Physics 224 Midterm 1 Study Guide
Physics 224 Midterm 1 Study Guide PHYS 224 A
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This 14 page Study Guide was uploaded by Jeremy Dao on Tuesday April 26, 2016. The Study Guide belongs to PHYS 224 A at University of Washington taught by COBDEN,DAVID in Spring 2016. Since its upload, it has received 121 views. For similar materials see THERMAL PHYSICS (NW) in Environmental Science at University of Washington.
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Date Created: 04/26/16
Physics 224 Midterm 1 Study Guide ▯ Heat: Energy in a "disordered" form, spread between many microscopic degrees of freedom. – Heat ﬂows from hot to cold because it is moving toward a more likely energy state (see interacting systems below) ▯ Equilibrium (eqm): Any closed system will reach a steady state after a suﬃciently long time. – If you have 2 systems and they are brought into contact, they will eventu- ally reach joint equilibrium state, and can be treated as one joint closed system. – Contact means free exchange of energy, sometimes particles as well – Thermal Equilibrium: The case when only energy is exchanged. The net ﬂow of energy between systems is zero (on average). ▯ "Zeroth Law of Thermodynamics": If A and B are each in equilibrium with C, then A in in equilibrium with C. ▯ Temperature: Property of system such that if T A T aCd T = B , thCn T = T . Temperature can be ordered. A C – Celsius: Deﬁned 0°C to be temperature of ice and water at 1 atm. 100°C deﬁned to be boiling point of water at 1 atm. – Kelvin: T(K) = 273 + T(°C) – Farenheit: T(°F) = 5(°C) + 32 ▯ Ideal Gas Law nRT NkT P = V = V – nR = Nk – T = absolute temperature (K) – V = volume of containing vessel – R = gas constant 8.314 J▯mol▯1▯K▯1 23 – n = number of moles of gas (1 mole = 6:02 ▯ 10 = N atoms) – N = nN =anumber of molecules ▯23 ▯1 – k = Boltzmann’s constant = R/N =a1:38 ▯ 10 J▯K – P = pressure: amount of stress on a static ﬂuid. ▯ Intensive variable: Doesn’t depend on amount of stuﬀ (P,T) ▯ Extensive variable: Scales with amount of stuﬀ (n, N, V) ▯ Thermal Expansions – Most matter expands as T increases 1 2 – Thermal coeﬃcient ▯: ▯ ▯ 1 @V ▯ = V @T At constant P P – In the case of ideal gas: 1 Nk 1 ▯ = = V P T – For solids, deﬁne the linear expansion coeﬀ. ▯ 1 ▯ @L▯ ▯ = L @T P ▯ Kinetic Theory – A gas is many particles moving independently and chaotically. Pressure is the average force due to impacts on container wall. – Consider force F exerted on piston. At low enough P, particles rarely collide with each other, so can look at each particle individually and sum up their contributions. 3 – Finding average Kinetic Energy d~ 2mu mu 2 Fav= x= = 4t 2L=u L 2 2 F av mu mu ) P = A = LA = V m X m For N molecules P = ui= Nhvxi V i=1 V Gas is isotropic sx hv hvy = hvzi ! hx i = 1=3hv i 2 N mhv i N 2 1 2 ) P = = h mv i V 3 V 3 2 N 2 P = V 3hE1i hE1i = avg. kinetic energy 3 hE1i = kT 2 ▯ Equipartition Theorem: Every accessible (not frozen out quantumly) mi- croscopic quadratic degreee of freedom (DOF) has avg. energy equal to 1=2kT in thermal eqm. – If there are N identical molecules, each with f accessible quadratic DOFs, then: NfkT hEtotal= 2 – Counting DOFs: ▯ 1 free particle (e.g. gas) f = 3 b/c v ;v ;v movement in 3 dimen- x y z sions ▯ 1 particle in solid f = 6, b/c there are restoring forces in each direction (3 ▯ 2 = 6) ▯ 1 water molecule in vapor f = 12 positional, rotational, vibrational (stretching, bending) ▯ 1 water molecule in liquid f = 18 restoring forces in liquid ▯ Depending on temperature, some DOFs are "frozen out". – Equipartition theorem also applies to macroscopic degrees of freedom ▯ Internal Energy: Deﬁne U = total energy in a closed system, then: NfkT U = 2 4 ▯ First Law of Thermodynamics (energy is conserved) – When a system undergoes a process, U increases by the amount of energy transferred in. 4U = Q + W where Q = heat ﬂow in conduction or radiation and W = work done by macroscopic forces (i.e. compression, stretching/bending, electrical