Thermodynamics Heat Transfer The Kinetic Theory of Gases Molecular Model

Transcription

1 Thermodynamics Heat Transfer The Kinetic Theory of Gases Molecular Model Lana Sheridan De Anza College May 1, 2017

2 Last time more about phase changes work, heat, and the first law of thermodynamics P-V diagrams applying the first law in various cases heat transfer, conduction, Newton s law of cooling

3 tion. int does Warm Upgas Question and (b) the net energy transferred as heat Q positive, negative, or zero? ered any iliband p V For one complete cycle as shown in the P-V diagram here, E int for the gas is (A) positive (B) negative rnal energy change (C) zero KEY IDEA 1 Halliday, Resnick, Walker, page 495.

4 tion. int does Warm Upgas Question and (b) the net energy transferred as heat Q positive, negative, or zero? ered any iliband p V For one complete cycle as shown in the P-V diagram here, the net energy transferred as heat Q is (A) positive (B) negative rnal energy change (C) zero KEY IDEA 1 Halliday, Resnick, Walker, page 495.

5 Overview continue heat transfer mechanisms conduction over a distance convection radiation and Stephan s law modeling an ideal gas at the microscopic level pressure and temperature from microscopic model

6 Heat Transfer When objects are in thermal contact, heat is transferred from the hotter object to the cooler object There are various mechanisms by which this happens: conduction convection radiation

7 Thermal Conduction over distance For Newton s law of cooling, we assumed we have a system at one temperature throughout, T, and an environment at another temperature T. What if we have a system that is in contact with two different environments (thermal reservoirs) at different temperatures? The system will conduct heat from one reservoir to the other. The system will not be the same temperature throughout.

8 Transfer Thermal Mechanisms Conduction in Thermal Processes over distance609 s your hand by means ame, the microscopic ositions. As the flame begin to vibrate with lide with their neighly, the amplitudes of m the flame increase ected. This increased he metal and of your The opposite faces are at different temperatures where T h T c. T h A ties of the substance estos in a flame indefough the asbestos. In uch as asbestos, cork, Energy transfer for T h T c Figure Energy transfer r conductors Rate of because heat transfer between surfaces: tals are good thermal ness Dx. that are relatively free power, P = Q T = ka arge distances. There- t x x T c through a conducting slab with a cross-sectional area A and a thick-

9 Thermal Conduction over distance Fourier s Law Imagining a subsection of the slab with an area A and an infinitesimal thickness dx: P = ka dt dx where k is the thermal conductivity and dt dx is called the temperature gradient. If k is large for a substance, the substance is a good conductor of heat. The units of k are W m 1 K 1.

10 P 5 kaa T h 2 T c b Thermal Conduction over distance The opposite ends of the rod are in thermal contact with Imagine a uniform energy rod of length reservoirs L, that different has been placed between two thermal reservoirs temperatures. for a long time. Assume for this bar k does not depend on temperature or position. L T h Energy transfer T c T h T c Insulation Figure Conduction of energy through The temperature at each point is constant in time and the a uniform, insulated rod of length L. gradient everywhere is dt dx = T h T c L the rate of energy transfer by conduction through the rod is

11 are in thermal contact with energy reservoirs at different temperatures. Thermal Conduction over distance L T h Energy transfer T c T h T c Insulation Then, Figure Conduction of energy through a uniform, insulated rod ( of length ) Th T L. c P = ka L the rate of energy transfer by conduction through the rod is P 5 kaa T h 2 T c b L

12 are in thermal contact with energy reservoirs at different temperatures. Thermal Conduction over distance L T h Energy transfer T c T h T c Insulation Then, Figure Conduction of energy through a uniform, insulated rod ( of length ) Th T L. c P = ka L the rate of energy transfer by conduction through the rod is What if there are many different bars for heat to be transferred through? P 5 kaa T h 2 T c b L

13 Thermal Conduction through multiple materials P 5 k For a compound slab containing s thermal conductivities k 1, k 2,..., the steady For situation state is (a): T h Rod 1 Rod 2 T c P = A(T h T c ) (L 1 /k 1 ) + (L 2 /k 2 ) P 5 a Rod 1 (See ex. 20.8) where T h and T c are the temperature stant) and the summation is over all s results For situation from a (b): consideration of two th b T h Rod 2 T c Figure (Quick Quiz 20.5) In which case is the rate of energy transfer larger? Q P uick = Quiz P P ( 2 You have two rods o formed from k1 A 1 different materials. T = + k ) 2A 2 (T different temperatures so h T that ener c ) L 1 L 2 can be connected in series as in Fig In which case is the rate of energy when the rods are in series. (b) The (c) The rate is the same in both cas

14 Thermal Conduction through multiple materials Compare: P = I = ( ) ka T L ( ) 1 V R On the LHS we have transfer rates, on the RHS differences that propel a transfer. L You can think of ka as a kind of resistance. k is a conductivity, like σ (electrical conductivity). Recall, R = ρl A = L σa.

