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Lecture 4:Lecture 4: Temperature, Heat Temperature, Heat

Transfer & Thermal ExpansionTransfer & Thermal Expansion

� What is "temperature"?

� Thermal expansion.

� What is "heat"?

� Heat capacity, specific heat & latent heat

� The First Law of Thermodynamics

� Heat Transfer

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Temperature & the "0Temperature & the "0thth Law of Thermodynamics"Law of Thermodynamics"

The "0The "0thth Law of Thermodynamics:"Law of Thermodynamics:"

If two systems are in THERMAL EQUILIBRIUM, they must be at the SAME TEMPERATURE

Some common sense notions:Some common sense notions:• There exists such a property as "temperature" that tells us

whether an object is "hot" or "cold"• Temperature has something to do with energy• Many physical properties depend upon temperature:

volume, pressure, "phase" (solid, liquid, gas), length, electrical resistance, etc.

• If we place two objects in "thermal contact," their temperatures eventually equalize: some sort of exchange of energy takes place and we reach a equilibrium

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Temperature Scales:Temperature Scales: Fahrenheit, Roemer, Celsius, Kelvin Fahrenheit, Roemer, Celsius, Kelvin

T(0C) =T(K)−273.15

0

Celsius scale:000 32)(5/9)( +⋅= CTFT

Fahrenheit scale:

Fundamental scale: Kelvin (K)• Lowest possible temperature: 0 K (absolute zero)

• Lowest lab temp. Liquid helium: 4.2 K• Liquid nitrogen: 77 K• Room temp: ~300 K• Solar surface: ~6000 K• Helium fusion: ~108 K• Nucleus-nucleus collisions: ~1012 K• “Big Bang": ~1039 K

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Consequence of temperature change.Consequence of temperature change. Thermal Expansion.Thermal Expansion.

L0

T

L0 + ∆L

T + ∆T

∆∆∆∆L = αααα L0 ∆∆∆∆T

α= coefficient of linear expansion

Similarly: coefficients of areal and volume expansion

∆∆∆∆A = γ γ γ γ A0 ∆∆∆∆T∆∆∆∆V = β β β β V0 ∆∆∆∆T

Application & demo:

Bimetallic strip

• two metal strips with

differing values of α• Strip bends since one metal

expands more than the other

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Thermal Expansion:Thermal Expansion: Sample Problem.Sample Problem.

Aluminum washer has an hole with diameter = 0.5 cm at 20 0C. If the washer is heated to 100 0C, what is the new diameter of the hole? (Aluminum α = 23 x 10-6/ 0C)

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Heat. Heat. Heat capacity and specific heatHeat capacity and specific heat

• Heat: energy exchanged due to difference in temperature.

• Units: Joules (after all it is energy!)

• Historically important unit: calorie

• One calorie = energy required to raise the temperature of 1 g of water

by 10C= 4.186 J

• (Note: nutritionists: "just one calorie" refers to 1000 calories)

• Heat capacity C of a system: heat energy required to raise the

temperature of the system by 1 0C;

Q = C ∆∆∆∆T

• Specific heat c of a substance: heat energy required to raise the

temperature of unit mass of the substance by 1 0C;

Q = m c ∆∆∆∆T

• Latent heat L of a substance: heat energy required to change the

phase of the substance (e.g. solid to liquid, liquid to vapor)

Q = m L

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Thermal equilibrium.Thermal equilibrium. Sample problem.Sample problem.

An ice cube of mass 10 g at 00C is dropped into a glass containing 250 g of water at 20 0C. Describe the final equilibrium state of the system. Latent heat of fusion of ice = 80 cal/g; specific heat of water = 1 cal/g 0C. (Neglect the heat capacity of the glass.)

Possible scenarios:

• All the ice melts; 0 0C <= Tfinal < 20 0C

• Some ice melts; Tfinal = 0 0C

• All the water freezes; Tfinal = 0 0C

• What is Tfinal?

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� System of interest: an ideal gas in a container

� Walls of container may or may not conduct heat

� One of the walls may be moveable: it is a piston that exerts a force

� Relevant parameters:

� Pressure p, Volume V

� Temperature T

� Number of molecules N

� Number of moles n (one mole = 6.02 x 1023

molecules – Avogadro’s number)

� Ideal gas law: pV = nRT = NkT

(will derive next time)

� R = universal gas constant = 8.31 J/mol·K

� k = Boltzmann constant = 1.38 x 10 -23 J/K

Suppose we allow the gas to change volume in a way that the system is always in equilibrium("quasi-static process") and the piston exerts a uniform pressure p. How much work W is done BYthe gas?

Ideal gas.Ideal gas. Work.Work.

∫=2

1

V

V

pdVW

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• Equilibrium state: a point

• Quasi-static process: a line or curve

• Isobar: constant P

• Isochore: constant V

• Isotherm: constant T

P =nRT

V∝

1

V

p

V

pV = nRT

isobar

isotherm

isochoric

p

V

(p1,V1,T1)

(p2,V2,T2)

For a "path" on the p-V diagram, work done by gas = area under curve and hence depends upon the path!

∫=2

1

V

V

pdVW

Ideal gas.Ideal gas. pp--VV diagram.diagram.

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pV = nRTp

V

V0 10V0

An ideal gas is allowed to expand to An ideal gas is allowed to expand to

1010 times its original volume during a times its original volume during a

quasiquasi--static isothermal process. How static isothermal process. How

much work is done BY the gas?much work is done BY the gas?

