17. Thermal Behavior of Matter

1. Gases

2. Phase Changes

3. Thermal Expansion

What unusual property of water is evident in this photo?

Ice is less dense than water.

17.1. Gases

The Ideal Gas Law:

A piston-cylinder system.

p V N k T

k = 1.381023 J / K = Boltzmann’s constant

N = number of molecules

AN n N

NA = 6.0221023 = Avaogadro’s number

= number of atoms in 12 g of 12C.

n = number of moles (mol)

p V n R T

AR N k = 8.314 J / K mol = Universal gas constant

All gases become ideal if sufficiently dilute.

Example 17.1. STP

What volume is occupied by 1.00 mol of an ideal gas

at standard temperature & pressure (STP),

where T = 0C, & p = 101.3 kPa = 1 atm?

p V n R T

n R TV

p

3 322.42 10 m 22.42 L

3

1.00 8.314 / 273.2

101.3 10

mol J K mol K

Pa

( last figure subject to round-off error )

Kinetic Theory of the Ideal Gas

Kinetic theory ( Newtonian mechanics ):

1.Gas consists of identical “point” molecules of mass m.

2. No interaction between molecules, except when they collide.

3. Random motion.

4. Collisions with wall are elastic.

Molecule i collides with right-hand wall (RHW).

Momentum transfer to wall is 2x i x ip m v

No intermolecular collision

Next collision with RHW occurs at2

ix i

Lt

v

Average force of i on RHW: ii

i

pF

t

2x im v

L

Fp

A i

iF

A 2

ix im v

A L

2x

ii

mv

V

2x

mp N v

V

22 1xx i

i

vvN

Random motion 2 2 2x y zv v v 21

3v 22 1

3 2p V N m v

2

3N K

Ideal gas law is recovered if21 3

2 2K m v k T T ~ K

in

out

Example 17.2. Air Molecule

Find K of a molecule in air at room temperature ( 20C = 293K),

& determine the speed of a N2 molecule with this energy.

3

2K k T 233

1.38 10 / 2932

J K K 216.07 10 J

2

272 14 1.66 10Nm u kg 264.65 10 kg

2 2 Kv

m

21

26

2 6.07 10

4.65 10

J

kg

5 2 22.61 10 /m s

2v v 511 /m s

3th

k Tv

mThermal speed:

Distribution of Molecular Speeds

Maxwell-Boltzmann Distribution: (elastic collisions between free particles)

High-E tail extends rapidly with T

chemical reaction easier at high T

cooling of liquid

( by escape of high-E molecules)

80 K

vth

300K

vth

2

2 exp2

mvn v C v

k T

0

n v dv N

Real Gases

Important corrections to the ideal gas model:

1.finite size of molecules available V reduced.

2.Attractive interaction between molecules (van der Waals forces) reduced P.

van der Waals equation

minimum volume

2

2

a nP V nb nRT

V

nRTP

V

2

2

nRT a nP

V nb V

17.2. Phase Changes

Phase changes take place at fixed T = TC until whole system is in the new phase.

( breaking / building bonds raises U but keeps K unchanged )

Heat of transformation L = energy per unit mass needed to change phase.

Lf = Heat of fusion ( solid liquid )

Lv = Heat of vaporization ( liquid gas )

Ls = Heat of sublimation ( solid gas )

Q L m

Water: 334 /fL kJ kg 80 /cal g

1 / /C cal g K

Same E to melt 1 g ice

or heat water by 80 C

Conceptual Example 17.1. Water Phases

T vs t for a block of ice, initially at － 20 C, that is

supplied with constant power under atmospheric P.

ice warming

melting

water warming

boiling

steam warming

You put a block of ice initially at － 20C in a pan on a hot stove with a constant power output,and heat it until it has melted, boiled, and evaporated.

Make a sketch of temperature versus time for this experiment.