15 Thermal Conduction through multiple materials For multiple thermal transfer slabs in series: P = 1 i (L i/(k i A)) T For multiple thermal transfer slabs in parallel: ( ) k i A i P = T L i i Now for convenient comparison, let r i = L i k i A i. Then r i is a thermal resistance, for the ith slab.

16 Thermal Conduction through multiple materials For multiple resistors in series: ( ) 1 I = i R V i For multiple thermal transfer slabs in series: ( ) 1 P = i r T i For multiple resistors in parallel: ( ) 1 I = V For multiple thermal transfer slabs in parallel: ( ) 1 P = T i i R i r i

17 Thermal Conduction and Ohm s Law Fourier s work on thermal conductivity inspired Ohm s model of electrical conductivity and resistance!

18 L 1 /k 1 L 2 /k 2 Thermal Conductivity Question Eq to apply to any number n of materials making up The figure P cond shows A(T H T the face C ) and. interface temperatures (18-37) of a composite slab consisting (L/k) of four materials, of identical gn in the thicknesses, denominator through tells which us to the add heat the transfer values is of steady. L/k for Rank all the materials according to their thermal conductivities, greatest first. T 7 the face and 25 C 15 C 10 C 5.0 C 10 C res of a comting of four a b c d al thicknesses, through which the heat transfer is steady. Rank the matheir thermal conductivities, greatest first. (A) a, b, c, d (B) (b and d), a, c (C) c, a, (b and d) the flame (D) (b, of c, a candle d), aor a match, you are watching thermal sported upward by convection. Such energy transfer occurs as air or water, comes in contact with an object whose tem- 1 Halliday, Resnick, Walker, page 495.

19 Thermal Conduction and Insulation Engineers generally prefer to quote R-values for insulation, rather than using thermal conductivity, k. For a particular material: R = L k This is its length-resistivity to heat transfer. A high value of R indicates a good insulator. The units used are ft 2 F h / Btu. (h is hours, Btu is British thermal units, 1 Btu = 1.06 kj)

20 Convection In liquids and gases convection is usually a larger contributor to heat transfer. In convection, the fluid itself circulates distributing hot (fast moving) molecules throughout the fluid. When there is gravity present, convection current circulations can occur.

21 Convection Hot fluid expands, and since it is less dense, it will have a greater buoyant force and rise. Cooler, denser fluid will tend to sink.

22 Convection Heat loss by convection from a person s hand: This type of convection is called free convection.

23 Forced Convection External energy can also drive convection by means of a pump or fan. This is used in convection ovens to evenly heat food. It is also used in cooling systems to keep cool air flowing over hot components.

24 Radiation Heat can also be transferred across a vacuum by radiation. This radiation takes the form of electromagnetic (em) radiation, or light. However, most of this radiation is not in the range of wavelengths that are visible to us. Light carries energy, so this an energy transfer. Dark colored surfaces absorb and emit radiation more readily that white or shiny surfaces.

25 Stefan s Law How fast does a hot object radiate away energy? where P is power P = σaet 4 σ = W m 2 K 4 A is the surface area of the object e is the emissivity (0 e 1 always) T is temperature The net rate of energy change depends on the difference in temperature T, between an object and its environment. P net = σae(t 4 o T 4 e )

26 Emissivity and Absorptivity Properties of a surface tell us how it will interact with light. emissivity, e (or ɛ), ratio of emitted energy to the amount that would be radiated if the object were a perfect black body absorptivity, α, the ratio of energy absorbed to energy incident These are not independent of each other, and both depends on the wavelength of light.

27 Emissivity and Absorptivity Kirchhoff s law of thermal radiation can be stated (this isn t how Kirchhoff stated it): For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity. This is true for every wavelength: e(λ) = α(λ). For perfect blackbodies e = 1. Very reflective surfaces have low e and α.

