Category: Thermodynamics1

Updated:2014March13

II. Heat and Phase changes

Outline

A. Heat Capacity (Storage of Heat)

B. Heat Flow

C. Phases of Matter

D. References

2

A. Heat Capacity

1. Heat is Energy

2. Storage of Energy: Heat Capacity

3. Atomic Theory of Specific Heats

3

1. Heat is Energy

(a) Caloric Theory

(b) Friction creates heat

(c) Equivalence of heat and energy

4

1a. Caloric Theory (1783)

• “Caloric”: the substance of heat

• It flows like a fluid from hot to cold objects until “level” (same temperature)

• It is conserved (heat cannot be created, only moved around)

• 1 calorie raises 1 gm of water by 1C (Nicolas Clement 1824)

5

Antoine Lavoisier1743-1794

(beheaded-French Revolution)

1b. Mechanical Equivalence of Heat

• Count Rumford: Inventor of the “drip” coffee pot

• 1798 shows that heat is NOT conserved (boring a cannon creates enough heat via friction to boil water)

6

Sir Benjamin ThompsonCount Rumford

1753-1814

1c. Heat is Energy (Joule experiment)

• 1843-1845 experiment, dropping weight drives paddlewheel that heats fluid.

• 1 calorie of heat is equivalent to 4.184 Joules of work

• 1847 Helmholtz states principle of conservation of total energy

7

James Prescott Joule 1818-1889

2. Storage of Heat

(a) Heat Capacity

(b) Specific Heat

(c) Calorimetery

8

2a. Heat Capacity

• Heat is a form of energy that can be stored

• Heat Capacity: the amount of heat Q required to raise temperature of system by T=1

• SI Units: Joule/C

• Extensive quantity

9

T

QC

2b. Specific Heat

• Specific Heat is the (intensive) measure of heat capacity per unit (thermal) mass

• Joseph Black (1760s) shows materials have different values of specific heat

10

Joseph Black1728-1799

Tm

Q

m

Cc

Ckg

Jc ][

Material J/kg C cal/gm C

Copper 386 0.092

Aluminum 900 0.215

Air 1050 0.251

Ice 2220 0.530

Water 4186 1.000

2c. Calorimetry

• First calorimeter used by Lavoisier & Laplace 1782

• Measure specific heat “c” of block of metal (mass M) by heating it to temperature Th, drop into mass “m” of cold water (Tc), measure final temperature T

11

)()( cwh TTmcTTcM

3. Atomic Theory of Specific Heat

(a) Specific Heat of Gas

(b) Specific Heat of Solids, Dulong-Petit Law

(c) Low Temperature Behavior

12

3a. Specific Heat of a Gas

• At constant volume*, heat added to a monatomic gas goes into random kinetic energy of molecules. All monatomic gasses have same specific heat per mole (R=gas constant) !

• For diatomic gasses, heat also goes into two axes of rotational kinetic energy. Each mode contributes (1/2)R per mole

13

Rcmol 25 :Diatomic

Cmol

Jc

Rc

TRnQ

mol

mol

][

23

23

*If instead measured at constant pressure, the specific heat is more because energy goes into work of expanding the gas.

3b. Specific Heats of Solids

• Dulong & Petit Law (1819) show that many solids have the same specific heat if measured per mole (instead of per mass) of 25 J/mol C

• Later it was realized this is related to gas constant c=3R

14

Material J/kg C gm/mol J/mol C

Aluminum 900 26.98 24.3

Copper 386 63.55 24.5

Silver 236 107.87 25.5

Tungsten 134 183.84 24.6

Lead 128 207.2 26.5

3c. Einstein and Debye Models

• Einstein (1907) and Debye (1912) show that specific heats all approach 3R at high enough temperature

• At low temperature go to zero (this prevents being able to go below absolute zero!)

15

3d. Einstein and Debye Models• Einstein’s Model (1907) works well at

intermediate and high temperatures, but not at low temperature.

16

22

13

TET

TET

e

e

T

T

Nk

C Ev

34

0 2

43

5

4

3

13

3

D

v

x

x

D

v

T

T

Nk

C

e

exdx

T

T

Nk

C DTT

• Debye’s Model (1912) works well at low temperature and high, but not in the middle.

• The Debye Temperature: TD

• At low temperatures Debye shows that the specific heat increases with the cube of the temperature.

B. Heat Flow

There are three mechanisms by which heat flows:

1. Convection

2. Conduction

3. Radiation

17

1. Convection

• Heat moved by physical flow of gas/liquid. Often creates “convection cells” (can be seen on surface of sun!).

• No good formula!

18

2. Conduction(a) Phenomena: Heat in a solid is

transported by collisions between electrons. Generally good conductors of electricity are also good conductors of heat because the electrons are free to move.

(b) Newton’s Law of Cooling• Loss of heat is proportional to the

difference in temperature of system to environment [R=thermal resistance]

• Temperature of system will follow an exponential decay law [C=heat capacity]

19

RCt

eTtT 0)(

)(1

ch TTRt

Q

(c) Fourier’s Theory of thermal conduction

• Heat Flux: heat flow per time (Watts)=Q/t

• Amount of heat flow is inversely proportional to the “Thermal Resistance R” of the system (units of degree per Watt).