Ideal gas.Ideal gas. Sample problem.Sample problem.

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An ideal gas is allowed to expand to 10 times its original volume

during a quasi-static isothermal process. Since the gas does work,

how is energy conserved?

• One possibility: the gas absorbs heat Q from the environment!

• Another possibility: energy ∆∆∆∆U from some "internal source" e.g.

change in kinetic energy of molecules

W = Q - ∆∆∆∆U

• U = "internal energy"

• Unique function of temperature ("function of state")

• Both Q and W depend on the details of a process -- i.e. the path

taken on a p-V diagram

Ideal gas.Ideal gas. Fist Law of Thermodynamics.Fist Law of Thermodynamics.

(conservation of energy)(conservation of energy)

Q = ∆∆∆∆U+Wor

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p

V

An ideal gas is taken through a quasi-

static cyclic process as shown. During

the cycle, the gas absorbs 10 J of heat

from the surroundings.

• What is the change in internal energy

during one cycle?

• How much work is done BY the gas

during one cycle?

Ideal gas.Ideal gas. Cyclic process.Cyclic process.

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An ideal gas is compressed during a quasi-

static adiabatic process as shown.

("Adiabatic" means there is NO HEAT

EXCHANGED.) During the process, an

external agent does 10 J of work.

• What is the change in the internal energy of

the gas?

• Do you expect the temperature of the gas to

remain fixed, increase or decrease?

p

V

Ideal gas.Ideal gas. Adiabatic process.Adiabatic process.

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1 g of water is boiled into steam at a constant pressure of 1 atm

(1.01 x 105 N/m2). Given that 1 g steam occupies about 1600 cm3 at

1 atm, estimate the change in internal energy during this process.

(Assume you can treat this as a quasi-static process for an ideal gas.)

Density of water = 1g/cm3. Latent heat of vaporization = 2260 J/g.

Ideal gas.Ideal gas. Isobaric process.Isobaric process.

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Gas within a chamber passes through the cycle

shown in the adjacent figure.

• AB: isochoric

• BC: adiabatic

• CA: isobaric

• During process AB, QAB = 20 J.

• Net work done BY gas during the cycle = 15 J

Determine the heat absorbed during process CA.

p

V

A

B

C

Ideal gas.Ideal gas. One more cyclic process.One more cyclic process.

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A compressed gas is allowed to

suddenly expand into the atmosphere

through a nozzle.

• Can you represent this process on a

P-V diagram?

• What can you conclude about the

final temperature of the gas from the

first law of thermodynamics?

p

V

A

B

NonNon--equilibrium process.equilibrium process. JouleJoule--Thomson.Thomson.

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Heat Transfer.Heat Transfer.

Question: Good conductors of electricity are also good conductors of heat. What might you conclude from this observation?

ConductionConduction(exchange of kinetic energy)(exchange of kinetic energy)

ConvectionConvection(displacement of moving substance)(displacement of moving substance)

RadiationRadiation(electromagnetic waves)(electromagnetic waves)

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Thermal conduction.Thermal conduction. Conduction rate.Conduction rate.

Conduction rate (Watts)Conduction rate (Watts)

L

TTkAPcond

)( 21 −=

k = thermal conductivity (units: W/K· m)

T1T2

L

cross-sectional area = A

T1 > T2; both are constant in time

Conduction rate Conduction rate

must be same for must be same for

both rods!both rods!

P = k1A(T1 − TX )

L1

= k2A(T2 − TX )

L2

Given values of T1 > T2, what is the conduction rate?

[ ]21

21

2

2

1

1

21 )()(

RR

TTA

k

L

k

L

TTAP

+

−=

+

−=

Practical interest:R = L/k"Thermal resistance"

Thermal conduction.Thermal conduction. Composite rod.Composite rod.

T1T2

L1 L2

TXk1 k2

cross-sectional area = A

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4ATPrad σε=

Exchanging energy via EM wavesExchanging energy via EM waves

Emission rate:Emission rate: Prad = Power (W)

σ = Stephan-Boltzmann constant

)/(106703.5 428KmW

−×=

ε = Emissivity 0<ε<1

)( 2mAreaA =

T=Temperature in K

ε = 1 is a black body radiator4

envabs ATP σε=

Radiation. Radiation. StephanStephan--BoltzmannBoltzmann LAwLAw

Absorption rate:Absorption rate:

Process by which hot air rises and cold air descends in a roomProcess by which hot air rises and cold air descends in a room

Convection.Convection.

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If planet A (surface at room temperature) has twice the area as

planet B and has the same measured radiative power, what is the

temperature at the surface of planet B?

Radiation.Radiation. Sample problem.Sample problem.

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�� Temperature is not HeatTemperature is not Heat

�� Expanding with temperature Expanding with temperature

�� Specific heat & latent heatSpecific heat & latent heat

�� First Law of ThermodynamicsFirst Law of Thermodynamics

�� Heat TransferHeat Transfer

∆∆∆∆L = α α α α L0 ∆∆∆∆T

Q = m c ∆∆∆∆T Q = m L

∆∆∆∆U = Q - W

L

TTkAP cond

)( 21 −=

What we learned.What we learned.

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NEXT LECTURE: Chap. 19NEXT LECTURE: Chap. 19

�� Kinetic theory of gasesKinetic theory of gases

�� Distribution of speedsDistribution of speeds

�� Equipartition TheoryEquipartition Theory