Physics 1501: Lecture 34 Today ’ s Agenda

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

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

3. Water boils 32 0 273.15 Water freezes -459.67 -273.15 0 Absolute Zero Temperature scales Farenheit Celcius Kelvin • Three main scales 212 100 373.15

4. T (K) 108 107 106 105 104 103 100 10 1 0.1 Some interesting facts Hydrogen bomb Sun’s interior • 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 30oF), and a third (set at 96oF) “when placing the thermometer in the mouth of a healthy man”. • On that scale, water boiled at 212. • Later, Fahrenheit moved the freezing point of water to 32 (so that the scale had 180 increments). • 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 (1701-1744) 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. Solar corona Sun’s surface Copper melts Water freezes Liquid nitrogen Liquid hydrogen Liquid helium Lowest T ~ 10-9K

5. L0 L V V + V Thermal expansion • In most liquids or solids, when temperature rises • molecules have more kinetic energy • they are moving faster, on the average • consequently, things tend to expand • amount of expansion L depends on… • change in temperature T • original length L0 • coefficient of thermal expansion • L0 + L = L0 +  L0 T • L =  L0 T (linear expansion) • V =  L0 T (volume expansion)

6. Lecture 34,ACT 1Thermal expansion • As you heat a block of aluminum from 0 oC to 100 oC, its density (a) increases(b) decreases(c) stays the same

7. Lecture 34: ACT 2Thermal expansion • An aluminum plate (=2410-6) has a circular hole cut in it. A copper ball (solid sphere, =1710-6) 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

8. Thermal expansion 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.

9. (kg/m3) T (oC) Special system: Water • Most liquids increase in volume with increasing T • water is special • density increases from 0 to 4 oC ! • ice is less dense than liquid water at 4 oC: hence it floats • water at the bottom of a pond is the denser, i.e. at 4 oC Water has its maximum density at 4 degrees. • Reason: chemical bonds of H20 (see your chemistry courses !)

10. Lecture 34: ACT 3 • 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 oC(b) 4oC(c) 32 oC (d) 100 oC (e) 212 oC

11. Heat • Solids, liquids or gases have internal energy • Kinetic energy from random motion of molecules • translation, rotation, vibration • At equilibrium, it is related to temperature • Heat: transfer of energy from one object to another as a result of their different temperatures • Thermal contact: energy can flow between objects T2 T1 > U2 U1

12. Heat: Q = C  T • Q = amount of heat that must be supplied to raise the temperature by an amount  T . • [Q] = Joules or calories. • energy to raise 1 g of water from 14.5 to 15.5 oC • James Prescott Joule found mechanical equivalent of heat. • C : Heat capacity 1 cal = 4.186 J 1 kcal = 1 Cal = 4186 J +Q : heat gained - Q : heat lost • Sign convention: Heat • Q = c m  T • c: specific heat (heat capacity per units of mass) • amount of heat to raise T of 1 kg by 1oC • [c] = J/(kg oC)

13. Substance c in J/(kg-C) aluminum 900 copper 387 iron 452 lead 128 human body 3500 water 4186 ice 2000 Specific Heat : examples • 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 • the same

14. Lf (J/kg) Lv (J/kg) water 33.5 x 104 22.6 x 105 Latent Heat • Latent heat: amount of energy needed to add or to remove from a substance to change the state of that substance. • Phase change: T remains constant but internal energy changes • heat does not result in change in T (latent = “hidden”) • e.g. : solid  liquid or liquid gas heat goes to breaking chemical bonds • L = Q / m • Heat per unit mass [L] = J/kg • Q =  m L + if heat needed (boiling) - if heat given (freezing) • Lf : Latent heat of fusion solid  liquid • Lv : Latent heat of vaporization liquid gas

15. T (oC) 120 100 80 60 40 Water + Steam Steam 20 0 Water + Ice Water -20 -40 62.7 396 815 3080 Energy added (J) Latent Heats of Fusion and Vaporization

16. Energy in Thermal Processes • Solids, liquids or gases have internal energy • Kinetic energy from random motion of molecules • translation, rotation, vibration • At equilibrium, it is related to temperature • Heat: transfer of energy from one object to another as a result of their different temperatures • Thermal contact: energy can flow between objects T2 T1 > U2 U1

17. A Th Tc Energy flow Dx Energy transfer mechanisms • Thermal conduction (or conduction): • Energy transferred by direct contact. • E.g.: energy enters the water through the bottom of the pan by thermal conduction. • Important: home insulation, etc. • Rate of energy transfer • through a slab of area A and thickness Dx, with opposite faces at different temperatures, Tc and Th • k : thermal conductivity  =Q/t = k A (Th - Tc ) / x

