Short Version : 17. Thermal Behavior of Matter

17.1. Gases

The Ideal Gas Law:

A piston-cylinder system.

pV 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)

pV 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?

pV 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 2pV 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.

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

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.