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# Week 2 Book Notes - Thermal Physics Physics 60

UCI

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This 5 page Class Notes was uploaded by Hazel Medina on Thursday October 8, 2015. The Class Notes belongs to Physics 60 at University of California - Irvine taught by Feng, J. in Fall 2015. Since its upload, it has received 46 views. For similar materials see Thermal Physics in Physics 2 at University of California - Irvine.

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Date Created: 10/08/15

Physics 60 10052015 WG 9 Ch 2 The Second Law 0 Combinatorics math of counting ways of arranging things Ch 21 TwoState Systems 0 Imagine a penny nickel and dime if all flipped for heads or tails and all are fair each of eight possible outcomes is equally probable Microstate each of the eight different outcomes in this example I Must specify state of individual particles here it would be heads or tails Macrostate how many heads or tails there are in this example I If you know the microstate of the system HHT then you know its macrostate two heads Multiplicity number of microstates corresponding to a given macrostate in this example the multiplicity for the microstate of two heads and one tails is three because two heads and one tail can only happen in three out of the eight possible outcomes I Multiplicity Q in the example of the three coins 03 heads 00 heads 1 and 02 heads 01 head 3 notice total of all 4 macrostates is 11338 which is the total number of outcomes I Probability of n heads Qn Qall 0 Now imagine 100 coins instead of the three coins in the previous example Total number of microstates 2100 total number of macrostates 101 Multiplicities of the macrostates Qn 2111 130 N N For n objects out of N QN n n 0 v The TwoState Paramagnet o Paramagnet material in which constituent particles act in ways that tend to align parallel to externally applied magnetic field lasts only as long as the external field is applied Ferromagnet particles interact so strongly with each other that the material can magnetize even without an externally applied magnetic field Recall dipoles each individual magnetic particle has its own magnetic dipole moment vector I In simplest case only two values are allowed for the moment vector positive and negative which is then known as a twostate paramagnet where they can either be parallel or antiparallel to the applied field I Dipole pointing one way N1 and dipole pointing another N1 total number of dipoles N N1 N1 Multiplicity for N1 is QN1 NN1N1 Note macrostate of a system is characterized at least partially by its total energy Ch 22 The Einstein Model of a Solid 0 Any quantummechanical harmonic oscillator has equalsized energy units whose potential energy function has the form 12ksx2 where kS is the spring constant Size of the energy unit hf h is Planck s constant 66310quot 34 15 and f is the frequency 12nk m where k is the spring constant Remember that vibrational motions of diatomic and polyatomic gas molecules are examples of quantum oscillators but even more common are the oscillation of atoms in a solid I Einstein solid model of a solid as a collection of identical oscillators with quantized energy units 0 General formula for the multiplicity of an Einstein solid with N oscillators and q q N 1 qN 1 energy units QN q lt q qN1 Ch 23 Interacting Systems 0 Consider system of two Einstein solids A and B that share energy back and forth Assume the two solids are weakly coupled meaning that they exchange energy between them more slowly than the atoms within each solid exchange nergy I Macrostate refers to the state of the combined system temporarily constrained by the individual energies of the solid UA and U3 I Now make assumption that all microstates of the combinations of U and q are all equally probable o Fundamental assumption of statistical mechanics in an isolated system in thermal equilibrium all accessible microstates are equally probable 0 Assume that transitions are random in how they have no discernable pattern 0 Second law of thermodynamics spontaneous flow of energy stops when a system is at or near its most likely macrostate ie the macrostate with the greatest multiplicity Ch 24 Large System 0 Multiplicity is much sharper with more oscillators in Einstein solids eg 1020 oscillators 3 Very Large Numbers 0 Three kinds of numbers occurring commonly in statistical mechanics Small numbers easy to manipulate small numbers Large numbers frequently made by exponentiating small numbers for example Avogadro s number 1023 I You can add small numbers to large numbers without changing the small numbers for example 1023 23 1023 Very large numbers made by exponentiating large numbers I You can multiply very large numbers with large numbers without changing them for example 1010A23 X 1023 1010quot2323 1010A23 339 Stirling s Approximations o N z NNexZn N Accurate in limit where Ngtgt1 0 v Multiplicity of Large Einstein Solid 0 Consider and Einstein solid with a large number of oscillators and energy units where qgtgtN the