CCh

ap. 1

5: p

V =

nR

T

•M

ole and Avogadro’s num

ber. •

Equations of state. •

Kinetic theory of an ideal gas. •

Heat capacities.

•First Law of Therm

odynamics.

•Therm

odynamic processes.

•Properties of an ideal gas.

1

3 E

xam

s an

d 2

More

2

1)D

EC 1 (Tue): Lecture 2)

DEC 3 (Thu): Exam

4 – Chap. 12, 14 and 15 1)

P.1 … Five quick quizzes from 12, 14 and 15

2)P.2 … 12

3)P.3 … 14

4)P.4 … 15

3)D

EC 8 (Tue): Last class 4)

DEC 8 (Tue): Com

mon M

akeup Test at 7 pm

5)D

EC 11 (Fri): Final Exam – Com

prehensive (all chapters) – Please start reviewing Eaxm

1~Exam3

materials now!

6)BTW

: Your TA has Exam

3. It should have been returned to you on N

ov 16 (Mon) at Recitation if you

didn’t pick up on Nov. 10 and 12.

Calen

dar

3

Final Exam

Exam4

LAST Class

Makeup4

Class

You don’t need to work on W

eek 13 & 14

TTopics in

Ch

ap. 1

5 a

t First G

lan

ce �

Limited subjects – KEEP

checking my PH

YS201 website.

�Special arrangem

ent for MP –

you are not required to do all assigned H

Ws.

4

55

1

EEqu

atio

n o

f Sta

te: pV

= n

RT

�

You are now familiar with “Equation of

motion”: F = m

a

�N

ow, imagine a device that we could

vary temperature (T), volum

e (V), pressure (p), and the am

ount of sam

ple.

�It would show that V is proportional to the m

oles of sample (n), that V varies

inversely with p, and that p and/or V vary in proportion to T.

�Results lead to form

one equation to describe the overall behavior of an ideal gas: pV = nRT

6

Visualize the equation with the figure.

KKey N

um

bers a

nd E

qu

atio

ns

7

Equation of State in “Per mole” basis

in “Per molecule” basis

P V = n R T ; R = ideal gas constant

Two types of equations.

HH2 O

Molecu

les

8

M = N

A x mH2O

mH2O

= 29.9x10-24 g/m

olecule from

Example 15.1

Be familiar with M

olar mass.

MMolecu

lar S

peed

Distrib

utio

ns

�M

olecules move at a distribution of speeds

around a mean velocity for any given

temperature (T). See Exam

ples 15.5 – 15.6.

9

Check visually the locations of “m

p”, “avg” and “rms”.

KKin

etic Molecu

lar T

heo

ry

�Gases m

ay be treated as point particles undergoing rapid elastic collisions with each other and the container�

Pressure

10

Chap. 8

Check two equations

PP, V

, an

d T

Plo

ts �

Especially useful to plot p and V at constant T for a range of tem

peratures. In this way we can generate a 3-D

surface of isotherm

al lines and make

predictions of an ideal gas’s behavior.

11

�Adiabatic: have no heat transfer in

or out of the system

�Isochoric: have no volum

e change. �

Isobaric: no pressure change. �

Isothermal: no tem

perature change.

Read p-V graphs

ppV

= n

RT

(Idea

l Gas)

12

M = N

A x mH2O

mH2O =29.9x10

-24 g/molecule

VVisu

al E

xam

ples o

f “Work

”

13

Note: they are signed num

bers

F x �x = p (A �x) = p �V

Check visually the “work” corresponds to the “area” in p-V graph.

114

[extra Q] total work from

d to a?

115

[Q1] total work from

a � b �

d? [Q

2] total work from d �

b � a?

EExa

mple Q

uestio

ns

16

Note that there is no

dependence on “m”

EExa

mple 1

5.3

17

3

4

p = gauge pressure + 1 atm

1

2 M

olar mass

�Absolute pressure is zero-referenced against a perfect vacuum

, so it is equal to gauge pressure plus atmospheric

pressure. �

Gauge pressure is zero-referenced against ambient air pressure, so it is equal to absolute pressure m

inus atm

ospheric pressure.