• Thermal resistance “R” depends upon geometry (length L, area A, hence its “extensive”) and the thermal conductivity “k” of the material

2c. Law of Thermal Conduction 20

Ak

LR

TTRt

Qch

1

Th Tc

2c. Thermal Conductivity

Thermal conductivity (intensive quantity!) of solids are better than liquids or gasses. In the latter, convection will usually dominate.

• Aluminum 240• Water 0.57• Air 0.024• Vacuum 0

21

Ck meter

Watt][

3. Radiation

(a) Stefan’s Law

(b) Emissivity

(c) Black Body Radiation Curve

22

3a. Josef Stefan’s Law 1879• Experimentally shows total output of light

of a hot dense (black) body is proportional to 4th power of the temperature (in Kelvin)

• Power (watts)=AT4

• =5.67x10-8 Watts/(m2-K4)• A=surface area

• 1884 Ludwig Boltzmann (formerstudent of Stefan) derives formulafrom thermodynamics.

• I was a guest speaker (Sept 2005) at theJosef Stefan Institute in Slovenia.

23

3a ii Measure Temperature of Sun

• 1604 Kepler proposes intensity of light drops of with square of distance (?)

24

•Charles Soret measures solar flux to be about 1400 Watts/m2 at surface of the earth.

•Stefan uses this to estimatetemperature of sun to be 5700 K.

3b Leslie Cube Experiment (1804)• A cube with different emissitivites on different

faces. One usually shows that the “black” surface radiates better than “white” surface.

25

3bii Emissivity• A perfect “black body” will radiate according to

Stefan’s law.

• Most systems are not perfect, and so we include a “fudge factor” called the “emissivity”: 0<e<1

• Further, the environment at temperature Te will radiate energy back into the system, so the rate at which system loses heat is:

26

44eTTeA

t

QP

3c.i. Wien’s Displacement Law

• 1893 shows that the “color” of black body is inversely proportional to temperature

• Wien’s constant=2,898,000 nm-K

• So T=6000K gives=483 nm

27

T

3c.ii. Black Body Curve• Willhelm Wien gets

Nobel Prize 1911

• 1894 coins term“black body”

• The black body emitsall colors, but whereit peaks is describedby Wien’s law

28

3.c.iii. Black Body Theory• Maxwell: hot atoms vibrate, acting

like small antennas, radiating electromagnetic waves

• Wien tries to give theory to explain shape of curve, but it fails in IR

• Rayleigh (1900) & Jeans (1905) have another theory, but it fails in UV, blowing up to infinite energy (the “ultraviolet catastrophe”).

29

3c.iv. Max Planck’s Theory

• 1900 Max Planck ad-hoc proposes that vibrations are “quantized”, i.e. come in steps of n=1, 2, 3, rather than continuous.

• Energy: E=nhfn=integer quantum numberf=frequency of oscillationh is “Planck’s Constanth=6.626x10-34 Joule-Sec

30

3cv. Planck Radiation Law• His theory exactly matched the experimental

measurements of the black body radiation curve

31

k = Boltzmann Constant (1.3810-23 Joule/Kelvin)

C. Phases of Matter

1. Phases of matter

2. Latent Heats of Phase Change

3. Phase Diagrams (PVT diagrams)

32

1. Phases

There are 4 states of matter (called “Phases”)

1.Solid

2.Liquid

3.Gas

4.Plasma (hot gas ionizes)

33

1b. Plasma

• Superheating a gas, it will ionize (electrons separate from rest of atom) into a 4th state of matter called a “plasma”.

34

1c. Van der Waals equation• Why are there phases of matter?

• Van der Waals (Nobel Prize 1910) modified gas equation predicts solid, liquid and gas phases.

• “a” represents attractive forces between molecules and “b” the volume of a mole of molecules.

35

Johannes van der Waals1837-1923

nRTnbVV

anP

2

2

2. Latent Heats

(a) Black (1761) measures the amount of heat it takes to change phase of water. Defines:

• Latent Heat of fusion (melt ice) is 80 cal/gm

• Latent Heat of vaporization (boil) 597 cal/gm

36Joseph Black

1728-1799

2b. Latent Heats-continued

• Some substances, n.b. dry ice (CO2), go straight from solid to gas, so we have a latent heat of “sublimation” (opposite process is “deposition”)

• Dry ice latent heat of sublimation is:571 J/gm

• Much better than using water ice(no melted water!)

37

3(a) PT Diagram for Pure Substance 38

• Triple Point: all 3 phases coexist.

• Above critical point it’s a “supercritcal fluid”, no distinction between gas and liquid

• Clausius-Clapeyron equation tells us the slope of the phase change curve, where “L” is the latent heat, and v is change in volume per mole of phase transition.

vT

L

dT

dP

PT Diagram for Water 39

• Expands when freezes, hence ice under pressure will melt (ice skates!)

• Note from A to D the slope is negative!

PT Diagram for Helium 40

• Helium: Has two different liquid states, hence two “triple points”

PVT diagram

• Van der Waals equation

• PVT diagrams

41

PVT diagram 42

Misc Notes• MCAT includes Van der Waals equation.• Condensation vs evaporation• Evaporation takes place at surface, boiling is throughout fluid.• Regelation (melting under pressure of ice)• Heat released by freezing, e.g. center of earth.• Change of phase of hydrogen to a metal in jupiter and saturn (liquid or solid?)• Ice III (what happened to Ice II ?), what conditions for change, glaciers?

43