PHY 113 A Fall 2012 -- Lecture 31 111/16/2012

PHY 113 A General Physics I9-9:50 AM MWF Olin 101

Plan for Lecture 31:

Chapter 21: Ideal gas equations

1. Molecular view of ideal gas

2. Internal energy of ideal gas

3. Distribution of molecular speeds in ideal gas

PHY 113 A Fall 2012 -- Lecture 31 211/16/2012

PHY 113 A Fall 2012 -- Lecture 31 311/16/2012

Review:Consider the process described by ABCA

iclicker exercise:What is the net work done on the system in this cycle?

A. -12000 JB. 12000 JC. 0

PHY 113 A Fall 2012 -- Lecture 31 411/16/2012

Equation of “state” for ideal gas(from experiment)

nRTPV

pressure in Pascals

volume in m3 # of moles

temperature in K

8.314 J/(mol K)

PHY 113 A Fall 2012 -- Lecture 31 511/16/2012

Ideal gas -- continued

..............................

diatomicfor

monoatomicfor

gas ideal of on type dependingparameter

1

1

1

1 :energy Internal

:state ofEquation

57

35

int

PVnRTE

nRTPV

Note that at this point, the above equation for Eint is completely unjustified…

PHY 113 A Fall 2012 -- Lecture 31 611/16/2012

From The New Yorker Magazine, November 2003

PHY 113 A Fall 2012 -- Lecture 31 711/16/2012

Microscopic model of ideal gas:

Each atom is represented as a tiny hard sphere of mass m with velocity v. Collisions and forces between atoms are neglected. Collisions with the walls of the container are assumed to be elastic.

PHY 113 A Fall 2012 -- Lecture 31 811/16/2012

Proof: Force exerted on wall perpendicular to x-axis by an atom which collides with it:

average over atoms

What we can show is the pressure exerted by the atoms by their collisions with the walls of the container is given by:

avgavgK

V

Nvm

V

NP

3

2

3

2 22

1

t

vm

t

pF ixiixix

2

d

x

ixvdt /2

22

2

/22

xii

ixi

i

ix

ixi

ix

ixiix

vmVN

dAvm

AF

P

dvm

vdvm

F

vix

-vix

number of atoms

volume

PHY 113 A Fall 2012 -- Lecture 31 911/16/2012

22122

22222

2222

222

22

2

3

2

3

3

that note Also

:,, along move likely toequally are molecules Since

/2

2

iiiiixi

ixiziyixi

iziyixi

iziiyiixi

ixii

ixi

i

ix

ixi

ix

ixiix

vmV

Nvm

V

Nvm

V

NP

vvvvv

vvvv

vmvmvm

zyx

vmV

N

dA

vm

A

FP

d

vm

vd

vmF

PHY 113 A Fall 2012 -- Lecture 31 1011/16/2012

iclicker question:What should we call ?

A. Average kinetic energy of atom.B. We cannot use our macroscopic equations

at the atomic scale -- so this quantity will go unnamed.

C. We made too many approximations, so it is not worth naming/discussion.

D. Very boring.

221

iivm

PHY 113 A Fall 2012 -- Lecture 31 1111/16/2012

atoms of moles ofnumber

atom) Hefor kg (0.004 massmolar thedenotes M where

)atom Hefor kg106.6( atom of mass

atoms ofnumber :Note3

2

27-

221

n

nMNm

m

N

vmV

NP

i

i

ii

atoms gas ideal of mole ofenergy kinetic average

2

3or

3

2

3

2

:law gas ideal toConnection

221

2212

21

221

i

ii

i

Mv

RTMvRTMv

nRTMvnPV

nRTE2

3int

PHY 113 A Fall 2012 -- Lecture 31 1211/16/2012

Average atomic velocities: (note <vi>=0)

M

RTv

RTMv

i

i

3

2

3

2

221

Relationship between average atomic velocities with T

PHY 113 A Fall 2012 -- Lecture 31 1311/16/2012

nRTE2

3int

For monoatomic ideal gas:

General form for ideal gas (including mono-, di-, poly-atomic ideal gases):

..............................

diatomicfor

monoatomicfor

1

1

57

35

int

nRTE

PHY 113 A Fall 2012 -- Lecture 31 1411/16/2012

RT-n

Tk-N

TkNvmNE BB 1γ1γ23

21

2int

Internal energy of an ideal gas:

derived for monoatomic ideal gas more general relation for

polyatomic ideal gas

Gas g (theory) (g exp)

He 5/3 1.67

N2 7/5 1.41

H2O 4/3 1.30

Big leap!

