Physics 1501 Lecture 35

Transcription

1 Physics 1501: Lecture 35 Todays Agenda Announcements Homework #11 (Dec. 2) and #12 (Dec. 9): 2 lowest dropped Honors students: see me after the class! Todays topics Chap.16: Temperature and Heat» Latent Heat Heat transfer processes» Conduction, convection, radiation» Application Chap. 17: ideal gas» Kinetic theory Physics 1501: Lecture 35, Pg 1 Three main scales Temperature scales Farenheit Celcius Kelvin Water boils Water freezes Absolute Zero T F = 9 5 T C + 32 o F T C = T " K T C = 5 ( 9 T F " 32 o F) T = T C K Physics 1501: Lecture 35, Pg 2 Page 1

2 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 ΔL depends on change in temperature ΔT original length L 0 coefficient of thermal expansion» L 0 + ΔL = L 0 + α L 0 ΔT» ΔL = α L 0 ΔT (linear expansion)» ΔV = β L 0 ΔT (volume expansion) L 0 V ΔL V + ΔV Physics 1501: Lecture 35, Pg 3 Special system: Water Most liquids increase in volume with increasing T water is special density increases from 0 to 4 o C! ice is less dense than liquid water at 4 o C: hence it floats water at the bottom of a pond is the denser, i.e. at 4 o C ρ(kg/m 3 ) Density T ( o C) Water has its maximum density at 4 degrees. Reason: chemical bonds of H 2 0 (see your chemistry courses!) Physics 1501: Lecture 35, Pg 4 Page 2

3 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 T 1 > T 2 U 1 U 2 Physics 1501: Lecture 35, Pg 5 Heat Heat: Q = C Δ T Q = amount of heat that must be supplied to raise the temperature by an amount Δ T. 1 cal = J [Q] = Joules or calories. 1 kcal = 1 Cal = 4186 J 1 cal = energy to raise 1 g of water from 14.5 to 15.5 o C James Prescott Joule found mechanical equivalent of heat. C : Heat capacity Q = c m Δ T c: specific heat (heat capacity per units of mass) amount of heat to raise T of 1 kg by 1 o C [c] = J/(kg o C) Sign convention: +Q : heat gained - Q : heat lost Physics 1501: Lecture 35, Pg 6 Page 3

4 L = Q / m Heat per unit mass [L] = J/kg Q = ± m L + if heat needed (boiling) - if heat given (freezing) L f : Latent heat of fusion solid liquid L v : Latent heat of vaporization liquid gas 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 f (J/kg) L v (J/kg) water 33.5 x x 10 5 Physics 1501: Lecture 35, Pg 7 Latent Heats of Fusion and Vaporization T ( o C) Water + Ice Water Water + Steam Steam Energy added (J) Physics 1501: Lecture 35, Pg 8 Page 4

5 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 T 1 > T 2 U 1 U 2 Physics 1501: Lecture 35, Pg 9 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 Δx, with opposite faces at different temperatures, T c and T h T h A T c =Q/Δt = k A (T h - T c ) / Δx k : thermal conductivity Energy flow Δx Physics 1501: Lecture 35, Pg 10 Page 5

6 Thermal Conductivities J/s m 0 C J/s m 0 C J/s m 0 C Aluminum 238 Air Asbestos 0.25 Copper 397 Helium Concrete 1.3 Gold 314 Hydrogen Glass 0.84 Iron 79.5 Nitrogen Ice 1.6 Lead 34.7 Oxygen Water 0.60 Silver 427 Rubber 0.2 Wood 0.10 Physics 1501: Lecture 35, Pg 11 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 Stephans law = Q/Δt = σae T 4 : Power σ = W/m 2 K 4, T is in Kelvin, and A is the surface area e is a constant called the emissivity Physics 1501: Lecture 35, Pg 12 Page 6

7 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 Physics 1501: Lecture 35, Pg 13 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 PV = nrt R is called the universal gas constant In SI units, R =8.315 J / mol K n = m/m : number of moles Physics 1501: Lecture 35, Pg 14 Page 7

8 Boltzmanns constant Number of moles: n = m/m In terms of the total number of particles N PV = nrt = (N/N A ) RT PV = N k B T m=mass M=mass of one mole One mole contains N A =6.022 X particles : Avogadros number = number of carbon atoms in 12 g of carbon-12 k B = R/N A = 1.38 X J/K k B is called the Boltzmanns constant P, V, and T are the thermodynamics variables Physics 1501: Lecture 35, Pg 15 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 u = "10 #27 kg The atomic mass is given in atomic mass units. For example, a Li atom has a mass of 6.941u. Physics 1501: Lecture 35, Pg 16 Page 8

