Physics 1501: Lecture 34, Pg 1 Physics 1501: Lecture 34 Today’s Agenda l Announcements çHomework #11 (Dec. 2) and #12 (Dec. 9): 2 lowest dropped l Honors’ students: see me at 2:30 today ! l Today’s topics çChap.16: Temperature and Heat »Thermal expansion »Heat transfer »Latent Heat çHeat transfer processes »Conduction, convection, radiation »Application

Physics 1501: Lecture 34, Pg 2 Chap. 16: Temperature and Heat l Temperature: measure of the motion of the individual atoms and molecules in a gas, liquid, or solid. çrelated to average kinetic energy of constituents l High temperature: constituents are moving around energetically çIn a gas at high temperature the individual gas molecules are moving about independently at high speeds. çIn a solid at high temperature the individual atoms of the solid are vibrating energetically in place. l The converse is true for a "cold" object. çIn a gas at low temperature the individual gas molecules are moving about sluggishly. çThere is an absolute zero temperature at which the motions of atoms and molecules practically stop. l There is an absolute zero temperature at which the classical motions of atoms and molecules practically stop

Physics 1501: Lecture 34, Pg 3 Temperature scales l Three main scales 212 Farenheit 100 Celcius Kelvin Water boils Water freezes Absolute Zero

Physics 1501: Lecture 34, Pg 4 Some interesting facts l In 1724, Gabriel Fahrenheit made thermometers using mercury. The zero point of his scale is attained by mixing equal parts of water, ice, and salt. A second point was obtained when pure water froze (originally set at 30 o F), and a third (set at 96 o F) “when placing the thermometer in the mouth of a healthy man”. çOn that scale, water boiled at 212. çLater, Fahrenheit moved the freezing point of water to 32 (so that the scale had 180 increments). l In 1745, Carolus Linnaeus of Upsula, Sweden, described a scale in which the freezing point of water was zero, and the boiling point 100, making it a centigrade (one hundred steps) scale. Anders Celsius ( ) used the reverse scale in which 100 represented the freezing point and zero the boiling point of water, still, of course, with 100 degrees between the two defining points. T (K) Hydrogen bomb Sun’s interior Solar corona Sun’s surface Copper melts Water freezes Liquid nitrogen Liquid hydrogen Liquid helium Lowest T ~ K

Physics 1501: Lecture 34, Pg 5 Thermal expansion l In most liquids or solids, when temperature rises çmolecules have more kinetic energy »they are moving faster, on the average çconsequently, things tend to expand l amount of expansion  L depends on… çchange in temperature  T çoriginal length L 0 çcoefficient of thermal expansion »L 0 +  L = L 0 +  L 0  T »  L =  L 0  T (linear expansion) »  V =  L 0  T (volume expansion) L0L0 LL V V +  V

Physics 1501: Lecture 34, Pg 6 Lecture 34, ACT 1 Thermal expansion l As you heat a block of aluminum from 0 o C to 100 o C, its density (a) increases (b) decreases (c) stays the same

Physics 1501: Lecture 34, Pg 7 Lecture 34: ACT 2 Thermal expansion l An aluminum plate (  =24  ) has a circular hole cut in it. A copper ball (solid sphere,  =17  ) has exactly the same diameter as the hole when both are at room temperature, and hence can just barely be pushed through it. If both the plate and the ball are now heated up to a few hundred degrees Celsius, how will the ball and the hole fit ? (a) ball won’t fit (b) fits more easily (c) same as before

Physics 1501: Lecture 34, Pg 8 A hole in a piece of solid material expands when heated and contracts when cooled, just as if it were filled with the material that surrounds it. Thermal expansion

Physics 1501: Lecture 34, Pg 9 Special system: Water l Most liquids increase in volume with increasing T çwater is special çdensity increases from 0 to 4 o C ! çice is less dense than liquid water at 4 o C: hence it floats çwater at the bottom of a pond is the denser, i.e. at 4 o C Water has its maximum density at 4 degrees.  (kg/m 3 ) T ( o C) l Reason: chemical bonds of H 2 0 (see your chemistry courses !)

Physics 1501: Lecture 34, Pg 10 Lecture 34: ACT 3 l Not being a great athlete, and having lots of money to spend, Gill Bates decides to keep the lake in his back yard at the exact temperature which will maximize the buoyant force on him when he swims. Which of the following would be the best choice? (a) 0 o C (b) 4 o C (c) 32 o C(d) 100 o C (e) 212 o C

Physics 1501: Lecture 34, Pg 11 Heat l Solids, liquids or gases have internal energy çKinetic energy from random motion of molecules n translation, rotation, vibration çAt equilibrium, it is related to temperature l Heat: transfer of energy from one object to another as a result of their different temperatures l Thermal contact: energy can flow between objects T1T1 T2T2 U1U1 U2U2 >