PHY 113 A Fall 2012 -- Lecture 31 1511/16/2012

Determination of Q for various processes in an ideal gas:

Example: Isovolumetric process – (V=constant W=0)

In terms of “heat capacity”:

WQTR-

nE

RT-

nE

1γ

1γ

int

int

fififi QTR-n

E 1γ

int

1γ

1γ

-R

C

TnCTR-n

Q

V

fiVfifi

PHY 113 A Fall 2012 -- Lecture 31 1611/16/2012

Example: Isobaric process (P=constant):

In terms of “heat capacity”:

Note: = CP/CV

fifififi WQTR-

nE

1γ int

1γγ

1γ

1γ1γ

-R

R-R

C

TnCTnRTR-n

VVPTR-n

Q

P

fiPfifiifififi

PHY 113 A Fall 2012 -- Lecture 31 1711/16/2012

More examples:

Isothermal process (T=0)

DT=0 DEint = 0 Q=-W

WQTR-

nE

RT-

nE

1γ

1γ

int

int

i

fV

V

V

V V

VnRT

V

dVnRTPdVW

f

i

f

i

ln

PHY 113 A Fall 2012 -- Lecture 31 1811/16/2012

Even more examples:

Adiabatic process (Q=0)

TnRVPPV

nRTPV

VPTR-

n

WE

1γ

int

γγγ

γ

lnln

γ

1γ

ffiii

f

i

f VPVPP

P

V

V

PP

VV

VPPVVP-TnR

PHY 113 A Fall 2012 -- Lecture 31 1911/16/2012

VVP

P

VVP

P

ii

ii

:Isotherm

:Adiabat

PHY 113 A Fall 2012 -- Lecture 31 2011/16/2012

iclicker question:

Suppose that an ideal gas expands adiabatically. Does the temperature

(A) Increase (B) Decrease (C) Remain the same

1-γ

1-γ1-γ

γγ

f

iif

ffii

i

iiiii

ffii

V

VTT

VTVT

V

TnRPnRTVP

VPVP

PHY 113 A Fall 2012 -- Lecture 31 2111/16/2012

Review of results from ideal gas analysis in terms of the specific heat ratio g º CP/CV:

For an isothermal process, DEint = 0 Q=-W

For an adiabatic process, Q = 0

1γ ;

1γ int -

RCTnCTR

-n

E VV

1γγ-R

CP

i

fii

i

fV

V V

VVP

V

VnRTPdVW

f

i

lnln

1-γ1-γ

γγ

ffii

ffii

VTVT

VPVP

PHY 113 A Fall 2012 -- Lecture 31 2211/16/2012

Note:

It can be shown that the work done by an ideal gas which has an initial pressure Pi and initial volume Vi when it expands adiabatically to a volume Vf is given by:

1γ

11γ f

i

V

V

ii

V

VVPPdVW

f

i

PHY 113 A Fall 2012 -- Lecture 31 2311/16/2012

P (

1.01

3 x

105 )

Pa

Vi Vf

Pi

Pf

A

B C

D

Examples process by an ideal gas:

A®B B®C C®D D®A

Q

W 0 -Pf(Vf-Vi) 0 Pi(Vf-Vi)

DEint

1-γ

)( ifi PPV 1-γ

)(γ iff VVP 1-γ

)( iff PPV 1-γ

)(γ ifi VVP-

1-γ

)( ifi PPV 1-γ

)( iff PPV 1-γ

)( iff VVP 1-γ

)( ifi VV-P

Efficiency as an engine:

e = Wnet/ Qinput