28 Emissivity and Absorptivity Why are survival and rescue blankets now made out of shiny material?

29 Light (Electromagnetic Radiation) Light travels at this fixed speed, c = m s 1. For any wave, if v is the wave propogation speed: v = f λ For light: c = f λ So, if the frequency of the light is given, you also know the wavelength, and vice versa. λ = c f ; f = c λ

30 Electromagnetic spectrum

31 Electromagnetic spectrum

32 Blackbody Radiation All objects radiate light with characteristic wavelengths depending on the object s temperature. Hotter objects emit light with shorter wavelengths (on average). 1 Graph from Wikipedia, created by user Darth Kule.

33 Blackbody Radiation: Wien s Law Wien s (Displacement) Law relates the peak wavelength emitted by a blackbody to its temperature: λ max = b T b = m K is a constant. 1 Figure from HyperPhysics.

34 Radiation People emit light as well, but the wavelengths corresponding to our body temperature are longer than what we can see with the naked eye. Humans and warm-blooded animals radiate infrared radiation. ( below red )

35 Visible vs. Infrared radiation

36 Radiation Objects much hotter than the human body will radiate at shorter wavelengths that are visible to us.

37 The Greenhouse Effect Gardeners use greenhouses, huts made of glass or transparent plastic, to keep plants warm. The effect that causes this to work is called the greenhouse effect. The Sun is very hot and emits a lot of radiation in the visible (shorter) wavelengths. Objects on Earth absorb this radiation and emit there own. However, since object on the Earth are substantially cooler than the Sun, the wavelengths re-emitted are longer. The glass allows the shorter wavelengths through, but traps the longer wavelengths.

38 The Greenhouse Effect

39 The Greenhouse Effect 1 Figure from the National Academy of Sciences, America s Climate Choices

40 Kinetic Theory of Gases We have already started to study what happens in thermodynamic systems to bulk properties in various transformations. However, just from looking at macroscopic quantities it is not entirely clear how to interpret pressure, or what temperature or heat actually is.

41 Kinetic Theory of Gases We have already started to study what happens in thermodynamic systems to bulk properties in various transformations. However, just from looking at macroscopic quantities it is not entirely clear how to interpret pressure, or what temperature or heat actually is. Statistical thermodynamics seeks to explain how these macroscopic quantities arise from the microscopic behavior of particles, on average. We cannot model every the motion of every single particle in a substance, but we can say a lot about the ensemble of particles statistically.

42 its way toward a collision Molecular Model of an Ideal Gas l gas. A structural m that cannot be ple, we can only e the solar system as led to different l, with the Earth at f course, the latter to observe directly have been develodel (Section 42.4). de for experimensystem makes pre- Earth. It turns out z d moves with velocity v on with the wall. d n that We occurs model with the particles Figure of21.1 gas as A cubical small, box identical, with and obeying sides of length d containing an another Newton s model. laws, with no long range interactions. ideal gas. called kinetic thewith the We following assume all collisions are elastic. m 0 y For now, consider a single particle in a cubic box, of side length d. 1 Figure from Serway & Jewett. S v i v xi d x

43 ton s laws of motion, but as a whole their motion can move in any direction with any speed. nly by short-range forces during elastic colliconsistent with the ideal gas model, in which ng-range We start forces by on considering one another. the origin of stic collisions pressure. with the walls. Molecular Model of an Ideal Gas When a particle, mass m 0, rebounds off gas as consisting of single atoms, the behavior of a wall, its momentum change is: of ideal gases rather well at low pressures. Usus have no effect on the motions considered here. tic theory, let us relate p = the 2m macroscope 0 v x variable ities. Consider a collection of N molecules of an V. As The indicated averageabove, force on the the container particleis isa cube e shall first focus our attention F part = p on one of these it is moving so that its component of velocity in 1.2. (The subscript i here refers t to the ith molitial value. We will combine the effects of all the and by Newton III the force on the wall e collides elastically with any wall (property 2(c) pendicular is: to the wall F = 2m is reversed because the 0v x the mass of the molecule. The molecule i is modich the impulse from the wall causes a change in t e the momentum component p xi of the molecule v after the collision, the change in the x com- S v i v yi v xi S v i v xi v yi The molecule s x component of momentum is reversed, whereas its y component remains unchanged. Figure 21.2 A +x molecule makes

44 Molecular Model of an Ideal Gas The gas particle bounces back and forth between the left and right walls. We do not know how long it interacts with the wall for each time, but what we can know, and is more useful, is what is the average force on the wall over a long period of time. The time between the particle hitting the right wall once and hitting it again is t = 2d v x So the average force on the wall from the one particle over long periods of time is F = m 0v 2 x d i

45 Molecular Model of an Ideal Gas In a gas, there will be many particles (assume each has mass m 0 ) interacting with each wall in this way. The ith particle exerts a force on the wall: F i = m 0vxi 2 i d (we ignore collisions between particles, since they are small and density is low) The magnitude of the total force on the wall is: F = m 0 d N i=1 v 2 xi where N is the number of particles.