current, magnetic, microwave, laser) ▯ Work due to compression (for gases) – Assumed system is at equilibrium ! P is a single, well-deﬁned quantity during the whole process – Quasistatic process: A process that’s done slowly enough for system to remain at internal eqm. – Cyclic Process: System is taken around a close path back to where is started H – First Law: 4U = Q + W = 0 !) Q = ▯W = P dV ▯ Ways of compressing the ﬂuid (which are quasistatic): (1)Isothermal: Very slowly, timescale >> ▯ (thermal equilibration time). T remains constant, set by surroundings. By def. T = T = T i f P = NkT ! P = NKT i = V iP V f Vf V f i Nf 4U = U ▯fU = i k(Tf▯ t i = 0 2 Z Vf Z Vf dV W = ▯PdV = ▯NKT POSITIVE V V V i i Q = ▯W NEGATIVE 5 (2) Adiabatic: Quickly, timescale << ▯. No heat ﬂows, Q=0. Z Vf 4U = W + Q = W = ▯PdV Vi ▯ NfkT ▯ NkT For a bit of process: dU = ▯PdV = ▯d = ▯ dV 2 V T dV = ▯ f dT! V T f=2= const: V 2 T f + 2 ▯▯! PV = const: = f (3) Other possible processes: 6 ▯ Heat Capacity: Relation between temperature rise and heat ﬂow Q into a system. – If Q is small, Q = C4T, C is some constant – C is extensive quantity, [C] =J°C▯1 – For ﬂuid deﬁne: ▯ Heat capacity at constant volume for a Vole, C ▯ ▯ Q ▯ 4U ▯ W = 0; 4U = Q CV= ▯ = ▯ 4T V 4T V ▯ ▯ C V @U Eval. at const. V @T V ▯ Heat capacity at constant pressPre, C ▯ ▯ ▯ Q ▯ 4U ▯ W ▯ 4U + P4V ▯ W = ▯P4V CP= ▯ = ▯ = ▯ 4T P 4T P 4T P ▯ ▯ ▯ ▯ C = @U + @V P P @T @T P P – C > C b/c ﬂuid does work against surroundings, so U doesn’t increase P V as much ▯ Speciﬁc Heat: c =C [c] = JK1g▯1 m – Heat capacity per mass – Depends on way heat is added (process) ▯ Case of ideal gas: NfkT NkT U = V = ▯ ▯ ▯ ▯ ▯ ▯ P @U f @U @U CV= = Nk = = CV For ideal gas ONLY @T V ▯ 2 ▯ ▯@T ▯ @T V ▯ ▯ @U @U f f ) C P + P = Nk + Nk = + 1 Nk @T P @T V 2 2 CP 2 ) C = 1 +f ▯ V ▯ First-Order Phase Transition: (melting, boiling, ...) – Heat Q ﬂows in with no change in temperature. – Energy goes into breaking bonds, not to K.E. 7 – Deﬁne latent heat L = Q = heat per unit mass to convert from one ▯1 phase to other at P.T constants. [L] = Jg ▯ Bulk Modulus: describes springiness of a ﬂuid. Depends on how you do compression. ▯ @P ▯ ▯ @P ▯ B = ▯V = ▯ @V @lnV (1) Slowly (Isothermal): ▯ ▯ ▯V @P PV = NkT = const: ! B = P @V P (2) Quickly (Adiabatic): ▯ ▯ @P ▯V PV = const: ! B = P @V Q=0 ▯ Enthalpy Deﬁne: H = U + PV ▯ ▯ ▯ @ ▯ @H – C P ▯ (U + PV ) = @T P @T P – Takes account of work that must be done given you’re at const. P against surroundings. – W = total work done = ▯P4V + Wother 4H = Q + W other If other 0: 4H = Q 8 – To ﬁnd heat -Q released by chemical reaction, use: ▯ ▯ ▯ ▯ ▯Q = enthalpy of formation▯ enthalpy of formation of products of reactants ▯ Heat Conduction: A "transport" phenomena. Heat ﬂows from higher to lower T. Consider slab of material with diﬀerent temps. on each side. Let Q = rate of heat ﬂow through slab. Deﬁne "heat current density" J =Q = heat ﬂow A per/unit area/unit time. [Q] = Js▯1 [J] = Jm1K ▯1 ▯ Fourier’s Law: Flow rate / temperature gradient – J = ▯k 4T t4x ▯1 ▯1 – ktis the thermal conductivity of material.t[k ] = WK – Generalize in 3-D:r) and J(r) J = ▯ktrT ▯ Conservation of Energy 9 – ▯ ▯ ▯ ▯ ▯ ▯ d energy in rate of work net heat ﬂow dt slab = done in slab+ into slab dT C dt= P + JA ▯ (J + 4J)A P = power C dT = P ▯ 4J V dt V 4x CV @T = PV ▯ @J ˆV means per unit volume @t @x – Generalize to 3-D: V V @T r ▯ J = P ▯ C @T – Combine with Fourier’s Law to get the heat (diﬀusion) equation: ▯ ▯ CV @T r T = kt @t Given a length scale L, you get a time scale (relaxation time): ▯ = CL 2 kt ▯ Diffusion: Concentrationr;t) of particles varies – Random motion ! net ﬂow tending to even out non-uniformity in n, i.e. net ﬂow of particles from higher n regions to lower – Fick’s Law JP= ▯Drn~ ~ Where JPis the ﬂux of particles ("particle current density"), and D=diﬀusion constant – Continuity Equation rJ P ▯ @n @t "ﬂow of particles out of small volume = - rate of change of number of particles in it" 10 – Combine Fick’s Law and Continuity Eq. like in heat ﬂow to get the diﬀusion equation: 