9 The Ideal Gas Law pv = nrt 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 V n = RT P 8.31 J / mol " K = ( ) 273 K 1.01x10 5 Pa = m 3 = 22.4! Physics 1501: Lecture 35, Pg 17 Example Beer Bubbles on the Rise Watch the bubbles rise in a glass of beer. If you look carefully, youll 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? PV = nrt P top V top = P bottom V bottom P top P bottom = V bottom V top P top < P bottom " V top > V bottom Physics 1501: Lecture 35, Pg 18 Page 9

10 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 Newtons laws (but move randomly as a whole) Short-range interactions during elastic collisions Elastic collisions with walls Pure substance: identical molecules Physics 1501: Lecture 35, Pg 19 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 particles speed. As a result, the atoms and molecules have different speeds. Physics 1501: Lecture 35, Pg 20 Page 10

11 Kinetic Theory The average force exerted by one wall F x = "p x "t = p x,after # p x,before "t p x, before = +mv x Time between successive collisions on the wall ( ) " ( +mv x ) F x = "mv x 2L v x Action-reaction gives 2 F x, wall = "F x = +mv x L p x, after = "mv x "t = 2L = "mv 2 x L = +mv2 3L v x v x 2 = v y 2 = v z 2 v 2 = v x 2 + v y 2 + v z 2 = 3v x 2 Physics 1501: Lecture 35, Pg 21 Pressure For N molecules, the average force is: " F = N %" % $ ' mv2 # 3 & $ # L ' & root-mean-square speed P = F A = F L = " N %" % $ ' mv2 2 # 3 & $ # L 3 ' & volume Physics 1501: Lecture 35, Pg 22 Page 11

12 Ideal gas law Pressure is " P = N %" % $ ' mv2 # 3 & $ # V ' & NkT KE PV = 1 N mv 2 3 rms ( ) = 2 N 1 mv 2 3 ( 2 rms) KE = 1 mv 2 2 rms = 3 2 kt Physics 1501: Lecture 35, Pg 23 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? Physics 1501: Lecture 35, Pg 24 Page 12

13 Air is primarily a mixture of nitrogen N 2 molecules (molecular mass 28.0u) and oxygen O 2 molecules (molecular mass 32.0u). m = v rms = Example: Speed of Molecules in Air 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. 1 mv 2 2 rms For nitrogen = 3 2 kt v rms = 3kT m 28.0g mol "10 23 mol #1 = 4.65 "10#23 g = 4.65 "10 #26 kg 3kT m ( ) 293K "10 #23 J K = 4.65 "10 #26 kg ( ) = 511m s Physics 1501: Lecture 35, Pg 25 Internal energy of a monoatomic ideal gas The kinetic energy per atom is KE = 1 mv 2 2 rms = 3 2 kt Total internal energy of the gas with N atoms U = N 3 2 kt = 3 2 nrt Physics 1501: Lecture 35, Pg 26 Page 13

14 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 Newtons 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 PV = N k B T PV = 2 3 N " 1 2 mv % $ 2 ' # & 1 2 mv 2 = 3 2 k B T Physics 1501: Lecture 35, Pg 27 Kinetic Theory of an Ideal Gas: summary Theorem of equipartition of energy v 2 x = v 2 y = v 2 z = 1 3 v mv 2 i = 1 2 k T B Each degree of freedom contributes k B T/2 to the energy of a system (e.g., translation, rotation, or vibration) Total translational kinetic energy of a system of N molecules " K tot trans = N 1 2 mv % $ 2 ' = 3 # & 2 Nk T = 3 B 2 nrt Internal energy of monoatomic gas: U = K ideal = K tot trans Root-mean-square speed: v rms = v 2 = 3k B T m Physics 1501: Lecture 35, Pg 28 Page 14

15 Lecture 35: ACT 1 Consider a fixed volume of ideal gas. When N or T is doubled the pressure increases by a factor of 2. pv = NkT 1) If T is doubled, what happens to the rate at which a single molecule in the gas has a wall bounce? a) x1.4 b) x2 c) x4 2) If N is doubled, what happens to the rate at which a single molecule in the gas has a wall bounce? a) x1 b) x1.4 c) x2 Physics 1501: Lecture 35, Pg 29 Diffusion The process in which molecules move from a region of higher concentration to one of lower concentration is called diffusion. Ink droplet in water Physics 1501: Lecture 35, Pg 30 Page 15

16 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! Physics 1501: Lecture 35, Pg 31 Comparing heat and molecule diffusion Both ends are maintained at constant concentration/temperature Physics 1501: Lecture 35, Pg 32 Page 16

17 Ficks law of diffusion For heat conduction between two side at constant T conductivity Q = (ka"t )t L temperature gradient between ends T h Energy flow A T c 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 L diffusion constant m = DA"C ( )t L concentration gradient between ends SI Units for the Diffusion Constant: m 2 /s Physics 1501: Lecture 35, Pg 33 Page 17