quothightemperature limit q N 1 qN 1qN 39 Q N39q q qN 1 qlN I M lnqN In q N lnq lnN qlN z qNlnqN qN qlnq q NlnN N q Nlnq N qlnq NlnN nqN In q1 Nq lnq n1 Nq lnq Nq an z NlnqN N NZq 39 0Nq e 39 quot eN eelNN 339 Sharpness of the Multiplicity Function 0 For a solid with N oscillators and total number of energy units q assuming that qgtgtN Q ecuNN quNN eN2N quBN Qmax eN2Nq22N I Now set qA q2 x and qB q2 x where x is any number much smaller than q then the equation becomes 0 eN2Nq22 x2N 0 Now taking logarithms and manipulating it lnq22 x2 Nlnq221 2xq2 Nno22 n1 2xq2 Nnq22 2xq2 Putting this back into the equation for 0 we get the Gaussian Q eN23939e39 q2A2e39WW 2 Qmax e39N2XqA2 which means it has a peak at x0 and a sharp falloff on both sides 0 Thermodynamic limit limit where system becomes infinitely large so that measurable fluctuations away from the likeliest macrostate never occur Ch 25 The Ideal Gas 0 Previous chapter on how only extremely few of the macrostates of a large interacting system can probably happen also applies to any pair of interacting objects not just Einstein solids Also can happen to ideal gases even though they re more complicated 339 Multiplicity ofa Monatomic Ideal Gas 0 Consider a single gas atom with kinetic energy U in a container of volume V Multiplicity should be proportional to momentum space the space where each point corresponds to momentum vector for the particle so 01 ocV Vp I V is the volume of ordinary space or position space Vp is volume of momentum space and 01 indicates the multiplicity of a gas of one molecule only 0 Constrained by U 12mvX2 vy2 v22 12mpx2 py2 p22 which can be written as pX2 py2 p22 2mU o A wavefunction describes the state of a system in quantum mechanics Heisenberg uncertainty principle AxApX z h I Axis the spread of x ApX is the spread of pX and h is Planck s constant 662610quot34 Js 413610quot15 evs o This means that the less spread out a wavefunction is in position space the more spread out it is in momentum space and vice versa Set L length of position space Lp length of momentum space Ax as length of peak of position space curve and ApX as length of peak of momentum space curve I Number of distinct position states is LAx and number of distinct momentum states is LAxLpApx LLph ol vvph3 0 As more molecules are added QN 1NVNh3N x area of momentum hypersphere area Z dZd21rd1 0 Q UVN fNVNU3N2 fN is a complicated function of N 3 Interacting Ideal Gases 0 Now consider two ideal gases Total multiplicity becomes Qtota fN2VAVB3939UAUB3N2 Width of peak UtotaIq 3N2 This system can exchange both energy and volume I Width of peak VtotaIxN Ch 26 Entropy Note how we ve seen that particles and energy tend to rearrange themselves to be at where multiplicity is at or very near its maximum value Second law of thermodynamics multiplicity tends to increase I Entropy S kan 0 Entropy defined as logarithm of number of ways a system can arrange things in the system times Boltzmann s constant in units JK S klneqNN NknqN1 0 Generally more particles means more energy and greater multiplicity and entropy Can also increase entropy by letting it expand into larger space or breaking large molecules or mixing together substances all of which increase possibly microstates Stota kantota klnQAQB kanA kanB SA SB Another component of the second law of thermodynamics entropy tends to increase 3 Entropy of an Ideal Gas 0 Equation for entropy of an ideal monatomic gas aka SackurTetrode equation 5 NknVN4Tth3Nh232 52 If volume changes while U and N are fixed AS NknVfVi I This equation works with quasistatic isothermal expansion considered in Section 15 0 Adding heat always increases entropy 0 Free expansion increasing entropy by puncturing a partition to allow the air within the partition to occuy all available space No work is done during free expansion 3 Entropy of Mixing 0 Consider two gases A and B separated by a partition both have equal energy volume and number of particles Removing the partition increases the entropy and the entropy becomes ASA ASB NklnVfVi Nkln2 Total entropy becomes AStota ASA ASE 2Nkln2 I This is the entropy of mixing I The SackurTetrode equation becomes S NklnV4Tth3Nh232 32 for distinguishable molecules 0 Gibbs paradox says that if this formula were true then one can decrease entropy of gas in a container by half by inserting a partition to divide the total volume of the container by half The best resolution is to assume that all atoms are indistinguishable 3 Reversible and Irreversible Processes o Irreversible used to describe processes that create entropy Reversible used to describe processes that don t change entropy Slow compression of gas does not cause entropy to increase 0 Think of flow of heat from hot object to cold object Heat flow is reversible but quotreversible heat flow refers to very slow heat flow between two objects of nearly the same temperature Larger entropy increases are highly irreversible and are much more common eg wood burning sunlight warming the earth and even metabolism of nutrients in a body

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