Physics 1501: Lecture 34, Pg 12 Heat l Heat: Q = C  T ç Q = amount of heat that must be supplied to raise the temperature by an amount  T. ç [Q] = Joules or calories. ç energy to raise 1 g of water from 14.5 to 15.5 o C ç James Prescott Joule found mechanical equivalent of heat. ç C : Heat capacity 1 cal = J 1 kcal = 1 Cal = 4186 J l Q = c m  T ç c: specific heat (heat capacity per units of mass) ç amount of heat to raise T of 1 kg by 1 o C ç [c] = J/(kg o C) l Sign convention: +Q : heat gained - Q : heat lost

Physics 1501: Lecture 34, Pg 13 Specific Heat : examples l You have equal masses of aluminum and copper at the same initial temperature. You add 1000 J of heat to each of them. Which one ends up at the higher final temperature ? a) aluminum b) copper c)the same Substance c in J/(kg-C) aluminum900 copper387 iron452 lead128 human body 3500 water 4186 ice 2000

Physics 1501: Lecture 34, Pg 14 Latent Heat l L = Q / m çHeat per unit mass [L] = J/kg çQ =  m L + if heat needed (boiling) - if heat given (freezing) çL f : Latent heat of fusion solid  liquid çL v : Latent heat of vaporization liquid  gas l Latent heat: amount of energy needed to add or to remove from a substance to change the state of that substance. çPhase change: T remains constant but internal energy changes ç heat does not result in change in T (latent = “hidden”) ç e.g. : solid  liquid or liquid  gas heat goes to breaking chemical bonds L f (J/kg) L v (J/kg) water33.5 x x 10 5

Physics 1501: Lecture 34, Pg 15 Latent Heats of Fusion and Vaporization Energy added (J) T ( o C) Water + Ice Water + SteamSteam

Physics 1501: Lecture 34, Pg 16 Energy in Thermal Processes l Solids, liquids or gases have internal energy çKinetic energy from random motion of molecules n translation, rotation, vibration çAt equilibrium, it is related to temperature l Heat: transfer of energy from one object to another as a result of their different temperatures l Thermal contact: energy can flow between objects T1T1 T2T2 U1U1 U2U2 >

Physics 1501: Lecture 34, Pg 17 Energy transfer mechanisms l Thermal conduction (or conduction): çEnergy transferred by direct contact. çE.g.: energy enters the water through the bottom of the pan by thermal conduction. çImportant: home insulation, etc. l Rate of energy transfer  through a slab of area A and thickness  x, with opposite faces at different temperatures, T c and T h çk : thermal conductivity xx ThTh TcTc A Energy flow  =Q/  t = k A (T h - T c ) /  x

Physics 1501: Lecture 34, Pg 18 Thermal Conductivities Aluminum238Air0.0234Asbestos0.25 Copper397Helium0.138Concrete1.3 Gold314Hydrogen0.172Glass0.84 Iron79.5Nitrogen0.0234Ice1.6 Lead34.7Oxygen0.0238Water0.60 Silver427Rubber0.2Wood0.10 J/s m 0 C

Physics 1501: Lecture 34, Pg 19 Energy transfer mechanisms l Convection: çEnergy is transferred by flow of substance çE.g. : heating a room (air convection) çE.g. : warming of North Altantic by warm waters from the equatorial regions çNatural convection: from differences in density çForced convection: from pump of fan l Radiation: çEnergy is transferred by photons çE.g. : infrared lamps çStephan’s law   = 5.7  W/m 2 K 4, T is in Kelvin, and A is the surface area ç e is a constant called the emissivity  = Q/  t =  Ae T 4 : Power

Physics 1501: Lecture 34, Pg 20 Resisting Energy Transfer l The Thermos bottle, also called a Dewar flask is designed to minimize energy transfer by conduction, convection, and radiation. The standard flask is a double-walled Pyrex glass with silvered walls and the space between the walls is evacuated. Vacuum Silvered surfaces Hot or coldliquid

Physics 1501: Lecture 34, Pg 21 Chap.17: Ideal gas and kinetic theory l Consider a gas in a container of volume V, at pressure P, and at temperature T l Equation of state çLinks these quantities çGenerally very complicated: but not for ideal gas PV = nRT R is called the universal gas constant In SI units, R =8.315 J / mol·K n = m/M : number of moles l Equation of state for an ideal gas çCollection of atoms/molecules moving randomly çNo long-range forces çTheir size (volume) is negligible

Physics 1501: Lecture 34, Pg 22 Boltzmann’s constant l In terms of the total number of particles N l P, V, and T are the thermodynamics variables PV = nRT = (N/N A ) RT k B is called the Boltzmann’s constant k B = R/N A = 1.38 X J/K PV = N k B T l Number of moles: n = m/M  One mole contains N A =6.022 X particles : Avogadro’s number = number of carbon atoms in 12 g of carbon-12 m=mass M=mass of one mole

Physics 1501: Lecture 34, Pg 23 Note on masses To facilitate comparison of the mass of one atom with another, a mass scale know as the atomic mass scale has been established. The unit is called the atomic mass unit (symbol u). The reference element is chosen to be the most abundant isotope of carbon, which is called carbon-12. The atomic mass is given in atomic mass units. For example, a Li atom has a mass of 6.941u.