1 @n r n = D @t Notice that is has the same form as the heat eq. – For length scale L, time scale: 2 ▯ = L D ▯ Thermal Conductivity of a gas 1 kt= C Lv ▯ 3 Where L is the "mean free path" and v ▯ is the mean particle velocity ▯ Distriution function p(x) – Let x be distance corresponding to travel between 2 consecutive collisions – Pick one collision, p(x)dx = probability that the distance to the next collision is between x and x+dx – p(x) is the probability density (probability per length) – Sum of probabilities of all possible outcomes = 1 Z 1 p(x)dx = 1 0 – Let P(x) = probability of getting to x (without scattering) Z x p(x)dx = ▯dP(x) ! P(x) = 1 ▯ p(x )dx0 0 ▯ Scattering Length l – Deﬁne l such that probability of scattering in dx ilx – prob. of getting to x+dx = prob. of getting to x times (1 - prob. not scattering in dx) 11 P(x + dx) = P(x)(1 ▯ dx=l) dx dP dx P(x) + dP = P(x) ▯ P(x) ▯ P(x) !) = ▯ l P l ! P(x) = P(0)e ▯x=l= e▯x=l ▯dP 1 ▯x=l p(x) = = e dx l R ▯ Mean Free Path x ▯ = hxi = xp(x)dx = l – Estimation of l dx prob. of scattering in dx, p(x)dx =l equals prob. of ﬁnding another molecule in cylinder of volume dx▯(2r) N V p(x)dx = ▯(2r) dx ! l = V N4▯r 2 ▯ Statistical Models – 2-state paramagnet: Simplest of all models N identical objects, having 2 states of diﬀerent energies. – A microstate is a speciﬁc conﬁguration of all the objects. P N P Total energy is E = i=1si4 = 4 si= q4, where q = number of up spins – A macrostate is a state where only macroscopic variables are speciﬁed. ▯ in this case, E or q 12 ▯ corresponds to a large number of microstates, called the "multi- plicity" ( ) ▯ ▯ ▯ N N N! = = Cq= q q!(N ▯ q)! Results in huge narrow spike at q = N/2 ▯ Fundamental Postulate of Statistical Mechanics: For an isolated sys- tem with known composition, every molecule consistent with that knowledge is equally probable. – Corollary: If you have an inﬁnite number of copies of the same system (an "ensemble") and we pick one, we could ﬁnd it in any microstate with equal probability – For two-state paramagnet if we assume that there’s no constraint on energy (! T = 1), then probability of ﬁnding it in a macrostate with energy E is: # microstates w/ q = E=4 N! P(E) = = total # microstates = 22 q!(N ▯ q)! ▯ Einstein model of a Solid – Collection of N identical harmonic oscillators – Quantum Mechanics:iE = (n + 1=2)~! N X X N ▯ E = Ei= ni~! + @~! IGNORE b.c const. (zero-point energy) i=1 ▯ @ ! X X ! ni ~! = q~!; let q ▯ni i q = total # of energy quanta in system 13 – To ﬁnd (N;q): # of distinct ways to arrange q quanta between N oscillators, draw row of q dots and N-1 lines: – Multiplicity of Einstein model of solid q+N▯1 (q + N ▯ 1)! (N;q) = CN▯1 = (N ▯ 1)!q! – High-temperature limit q >> N hnii >> 1 ▯ ▯ N ’ eq N ▯ ▯q – Low-temperature limit q << N l ’ eN q is a "very large number" ▯ Interacting systems: Why does heat ﬂow – Consider 2 identical Einstein solids in contact (energy can ﬂow) and are weakly coupled (energy transfer between them is slow compared with internal relaxation time). – For example, 2 blocks of copper separated by plastic slab Assume system is closed ! total energy E A E +AE = (a + B )~! Let q ▯ A + qB, then E = ▯ = constant – Consider high-temperature limia (b ;q >> N): ▯ ▯ ▯ ▯ eqa N eqB N A= ; B= N N For the combined system the multiplicity is: ▯ e 2N tot A B [qa(q ▯ a ] N 14 Maximum multiplicity: ▯ ▯ eq 2N max = 2N ▯ From the fundamental postulate of Statistical Mechanics: – In eqm. probability of ﬁnding A with energy EA = q A! is P(E A / tota ) – If start with A > EB;qA> q=2, then totqA) << max Energy transfers from A to B →hugely increases tot→more likely to happen Energy transfers from B to A →hugely decreases tot ) Transfer from A to B is far more likely. It will continue until ▯ max;EA= E ,Band then stay around totqa) – From this we can conclude the 2nd Law of Thermodynamics: Mul- tiplicity tends to increase whenever possible. – Result: Energy ﬂows to spread out amongst all microscopic D.O.F’s
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