the material discussed inChapters 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 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 thecollisions (analysis during regular lab periods).

• ✔ 11/24: complete discussion and comparisonof results with colliding partners and submitfinal results (velocities and errors) to professorWolfs.

• ✔ 11/27: professor Wolfs compiles results,determines momenta and kinetic energies, anddistributes the results.

• ✔ 11/28 + 12/5: office hours by lab TA/TIs tohelp 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 6

The Personal Response System (PRS).Quiz Lecture 24.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 7

The kinetic theory of gases.Thermodynamic variables.

• The volume of a gas is defined by the size of the enclosureof the gas. During a change in the state of a gas, thevolume may or may not remain constant (this depends onthe procedure followed).

• The temperature of a gas has been defined in terms of theentropy 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/m2.Another common unit is the atm (atmospheric pressure)which is the pressure exerted by the atmosphere on us (1atm = 1.013 x 105 N/m2).

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 8

A quick review:The equation of state of a gas.

• In order to specify the state of a gas, weneed to measure its temperature, itsvolume, and its pressure. The relationbetween these variables and the mass ofthe gas is called the equation of state.

• The equation of state of a gas was initiallyobtained on the basis of observations.

• Boyle’s Law (1627 - 1691):pV = constant for gases maintained atconstant temperature.

• Charle’s Law (1746 - 1823):V/T = constant for gases maintained atconstant pressure.

• Gay-Lussac’s Law (1778 - 1850):p/T = constant for gases maintained atconstant volume

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 9

The equation of state of a gas.

• Combining the various gas lawswe can obtain a single moregeneral relation betweenpressure, temperature, andvolume: pV = constant T.

• Another observation that needs tobe included is the dependence onthe amount of gas: if pressure andtemperature are kept constant, thevolume is proportional to themass m: pV = constant mT.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 10

The equation of state of a gas.

• The equation of state of a gas can be written as

pV = NkT

where• p = pressure (in Pa).• V = volume (in m3).• N = number of molecules of gas (1 mole = 6.02 x 1023 molecules or

atoms). Note the number of molecules in a mole is also known asAvogadro’s number NA.

• T = temperature (in K).

• Note: the equation of state is the equation of state of an idealgas. Gases at very high pressure and/or close to the freezingpoint show deviations from the ideal gas law.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 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 ofthe vessel.

• Each time a molecule collides with thewall, it will carry out an elasticcollision.

• Since the linear momentum of themolecule is changed, the linearmomentum of the wall will change too.

• Since force is equal to the change inlinear momentum per unit time, the gaswill exert a force on the walls.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 12

The molecular point of view of a gas.

• Consider the collision of a singlemolecule with the left wall.

• In this collision, the linearmomentum of the moleculechanges by mvx - (-mvx) = 2mvx.

• The same molecule will collidewith this wall again after a time2l/vx.

• The force that this singlemolecule exerts on the left wall isthus equal toΔp/Δt = (2mvx)/(2l/vx) = mvx

2/l

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 13

The molecular point of view of a gas.

• The force that this singlemolecule exerts on the left wall isthus equal to

Fleft = mvx2/l

• If the pressure exerted on the leftwall by this molecule is equal to

pleft = Fleft/A = mvx2/(lA)

where A is the area of the leftwall.

• The volume of the gas is equal tolA and we can thus rewrite thepressure on the left wall:

pleft = mvx2/V

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 14

The molecular point of view of a gas.

• The pressure that many moleculesexerts on the left wall is equal to

pleft = m(v1x2+v2x

2+v3x2+... )/V

• This equation can be rewritten interms of the average of the square ofthe x component of the molecularvelocity and the number of molecules(N):

pleft = mN(vx2)average/V

• Assuming that there is no preferentialdirection, the average square of the x,y, and z components of the molecularvelocity will be the same:

(vx2)average= (vy

2)average= (vz2)average

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 15

The molecular point of view of a gas.• The force on the left wall can be

rewritten in terms of the averagesquared velocity

pleft = mN(v2)average/3V

• Assuming there is no preferentialdirection of motion of themolecules, the pressure on allwalls will be the same and wethus conclude:

pV = mN(v2)average/3

• Compare this to the ideal gas law:pV = NkT

Kaverage = (1/2)m(v2)average = (3/2) kT

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 16

Simulating an ideal gas.

• Ideal gas simulations:• Assume elastic collisions

between the gas molecules.• Assume elastic collisions

between the gas molecules andthe walls.

• Results agree very wellwith measured values.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 17

2 Minute 48 Second Intermission.

• Since paying attention for 1 hourand 15 minutes is hard when thetopic is physics, let’s take a 2minute 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 18

The real equation of state.Different points of view.

Ideal gas law

Differenttemperatures

Critical temperature

Note:1. T > Tc: gas2. T < Tc: liquid

and/or vapor.Critical point

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 19

The real equation of state.Different points of view.

Direct change from solid to vapor.

Note the curvature of the solid-liquid line.Curvature to the left implies expansion on cooling.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 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 CO2

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 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 beequal to the heat added tot he system minus the work doneby the system: ΔU = Q - W (note: this is the work-energytheorem).

• 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 22

The first law of thermodynamics.Isothermal processes.

• An isothermal process is aprocess in which the temperatureof the system is kept constant.

• This can be done by keeping thesystem in contact with a largeheat reservoir and making allchanges slowly.

• Since the temperature of thesystem is constant, the internalenergy of the system is constant:ΔU = 0 J.

• The first law of thermodynamicsthus tells us that Q = W.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 23

The first law of thermodynamics.Adiabatic processes.

• An adiabatic process is a processin which there is no flow of heat(the system is an isolatedsystem).

• Adiabatic processes can alsooccur in non-isolated systems, ifthe change in state is carried outrapidly. A rapid change in thestate of the system does not allowsufficient time for heat flow.

• The expansion of gases differsgreatly depending on the processthat is followed (see Figure).

ΔU = 0 J

Q = 0 J

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 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 volumeof the gas:

dW = pAdl = pdV

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 25

Work done during expansion/compression. Isobaric and isochoric processes.

• Isobaric process:

• Processes in which the pressure iskept constant.

• WA->B = pdV = pA(VB - VA)

• Isochoric process:

• Processes in which the volume iskept constant.

• WA->B = pA(VB - VA) = 0

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 26

Work done during expansion/compression.Isothermal process.

• Isothermal process:

• The work done during the changefrom state A to state B is

p =

NkTV

W = pdVVA

VB

∫ = NkT 1V

dVVA

VB

∫

= NkT lnVB

VA

⎛

⎝⎜⎞

⎠⎟

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 24, Page 27

Work done during expansion/compression.• The work done during the

expansion of a gas is equal to thearea under the pV curve.

• Since the shape of the pV curvedepends on the nature of theexpansion, so does the workdone:

• Isothermal: W = NkT ln(VB /VA )

• Isochoric: W = 0

• Isobaric: W = pB (VB - VA)

• The work done to move state A tostate B can take on any value!

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

Done for today!

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