18. Thermal Conductivities J/s m 0C J/s m 0C J/s m 0C

19. Energy transfer mechanisms • Convection: • Energy is transferred by flow of substance • E.g. : heating a room (air convection) • E.g. : warming of North Altantic by warm waters from the equatorial regions • Natural convection: from differences in density • Forced convection: from pump of fan • Radiation: • Energy is transferred by photons • E.g. : infrared lamps • Stephan’s law • s =5.710-8 W/m2 K4 , T is in Kelvin, and A is the surface area • e is a constant called the emissivity  = Q/t = Ae T4 : Power

20. Resisting Energy Transfer • 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 cold liquid

21. Chap.17: Ideal gas and kinetic theory • Consider a gas in a container of volume V, at pressure P, and at temperature T • Equation of state • Links these quantities • Generally very complicated: but not for ideal gas • Equation of state for an ideal gas • Collection of atoms/molecules moving randomly • No long-range forces • Their size (volume) is negligible R is called the universal gas constant PV = nRT In SI units, R =8.315 J / mol·K n = m/M : number of moles

22. PV = N kBT Boltzmann’s constant m=mass M=mass of one mole • Number of moles: n = m/M • One mole contains NA=6.022 X 1023 particles : Avogadro’s number= number of carbon atoms in 12 g of carbon-12 • In terms of the total number of particles N • P, V, and T are thethermodynamics variables PV = nRT = (N/NA) RT kB = R/NA = 1.38 X 10-23 J/K kBiscalled the Boltzmann’s constant

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.

24. The Ideal Gas Law • What is the volume of 1 mol of gas at STP ? • T = 0 oC = 273 K • p = 1 atm = 1.01 x 105 Pa

25. Example • Beer Bubbles on the Rise • Watch 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?

26. Kinetic Theory of an Ideal Gas • Microscopic model for a gas • Goal: relate T and P to motion of the molecules • Assumptions for ideal gas: • Number of molecules N is large • They obey Newton’s laws (but move randomly as a whole) • Short-range interactions during elastic collisions • Elastic collisions with walls • Pure substance: identical molecules

27. Distribution of Molecular Speeds • The particles are in constant, random motion, colliding with each other and with the walls of the container. • Each collision changes the particle’s speed. • As a result, the atoms and molecules have different speeds.

28. Kinetic Theory • The average force exerted by one wall • Time between successive collisions on the wall • Action-reaction gives

29. root-mean-square speed volume Pressure • For N molecules, the average force is:

30. Ideal gas law • Pressure is

31. Concept of temperature • 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?

32. Example: Speed of Molecules in Air • Air is primarily a mixture of nitrogen N2 molecules (molecular mass 28.0u) and oxygen O2 molecules (molecular mass 32.0u). • Assume 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. • For nitrogen

33. Internal energy of a monoatomic ideal gas • The kinetic energy per atom is • Total internal energy of the gas with N atoms

34. Kinetic Theory of an Ideal Gas: summary • Microscopic model for a gas • Goal: relate T and P to motion of the molecules • Assumptions for ideal gas: • Number of molecules N is large • They obey Newton’s laws (but move randomly as a whole) • Short-range interactions during elastic collisions • Elastic collisions with walls • Pure substance: identical molecules • Temperature is a direct measure of average kinetic energy of a molecule

35.  Kinetic Theory of an Ideal Gas: summary • Theorem of equipartition of energy • Each degree of freedom contributes kBT/2 to the energy of a system (e.g., translation, rotation, or vibration) • Total translational kinetic energy of a system of N molecules • Internal energy of monoatomic gas: U = Kideal = Ktot trans • Root-mean-square speed:

36. a) x1.4 b) x2 c) x4 a) x1 b) x1.4 c) x2 Lecture 34: ACT 4 • 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? 2) If N is doubled, what happens to the rate at which a single molecule in the gas has a wall bounce?

37. Diffusion • The process in which molecules move from a region of higher concentrationto one of lower concentration is called diffusion. • Ink droplet in water

38. Why is diffusion a slow process ? • 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? • Many collisions !

39. Comparing heat and molecule diffusion • Both ends are maintained at constant concentration/temperature

40. A Th Tc Energy flow L Fick’s law of diffusion • For heat conduction between two side at constant T conductivity temperature gradient between ends • 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 diffusion constant concentration gradient between ends SI Units for the Diffusion Constant: m2/s