Honors Physics Notes Nov 16, 20 Heat Persans 1 Properties of solids Persans 2 Persans 3 Vibrations of atoms in crystalline solids Assuming only nearest neighbor interactions (+Hooke's law) Fs = C(us+1 ! us ) + C(us!1 ! us ) us is the displacement of the sth plane. The equation of motion of the sth plane is thus: 2 d us M = C (us+1 + us!1 ! 2us ) dt 2 We look for solutions where all atoms move as a travelling wave e!i"t eikx . 2 !M" us = C (us+1 + us!1 ! 2us ) !M" 2eiksa = C (eika + e!ika ! 2)eiksa 2 # 2C $ # 4C $ 2 ka " = % &(1! cos ka)= % &sin ' M ( ' M ( 2 1/ 2 # 4C $ ka " = % & sin ' M ( 2 Persans 4 Normal Mode Frequencies (Continued) Persans 5 Topics Heat and temperature Kinetic model Properties of materials Heat capacity Thermal expansion Transfer of heat First Law of thermodynamics: Heat and work Persans 6 What is temperature? • Temperature is a measure of the average kinetic energy of the particles in a space. It is proportional to the internal energy of the system. What is heat? • Heat is the energy that flows between a system and its environment by virtue of a temperature difference between them . Persans 7 Temperature scales • Common scales are based on temperatures at which everyday things happen. • The Fahrenheit scale is based on the freezing point of brine (~00F) and the temperature of the human body (~1000F). • The Centigrade or Celsius scale assigns the freezing point of pure water to 00C and the boiling point to 1000C. Celsius is the metric unit and is used worldwide, except in the US. • The Kelvin scale is based on thermodynamics and kinetic theory. 9 5 TF = TC + 32; TC = (TF ! 32); TK = TC + 273 5 9Persans 8 Units of Heat • Since heat is a form of energy, the SI unit is the Joule. • Another common unit of heat is the kilocalorie, which is the amount of heat energy needed to raise the temperature of 1 kg of water by 10C. 1kcal=4185 J • Some US engineers insist on using the BTU (British Thermal Unit) 1 BTU is the amount of heat needed to raise the temperature of 1 lb of water by 10F. 1BTU=0.252 kcal Persans 9 Kinetic Theory of Gases • A series of experiments on dry air led to the following equation of state for an ideal gas: pV = NkBT p = pressure; V = volume; N = number of gas molecules in that volume; !23 !5 kB = Boltzman's constant=1.38"10 J/K=8.65"10 eV/K pV = nRT n = number of moles of gas R = universal gas constant=N AkB Persans 10 Kinetic theory: a model 1. A gas consists of molecules 2. The molecules are in random motion and obey Newton’s laws. 3. The total number of molecules is very large (so we can use statistical averages) 4. The average distance between molecules is large compared to the size of the molecules. 5. Molecules experience only collisions forces. 6. Collisions are elastic and of short duration. Persans 11 Deducing pressure from kinetic energy • Find the average change in momentum when a molecule bounces elastically between two massive walls separated by distance L. • Assume that all molecules have the same average speed. • Compare the pressure we deduce with the ideal gas law. Persans 12 2mvx1 !momentum = "2mvx1; Fx1 = (2L / vx1) F mv2 p = x1 = x1 ; 1 A L3 m m For many atoms: p = # v2 = N v2 L3 xi L3 x 2 2 2 2 2 v = vx + vy + vz = 3vx 1 m 2 2 N $ 1 2 % p = N v = & mv ' 3 V 3 V ( 2 ) 2 $ 1 2 % pV = N & mv ' and comparing to pV = NkBT 3 ( 2 ) 1 3 we conclude that : mv2 = k T 2 2 B Persans 13 Properties of common matter • Phases – solid – liquid – gas – (plasma) • Phase changes – melting – vaporization (boiling) – dissociation of valence electrons from atoms • Thermal expansion Persans 14 • Anharmonic Forces Interatomic forces include higher order terms than linear term in atomic displacement. Figure 6-2 (a) Harmonic and (b) Anharmonic potential energy functions for the interaction of two atoms. R is the atomic separation , while E1, and E2 represent two possible vibrational energies. For (a) an increase in energy does not result in a change in the average atomic separation, given by the midpoints of constant energy lines. For (b) an increase in energy results in an increase in average separation. Persans 15 Thermal Expansion • To a good approximation over moderate temperature ranges, the change in length of a solid increases proportionately to the change in temperature. "L # !L"T L = original length, != linear expansion coefficient material α (10-6/C) Al 25 Steel 12 Glass 3 Concrete ~12 Persans 16 Thermal expansion of area and volume For solids, the amount by which the area of a plate or the volume of a block changes can be easily estimated from the linear expansion: 2 !L !A(simple square) = (L + !L) " L2 = 2L!L + !L2 # 2L!L = 2A L !A !L # 2 A L 3 3 !V (L + !L) " L !L = # 3 V L3 L Persans 17 Volume Thermal Expansion • The change in volume with temperature can be defined for solids, liquids and gases. #V $ !V #T ! $ 3" V = original volume, != volume expansion coefficient material β (10-6/C) Al 75 Steel 35 Glass 9 Concrete ~35 Water 210 Air 3400 Persans 18 Thermal Expansion: The special case of ideal gases • For many gases, it is found that the pressure, volume, and temperature are related in a simple way: pV = nRT p = pressure; V = volume; n = number of moles of gas; R= a constant R = 8.314 J/(mol K) mass (grams) n = molecular mass (g/mol) Persans 19 Exercise: 16 Nov 06 ________________ • An ideal gas is cooled from 300C to 00C. By what ratio does its volume change? Persans 20 Exercise: 16 Nov 06 • Two aluminum plates are held in place by a steel bridge as shown. The temperature for the figure on the left is 0oC. The plates heat in the sun to 40oC and expand. This causes a buckle as shown in the figure on the right. 20 m a) Find the differential expansion of Al and steel. b) Estimate the height of the buckle. Persans 21 a) Find the differential expansion of Al and steel. !(Steel) =11#10-6 ; !(Al) = 23#10"6 $$L = L$!$T = 20m#(12#10"6 K "1 )# 40K = 9.6x10"3 meters b) Estimate the height of the buckle. Use L=10 m for one side. 2 % $$L ( 2 2 & L + $LS + ) = (L + $LS ) + h ' 2 * 2 2 $$L L2 + $L + + 2$L L + L$$L + $L$$L S 4 S 2 2 $$L = (L + $LS ) + + L$$L + $L$$L 4 h2 + L$$L =10m#10"2 m = 0.1m2 , h + 0.3m Persans 22 Heat capacity and specific heat • When heat energy is added to or subtracted from an object, its temperature changes (unless it undergoes a phase change). In many cases, near room temperature, the temperature change is linear in energy added. !Q = C!T C = heat capacity • The specific heat takes the mass of material being heated into account. c = C / m ! "Q = mc"T Persans 23 Some approximate specific heats • Aluminum 0.9 kJ/(kg C) • Glass 0.84 • Steel 0.45 • Wood 1.7 • Water (liquid) 4.2 • Ice 2.1 • Air (const V) 1 Persans 24 Latent heat of phase change • When the phase of material changes, it takes up or gives off heat but the temperature does not change. The energy goes into breaking or making bonds between atoms. • The energy relationship is expressed using the latent heat of melting (or condensation, or vaporization, or freezing) !Q = Lm Persans 25 Persans 26 Example: Energy required to heat and boil a cup of water • Estimate the amount of energy that is required to heat a cup of water (0.2 kg) from room temperature to boiling. • Estimate the amount of energy that is required to convert that cup of water to vapor. Persans 27 Example continued To raise the temperature of 0.2 kg of water from 20°C to 100°C we use the approximate formula: J !Q = cm!T = 4190 •.2kg •80K = 67000J = 67kJ kgK To vaporize the same amount: kJ !Q = L m = 2256 ".2kg = 451kJ v v kg The total energy is thus: !QT = 518kJ Persans 28 Comparison of thermal and gravitational potential energy Let's estimate temperature change due to conversion of gravitational potential energy to heat energy from dropping a student from from the top of a 9 story building: m m !KE = mgh " 70kg # 4 #9 floors#10 = 25kJ (6 kcal). floor s2 If all of this energy were converted to heat the temperature of the water would increase by m J 10 #36m 360 !Q mgh gh 2 kg !T " = = = s = = 0.086K. cm cm c J J water 4200 4200 kgK kgK Persans 29 The flow of heat Persans 30 The flow of heat • Heat energy flows from warmer volume to colder volumes. • Convection: energy transfer by collective motion of a macroscopic volume of the fluid. • Conduction: energy transfer by transfer of vibrational kinetic energy on the atomic scale. • Radiative: energy transfer by emission of electromagnetic radiation from a hot material. Persans 31 Heat Conduction • Heat transfer by conduction can be represented using a simple and sensible relation: For a slab of thickness d and area A dQ ! TH #TC " ! dT " H = = kA$ % = kA$ % dt & xH # xC ' & dx ' A H= rate of heat flow ! W " k = specific thermal conductivity $ % & m • K ' A = area; dx TH (TC ) is the hot (cold) temperature Persans 32 Thermal conductivity of common materials Material ρ (W/m.K) BTU/(ft 0F hr) Still Air 0.026 0.011 Aluminum 235 140 Glass 1.0 0.50 Concrete 0.7 0.35 Wood 0.1 0.05 Persans 33 Solving a differential equation for temperature of an object • We can use what Q = cmTH we know about dQ dT # T !T $ heat flow and = !cm H = kA H C internal energy to % & dt dt ' xH ! xC ( deduce the dTC temperature of an TC is a constant, so = 0 object as a function dt of time.