physics 1501: lecture 34 today ’ s agenda

Physics 1501: Lecture 34, Pg 1

Physics 1501: Lecture 34Physics 1501: Lecture 34TodayToday’’s Agendas Agenda

AnnouncementsHomework #11 (Dec. 2) and #12 (Dec. 9): 2 lowest dropped

Honors’ students: see me at 2:30 today !

Today’s topicsChap.16: Temperature and Heat

» Thermal expansion

» Heat transfer

» Latent HeatHeat transfer processes

» Conduction, convection, radiation

» Application


Chap. 16: Temperature and HeatChap. 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 energeticallyIn 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


Temperature scalesTemperature scales Three main scales

212

Farenheit

100

Celcius

32 0 273.15

373.15

Kelvin

Water boils

Water freezes

0-273.15-459.67Absolute Zero


Some interesting factsSome interesting facts 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.

T (K)

108

107

106

105

104

103

100

10

1

0.1

Hydrogen bomb

Sun’s interior

Solar corona

Sun’s surface

Copper melts

Water freezes

Liquid nitrogen

Liquid hydrogen

Liquid helium

Lowest T~ 10-9K


Thermal expansionThermal expansion

In most liquids or solids, when temperature risesmolecules have more kinetic energy

» they are moving faster, on the averageconsequently, things tend to expand

amount of expansion L depends on…change in temperature Toriginal length L0

coefficient of thermal expansion

» L0 + L = L0 + L0 T

L = L0 T (linear expansion)

V = L0 T (volume expansion)

L0 L

V

V + V


Lecture 34Lecture 34,, ACT 1ACT 1Thermal expansionThermal expansion

As you heat a block of aluminum from 0 oC to 100 oC, its density

(a) increases (b) decreases (c) stays the same


Lecture 34: Lecture 34: ACT 2ACT 2Thermal expansionThermal expansion

An aluminum plate (=2410-6) has a circular hole cut in it. A copper ball (solid sphere, =1710-6) 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


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 expansionThermal expansion


Special system: WaterSpecial system: Water

Most liquids increase in volume with increasing T water is specialdensity 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.

(kg/m3)

T (oC)

Reason: chemical bonds of H20 (see your chemistry courses !)


Lecture 34: Lecture 34: ACT 3ACT 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) 4 oC (c) 32 oC (d) 100 oC (e) 212 oC


HeatHeat

Solids, liquids or gases have internal energyKinetic 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

T1T2

U1U2

>


HeatHeat 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 = 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)

Sign convention: +Q : heat gained- Q : heat lost


Specific Heat : examplesSpecific 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) aluminumb) copperc) the same

Substance c in J/(kg-C)aluminum 900copper 387iron 452lead 128human body 3500water 4186ice 2000


Latent HeatLatent Heat

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 fusionsolid liquid

Lv : Latent heat of vaporizationliquid gas

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

Lf (J/kg) Lv (J/kg)

water 33.5 x 104 22.6 x 105


Latent Heats of Fusion and VaporizationLatent Heats of Fusion and Vaporization

Energy added (J)

T (oC)

120

100

80

60

40

20

0

-20

-40Water

Water+

Ice

Water + Steam Steam

62.7 396 815 3080


Energy in Thermal ProcessesEnergy in Thermal Processes

Solids, liquids or gases have internal energyKinetic 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

T1T2

U1U2

>


Energy transfer mechanismsEnergy 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 transferthrough a slab of area A and

thickness x, with opposite faces at different temperatures, Tc and Th

k : thermal conductivity

x

Th Tc A

Energy flow

=Q/t = k A (Th - Tc ) / x


Thermal ConductivitiesThermal Conductivities

Aluminum 238 Air 0.0234 Asbestos 0.25

Copper 397 Helium 0.138 Concrete 1.3

Gold 314 Hydrogen 0.172 Glass 0.84

Iron 79.5 Nitrogen 0.0234 Ice 1.6

Lead 34.7 Oxygen 0.0238 Water 0.60

Silver 427 Rubber 0.2 Wood 0.10

J/s m 0C J/s m 0C J/s m 0C


Energy transfer mechanismsEnergy 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

= 5.710-8 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


Resisting Energy TransferResisting 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.