Making the Connection

If you start with 0.95 kg of ice at － 20C and supply heat at the rate of 1.6 kW,

how much time will it take until you’re left with only water vapor?Heat needed to warm ice to 0 C :

0.95 2.05 / 20kg kJ kg K K ice ice ice iceH m c T 39 kJ

Heat needed to melt ice at 0 C :

2i w ice fH m L 0.95 334 /kg kJ kg 317 kJ

Heat needed to vaporize water at 100 C :

Heat needed to warm water to 100 C :

water water water waterH m c T 0.95 4.18 / 100kg kJ kg K K 397 kJ

2w v water vH m L 0.95 2260 /kg kJ kg 2140 kJ

2900

1.6

kJt

kW Time

needed :1810 s 0.5 h

GOT IT? 17.2.

You bring a pot of water to boil & then forget about it.

10 min later you come back & find it still boiling.

Is its temperature

(a) less, (b) greater than, or (c) equal to

100 C ?

Example 17.3. Meltdown!

A nuclear power plant’s reactor vessel cracks, draining all cooling water.

Although nuclear fission stops, radioactive decay continues to heat the reactor’s

2.5105 kg uranium core at the rate of 120 MW.

Once the melting point is reached, how much energy will it take to melt the core?

How long will the melting take?

Q L m

82.8 /fL kJ kgTable 17.1: for U

582.8 / 2.5 10kJ kg kg 20.7 GJ

Time to melt the core: QT

P

9

6

20.7 10

120 10

J

W

173 s 3 m

Example 17.4. Enough Ice?

When 200 g of ice at 10 C are added to 1.0 kg of water at 15 C,

is there enough ice to cool the water to 0 C?

If so, how much ice is left in the mixture?

Q L m

1.0 4.184 / / 15waterQ kg kJ kg K C

Q m c T

Heat released to bring water down to 0 C

62.8 kJ

0.2 2.05 / / 10iceQ kg kJ kg K C Heat required to bring ice up to 0 C

4.1 kJ

0.2 334 /meltQ kg kJ kg

Heat required to bring ice up to 0 C

66.8 kJ more than enough ice

Ice needed: 62.8 4.1

334 /

kJm

kJ kg

0.176 kg ice left = 200 176 24g g g

Phase Diagrams

Phase diagram: P vs T

Sublimation: solid gas

e.g., dry ice ( s-CO2 )

AB: low P, s g

CD: medium P, s l g

EF: high P, s l / f

GH: medium T, l g

Caution: Phase transition doesn’t occur instantaneously

Triple point: s-l-g coexist

= 273.16K, 0.6 kPa for H2O

Solid

Gas

liquid

Melting

Sublimation

Boiling

C.P.

T.P.

壓力

TC

PC

Supercritical fluid : l-g indistinguishable

C.P. : Critical point

17.3. Thermal Expansion

Coefficient of volume expansion :

/V V

T

1 dV

V dT

Coefficient of linear expansion :

/L L

T

3

Prob. 69

Prob. 72

GOT IT? 17.3.

If a donut-shaped object is heated, will the hole get

(a) larger, or (b) smaller ?

L L T 2 2R R T 1R R T

Example 17.5. Spilled Gasoline

A steel gas can holds 20 L at 10C.

It’s filled to the brim at 10C.

If the temperature is now increased to 25C, by how much does the can’s volume increase?

How much gas spills out?

Table 17.2: 6 112 10steel K 6 136 10steel K

5 195 10gas K

/V V

T

V V T

6 120 36 10 25 10canV L K C C 0.0108 L

5 120 95 10 25 10gasV L K C C 0.285 L

Spilled gas: 0.285 0.0108 0.275L L L

Thermal Expansion of Water

Reason: Ice crystal is open ice water

ice floats

max water occurs at 4C

At 1C5 14.8 10water K

At fixed T Tm , ice melts if P .

Application: skating.

> 0 < 0

Application: Aquatic Life & Lake Turnover

Anomalous behavior of ice-water makes aquatic life in freezing weather possible.

If deep enough, bottom water stays at 4C even when surface is iced over.

In a lake where bottom water stays at 4C year round,

surface & bottom water can mix (turnover) only in spring time when both are at 4 C.