EExa

mple 1

5.3

(II)

18

PP1

5-1

3

19

PP1

5-1

3 (W

ork

Book

)

20

Fine print: Eq. 15-4

PP1

5-2

3

21

PP1

5-2

3 (W

ork

Book

)

22

MMore E

xam

ples

23

AAvo

gadro

’s Nu

mber

�6.022x10

23 molecules/m

ol: A

number to describe a set

count of atoms, like “dozen” is

a standard set for eggs.

�Approxim

ately 100 billion (1.0x10

11) stars in in Milky

Way galaxy.

�It would take a trillion (1.0x10

12) Milky W

ay galaxies to contain as m

any stars as there are particles in a m

ole.

24

�Two dozens of eggs �

24

�Two m

oles of gas � 12 x 10

23

�M

olar mass: M

= NA x m

molecule

�W

ater (a liquid at 18g/mol),

Nitrogen (a gas at 28g/m

ol), Table salt (sodium

chloride, a solid at 58g/m

ol).

�m

H2O =29.9x10

-24 g/molecule

from

Example 15.1

RRef. 1

: Model o

f Kin

etic Pro

perty

25

Chap. 8 Eq. 8.3

T2

T3

Fig. 15.12

Kav = (1/2) m

v2

Chap. 7 Eq. 7.3

RRef. 2

: Deta

ils of K

inetic P

roperty

26

V, N, T

�p = pf,i – p

i,x = (040 kg)(+30 m

/s) – (0.40 kg)(-30 m/s)

=2 (Mass) x |v

x | vi,x = 30 m

/s

vi,x = -30 m

/s

Chap. 8

Container

oN = N

umber of particles in

volume V

o

Num

ber of particles per unit volum

e is N/V

RRef. 3

: Deta

ils of K

inetic P

roperty

27

V, N, T, p

oN = N

umber of particles in

volume V

o

Num

ber of particles per unit volum

e is N/V

o

pV = n R T � pV = N

k T

�p = pf,i – p

i,x = (040 kg)(+30 m

/s) – (0.40 kg)(-30 m/s)

=2 (Mass) x |v

x | vi,x = 30 m

/s

vi,x = -30 m

/s

Chap. 8

Container

Kav = (3/2) k T

(3/2) nR T = N

Kav = K

trans

Kav = (1/2) m

v2 22

8

Ch

ap. 1

5 fo

r Exa

m4

Chap. 15.1 Chap. 15.2 Chap. 15.4 Chap. 15.5 (only Eq.15 & 15.16) Chap. 15.6

WWh

ere We A

re as o

f Nov 2

4?

29

Exam 4

330

[Q] H

ow do I study? Repeat MP problem

s? [A

] If you have completed M

Ps with your own notes on problem

s, then review my slides with your notes. M

y slides are reflection of what I was/am

thinking. See “reflection” for each exam

at URL.

[URL]

http://people.physics.tamu.edu/kam

on/teaching/phys201/class/2015C/2015C_U

ntil_Exam3.htm

l

Q&

A a

fter Nov.2

4 C

lass

331

2

m

c

Part 2

: (Reca

p) H

eat C

apacity

�Substances have an ability to “hold heat” that goes to the atom

ic level.

Q

= m c �T

[J] = [kg] [?] [K]

�c = specific heat capacity [J / (kg * K)]

�cwater = 4.19 x 10

3 J/(kg*K) vs. ccopper = 0.39 x 10

3 J/(kg*K)

[Q] W

hat is c? How

effectively the

substance can

hold heat. [Q

]W

hyQ

isproportional

tom?

Thevibration

ofeach

atomis

areason

forholding heat. “M

any atoms” m

eans holding more heat.

[Q] W

hy Q is proportional to �T?

It also

depends on

how much

the heat

transfer is made.

