04/19/23 PHY 113 -- Lecture 20 1

Announcements

1. Please bring your laptops to lab this week.

2. Physics seminar this week – Thursday, Nov. 20 at 4 PM – Professor Brian Matthews will discuss relationships between protein structure and function

3. Schedule –

Today, Nov. 18th: Examine Eint especially for ideal gases

Discuss ideal gas law

Continue discussion of first law of thermodynamics

Thursday, Nov. 20th: Review Chapters 15-20

Tuesday, Nov. 25th: Third exam

04/19/23 PHY 113 -- Lecture 20 2

From The New Yorker Magazine, November 2003

04/19/23 PHY 113 -- Lecture 20 3

First law of thermodynamics

Eint = Q W

For an “ideal gas” we can write an explicit relation for Eint.

What we will show:

is a parameter which depends on the type of gas (monoatomic, diatomic, etc.) which can be measured as the ratio of two heat capacities: =CP/CV.

f

i

V

VPdVW

Tk-N

RT-n

E B1γ1γ)gas ideal(

int

04/19/23 PHY 113 -- Lecture 20 4

Thermodynamic statement of conservation of energy –

First Law of Thermodynamics

Eint = Q - W

“Internal” energy of system

Heat added to system

Work done by system

Eint = Eint(f)int(i) independent on path

Qif depends on path

Wif depends on path

04/19/23 PHY 113 -- Lecture 20 5

How is temperature related to Eint?

Consider an ideal gas

Analytic expressions for physical variables

Approximates several real situations

Ideal Gas Law: P V = n R T

pressure (Pa)volume (m3)

number of moles

temperature (K)gas constant (8.31 J/(moleK))

04/19/23 PHY 113 -- Lecture 20 6

P V = n R T

Ideal gas – P-V diagram at constant T

Vi Vf

Pf

Pi

Pi Vi = Pf Vf

04/19/23 PHY 113 -- Lecture 20 7

04/19/23 PHY 113 -- Lecture 20 8

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.

04/19/23 PHY 113 -- Lecture 20 9

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

VN

vmVN

P32

32 2

21

tvm

tp

F 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

04/19/23 PHY 113 -- Lecture 20 10

222

21

32

31

vmVN

vmVN

vmVN

P x

Ideal gas law continued:

Recall that

Therefore:

Also:

TNkvmNPV

TNkTNR

NnRTPV

B

B

A

2

21

32

:model cMicroscopi

:relation cMacroscopi

mTk

vv

Tkvm

Brms

B

3

23

21

2

2

TkNvmNE

Tkvm

B

B

23

21

23

21

2int

2

04/19/23 PHY 113 -- Lecture 20 11

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 (theory) exp)

He 5/3 1.67

N2 7/5 1.41

H2O 4/3 1.30

Big leap!

04/19/23 PHY 113 -- Lecture 20 12

Determination of Q for various processes in an ideal gas:

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

In terms of “heat capacity”:

WQTR-n

E

RT-n

E

1γ

1γ

int

int

fififi QTR-n

E 1γ

int

1γ

1γ

-R

C

TnCTR-n

Q

V

fiVfifi

04/19/23 PHY 113 -- Lecture 20 13

Example: Isobaric process (P=constant):

In terms of “heat capacity”:

Note: = CP/CV

fifififi WQTR-n

E 1γ

int

1γγ

1γ

1γ1γ

-R

R-R

C

TnCTnRTR-n

VVPTR-n

Q

P

fiPfifiifififi

04/19/23 PHY 113 -- Lecture 20 14

More examples:

Isothermal process (T=0)

T=0 Eint = 0 Q=W

WQTR-n

E

RT-n

E

1γ

1γ

int

int

i

fV

V

V

V V

VnRT

VdV

nRTPdVWf

i

f

i

ln

04/19/23 PHY 113 -- Lecture 20 15

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

04/19/23 PHY 113 -- Lecture 20 16

VVP

P

VVP

P

ii

ii

:Isotherm

:Adiabat

04/19/23 PHY 113 -- Lecture 20 17

Peer instruction 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

04/19/23 PHY 113 -- Lecture 20 18

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

For an isothermal process, Eint = 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

04/19/23 PHY 113 -- Lecture 20 19

Extra credit:

Show 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

iV

V

ii

V

VVPPdVW

f

i

04/19/23 PHY 113 -- Lecture 20 20

Peer instruction questions

Match the following types of processes of an ideal gas with their corresponding P-V relationships, assuming the initial pressures and volumes are Pi and Vi, respectively.

1. Isothermal

2. Isovolumetric3. Isobaric

4. Adiabatic

(A) P=Pi (B) V=Vi (C) PV=PiVi (D)PV=PiVi

04/19/23 PHY 113 -- Lecture 20 21

P (

1.01

3 x

105 )

Pa

Vi Vf

Pi

Pf

A

B C

D

Examples process by an ideal gas:

AB BC CD DA

Q

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

Eint

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

04/19/23 PHY 113 -- Lecture 20 22