VacuumVacuum

SilveredSilveredsurfacessurfaces

Hot orHot orcoldcoldliquidliquid


Chap.17: Ideal gas and kinetic theoryChap.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 quantitiesGenerally 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

Equation of state for an ideal gasCollection of atoms/molecules moving randomlyNo long-range forcesTheir size (volume) is negligible


BoltzmannBoltzmann’’s constants constant

In terms of the total number of particles N

P, V, and T are the thermodynamics variables

PV = nRT = (N/NA ) RT

kB is called the Boltzmann’s constant

kB = R/NA = 1.38 X 10-23 J/K

PV = N kB T

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

m=mass M=mass of one mole


Note on massesNote on massesTo 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 atomicmass units. For example, a Li atom has a mass of 6.941u.


What is the volume of 1 mol of gas at STP ?T = 0 oC = 273 Kp = 1 atm = 1.01 x 105 Pa

The Ideal Gas LawThe Ideal Gas Law


ExampleExample 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?


Kinetic Theory of an Ideal GasKinetic Theory of an Ideal Gas

Assumptions for ideal gas:Number of molecules N is largeThey obey Newton’s laws (but move

randomly as a whole)Short-range interactions during elastic

collisionsElastic collisions with wallsPure substance: identical molecules

Microscopic model for a gas Goal: relate T and P to motion of

the molecules


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.


Kinetic TheoryKinetic Theory The average force exerted by one wall

Time between successive collisions on the wall

Action-reaction gives


For N molecules, the average force is:

root-mean-squarespeed

volume

PressurePressure


Ideal gas lawIdeal gas law Pressure is


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 temperatureConcept of temperature


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

Example: Example: Speed of Molecules in Air


Internal energy of a monoatomic ideal gasInternal energy of a monoatomic ideal gas The kinetic energy per atom is

Total internal energy of the gas with N atoms


Kinetic Theory of an Ideal Gas: summaryKinetic Theory of an Ideal Gas: summary

Assumptions for ideal gas:Number of molecules N is largeThey obey Newton’s laws (but move

randomly as a whole)Short-range interactions during elastic

collisionsElastic collisions with wallsPure substance: identical molecules

Temperature is a direct measure of average kinetic energy of a molecule

Microscopic model for a gas Goal: relate T and P to motion of the

molecules


Theorem of equipartition of energy

Each degree of freedom contributes kBT/2 to the energy of a system (e.g., translation, rotation, or vibration)

Kinetic Theory of an Ideal Gas: summaryKinetic Theory of an Ideal Gas: summary

Total translational kinetic energy of a system of N molecules

Internal energy of monoatomic gas: U = Kideal = Ktot trans

Root-mean-square speed:


Consider a fixed volume of ideal gas. When N or T is doubled the pressure increases by a factor of 2.

11) If T is doubled, what happens to the rate at which ) If T is doubled, what happens to the rate at which a single a single moleculemolecule in the gas has a wall bounce? in the gas has a wall bounce?

b) x2a) x1.4 c) x4

22) If N is doubled, what happens to the rate at which ) If N is doubled, what happens to the rate at which a single a single moleculemolecule in the gas has a wall bounce? in the gas has a wall bounce?

b) x1.4a) x1 c) x2

Lecture 34: Lecture 34: ACT 4ACT 4


DiffusionDiffusion The process in which molecules move from a region

of higher concentration to one of lower concentration is called diffusion.Ink droplet in water


Why is diffusion a slow process ?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 !


Comparing heat and molecule diffusionComparing heat and molecule diffusion Both ends are maintained at constant concentration/temperature


concentration gradientbetween ends

diffusion constant

SI Units for the Diffusion Constant: m2/s

FickFick’’s law of diffusions law of diffusion For heat conduction between two side at constant T

L

Th Tc A

Energy flow

conductivity temperature gradientbetween 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

physics 1501: lecture 34 today ’ s agenda

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