Physics 1501: Lecture 34, Pg 24 What is the volume of 1 mol of gas at STP ? T = 0 o C = 273 K p = 1 atm = 1.01 x 10 5 Pa The Ideal Gas Law

Physics 1501: Lecture 34, Pg 25 Example l Beer Bubbles on the Rise çWatch the bubbles rise in a glass of beer. If you look carefully, you’ll see them grow in size as they move upward, often doubling in volume by the time they reach the surface. Why does the bubble grow as it ascends?

Physics 1501: Lecture 34, Pg 26 Kinetic Theory of an Ideal Gas l Assumptions for ideal gas: çNumber of molecules N is large çThey obey Newton’s laws (but move randomly as a whole) çShort-range interactions during elastic collisions çElastic collisions with walls çPure substance: identical molecules l Microscopic model for a gas çGoal: relate T and P to motion of the molecules

Physics 1501: Lecture 34, Pg 27 Distribution of Molecular Speeds l The particles are in constant, random motion, colliding with each other and with the walls of the container. çEach collision changes the particle’s speed. çAs a result, the atoms and molecules have different speeds.

Physics 1501: Lecture 34, Pg 28 Kinetic Theory l The average force exerted by one wall l Time between successive collisions on the wall l Action-reaction gives

Physics 1501: Lecture 34, Pg 29 l For N molecules, the average force is: root-mean-square speed volume Pressure

Physics 1501: Lecture 34, Pg 30 Ideal gas law l Pressure is

Physics 1501: Lecture 34, Pg 31 l Does a Single Particle Have a Temperature? Each particle in a gas has kinetic energy. On the previous page, we have established the relationship between the average kinetic energy per particle and the temperature of an ideal gas. Is it valid, then, to conclude that a single particle has a temperature? Concept of temperature

Physics 1501: Lecture 34, Pg 32 l Air is primarily a mixture of nitrogen N 2 molecules (molecular mass 28.0u) and oxygen O 2 molecules (molecular mass 32.0u). çAssume that each behaves as an ideal gas and determine the rms speeds of the nitrogen and oxygen molecules when the temperature of the air is 293K. çFor nitrogen Example: Example: Speed of Molecules in Air

Physics 1501: Lecture 34, Pg 33 Internal energy of a monoatomic ideal gas l The kinetic energy per atom is l Total internal energy of the gas with N atoms

Physics 1501: Lecture 34, Pg 34 Kinetic Theory of an Ideal Gas: summary l Assumptions for ideal gas: çNumber of molecules N is large çThey obey Newton’s laws (but move randomly as a whole) çShort-range interactions during elastic collisions çElastic collisions with walls çPure substance: identical molecules l Temperature is a direct measure of average kinetic energy of a molecule l Microscopic model for a gas çGoal: relate T and P to motion of the molecules

Physics 1501: Lecture 34, Pg 35 l Theorem of equipartition of energy çEach degree of freedom contributes k B T/2 to the energy of a system (e.g., translation, rotation, or vibration) Kinetic Theory of an Ideal Gas: summary  l Total translational kinetic energy of a system of N molecules çInternal energy of monoatomic gas: U = K ideal = K tot trans l Root-mean-square speed:

Physics 1501: Lecture 34, Pg 36 l Consider a fixed volume of ideal gas. When N or T is doubled the pressure increases by a factor of 2. 1) If T is doubled, what happens to the rate at which a single molecule in the gas has a wall bounce? b) x2a) x1.4c) x4 2) If N is doubled, what happens to the rate at which a single molecule in the gas has a wall bounce? b) x1.4a) x1c) x2 Lecture 34: ACT 4

Physics 1501: Lecture 34, Pg 37 Diffusion l The process in which molecules move from a region of higher concentration to one of lower concentration is called diffusion. çInk droplet in water

Physics 1501: Lecture 34, Pg 38 Why is diffusion a slow process ? l A gas molecule has a translational rms speed of hundreds of meters per second at room temperature. At such speed, a molecule could travel across an ordinary room in just a fraction of a second. Yet, it often takes several seconds, and sometimes minutes, for the fragrance of a perfume to reach the other side of the room. Why does it take so long? çMany collisions !

Physics 1501: Lecture 34, Pg 39 Comparing heat and molecule diffusion l Both ends are maintained at constant concentration/temperature

Physics 1501: Lecture 34, Pg 40 concentration gradient between ends diffusion constant SI Units for the Diffusion Constant: m 2 /s Fick’s law of diffusion l For heat conduction between two side at constant T L ThTh TcTc A Energy flow conductivity temperature gradient between ends l The mass m of solute that diffuses in a time t through a solvent contained in a channel of length L and cross sectional area A is