32

“Per kg” basis

333

Mola

r Hea

t Capacities

C = Heat capacity for 6 x 10

23 molecules

“Per mole” basis

Q = m

c �T

334

Fin

din

g CV

Q = �K

35

CV =

(3/2) R

CV = (3/2) R

Q = �K

Monatom

ic molecules

Diatom

ic molecules

Why?

336

�M

olecules can store heat energy in

translation, rotation

and vibration.

�Table 15.3 to guide you through a calculation.

CV =

(3/2) R

+ R

= (5

/2) R

“Per molecule” basis

“Per mole” basis

337

3

PPart 3

: Th

ermodyn

am

ic Pro

cesses �

A process can be isochoric and have no volum

e change. �

A process can be isobaric and have no pressure change.

�A

process can be isothermal and have no tem

perature change. �

A process can be adiabatic and have no heat transfer in or out of the

system.

38

339

�In sim

ple terms, “the heat

added to a system will be

distributed between internal energy (�U

) and work (W)”.

�“W

ork” is defined differently than we did in earlier chapters, here it refers to a p�V (a pressure increasing a volum

e).

Th

e First L

aw

of T

herm

odyn

am

ics

Q = �U

+ W

Q = n C

V �T + p (V2 – V

1 )

440

Th

e First L

aw

of T

herm

odyn

am

ics

Q = n C

V �T + p (V2 – V

1 )

�In sim

ple terms, “the heat

added to a system will be

distributed between internal energy (�U

) and work (W)”.

�“W

ork” is defined differently than we did in earlier chapters, here it refers to a p�V (a pressure increasing a volum

e).

�Note on the sign convention.

Q = �U

+ W

441

�In sim

ple terms, “the incom

e to a household will be distributed between Saving and Spending”.

�“Spending” is defined differently than we did in earlier chapters, here it refers to a p�V (unit price tim

es volume).

Th

e First L

aw

of P

aych

eck

Q = n C

V �T + p (V2 – V

1 )

Paycheck = Saving + Spending

= +

442

Blank Page

443

“Work

” = A

rea in

p-V

graph

Incremental, reproducible

changes may be sum

med.

Example 15.8(a)

Example 15.9

MMore V

isual E

xam

ples

44

445

P1

5-7

8

�A

process can be adiabatic and have no heat transfer in or out of the system

�

Q = 0

�U = n C

V �T

Q = �U

+ W �

W = ���U

446

P1

5-7

8

�A

process can be adiabatic and have no heat transfer in or out of the system

�

Q = 0

�U = n C

V �T

Q = �U

+ W �

W = ���U

AAdia

batic P

rocess

47

448

P1

5-7

8 E

trxa

Can you find T(c) ?

T(b) = 1790.2 K

AAdia

batic C

om

pressio

n

49

550

Blank Page

551

“Work

” in Iso

therm

al P

rocess

W = n R

T ln(V2 / V

1 )

IIsoth

ermal E

xpan

sion

�

Exam

ples 15.8, 15.9, and 15.10

52

You heat a sample of air to tw

ice its original temperature

in a constant-volume container. The average translational

kinetic energy of the molecules is

A. half the original value. B

. unchanged. C

. twice the original value.

D. four tim

es the original value.

Kav = (3/2) k T

Q = n C

V ��T + p (V2 – V

1 )

53

You heat a sample of air to tw

ice its original temperature

in a constant-volume container. The average translational

kinetic energy of the molecules is

C. tw

ice the original value.

554

You compress a sam

ple of air slowly to half its original

volume, keeping its tem

perature constant. The internal energy of the gas

A. decreases to half its original value. B

. remains unchanged.

C. increases to tw

ice its original value.

Kav = (3/2) k T

Q = n C

V ��T + p (V2 – V

1 )

55

You compress a sam

ple of air slowly to half its original

volume, keeping its tem

perature constant. The internal energy of the gas

B. rem

ains unchanged.

556

557

Exa

mple

558

MC

Exa

mple

WWh

ere We A

re?

59

Final

660