46 Molecular Model of an Ideal Gas We can re-write the sum N i=1 v 2 xi in terms of the average of the x-component of the velocity squared: F = m 0 d N v 2 x We can already relate this force to a pressure, but first, let s relate it to the average translational kinetic energy of a particle.

47 Molecular Model of an Ideal Gas For a particle in 3-dimensions: v 2 i = v 2 xi + v 2 yi + v 2 zi If this is true for each individual particle, it is true for averages over many particles automatically: v 2 = v 2 x + v 2 y + v 2 z

48 Molecular Model of an Ideal Gas For a particle in 3-dimensions: v 2 i = v 2 xi + v 2 yi + v 2 zi If this is true for each individual particle, it is true for averages over many particles automatically: v 2 = v 2 x + v 2 y + v 2 z Is there any reason why the particles should have different motion in the y, z directions than the x direction? (We neglect long range forces, like gravity.)

49 Molecular Model of an Ideal Gas For a particle in 3-dimensions: v 2 i = v 2 xi + v 2 yi + v 2 zi If this is true for each individual particle, it is true for averages over many particles automatically: v 2 = v 2 x + v 2 y + v 2 z Is there any reason why the particles should have different motion in the y, z directions than the x direction? (We neglect long range forces, like gravity.) No! We assume isotropy: the gas behaves the same way in each direction. v 2 x = v 2 y = v 2 z and v 2 = 3v 2 x

50 Molecular Model of an Ideal Gas becomes: F = m 0 d N v 2 x F = 1 m 0 3 d N v 2 = 2 N 3 d K trans where K trans = 1 2 m 0v 2 is the average translational kinetic energy of a particle.

51 Pressure from the Molecular Model Using P = F /A, the pressure at the wall and throughout the gas is P = 2 N 3 Ad K trans which, since V = Ad = d 3 we can write as: P = 2 N 3 V K trans This relates the pressure in the gas to the average translational kinetic energy of the particles. More K.E., or less volume higher pressure.

52 Relation to Macroscopic view of an Ideal Gas Ideal gas equation: or equivalently: PV = nrt PV = Nk B T If we put our new expression for pressure into this equation: 2 3 N K trans = Nk B T We can cancel N from both sides and re-arrange: K trans = 1 2 m 0v 2 = 3 2 k BT

53 Temperature from the Molecular Model We can also relate temperature to molecular motion! T = 2 3k b K trans Temperature is directly proportional to the translational kinetic energy of the particles.

54 RMS Speed and Temperature K = 1 2 m 0v 2 = 3 2 k BT It would also be useful to express the average speed in terms of the temperature. Since the motion of the gas molecules are isotropic, the average velocity is zero. However, we can instead consider the root-mean-square (rms) speed. That is convenient here because we have the average of the squares of the speed, not the average speed itself.

55 RMS Speed and Temperature root-mean-square (rms) speed: 3kB v rms = v 2 T = m 0 Alternatively, it can be expressed 3RT v rms = M where M is the molar mass. rms speed is higher for less massive molecules for a given temperature.

56 RMS Speed Question An ideal gas is maintained at constant pressure. If the temperature of the gas is increased from 200 K to 600 K, what happens to the rms speed of the molecules? (A) It increases by a factor of 3. (B) It remains the same. (C) It is one-third the original speed. (D) It is 3 times the original speed. 1 Serway & Jewett, page 644, question 2.

57 Summary heat transfer mechanisms: conduction, convection, radiation molecular models Collected Homework! due Monday, May 8. Homework Serway & Jewett: Ch 20, onward from page 615. OQs: 11; CQs: 1, 9; Probs: 43, 45, 47, 51, 55 Ch 21, onward from page 644. OQs: 1, 3, 5; Probs: 1, 3, 5, 9, 13