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PHY 113 C Fall 2013 Lecture 22 1 PHY 113 C General Physics I 11 AM 1215 P M MWF Olin 101 Plan for Lecture 22 Chapter 21 Ideal gas equations Molecular view of ideal gas Internal energy of ideal gas ID: 224146

gas 2013 113 phy 2013 gas phy 113 lecture fall ideal energy system heat work volume note process webassign

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Slide1

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

1

PHY 113 C General Physics I

11 AM – 12:15

P

M MWF Olin 101

Plan for Lecture 22:

Chapter 21: Ideal gas equations

Molecular view of ideal gas

Internal energy of ideal gas

Distribution of molecular speeds in ideal

gas

Adiabatic processesSlide2

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

2Slide3

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

3

From

Webassign

(Assignment #19)

A combination of 0.250 kg of water at 20.0°C, 0.400 kg of aluminum at 26.0°C, and 0.100 kg of copper at 100°C is mixed in an insulated container and allowed to come to thermal equilibrium. Ignore any energy transfer to or from the container and determine the final temperature of the mixture.

387 J/(kg*

o

C

)

4186 J/(kg*

o

C

)

900 J/(kg*

o

C

)

(From Table 20.1)Slide4

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

4

From

Webassign

(Assignment #19)

A thermodynamic system undergoes a process in which its internal energy decreases by 465 J. Over the same time interval, 236 J of work is done on the system. Find the energy transferred from it by heat.

Note: Sign convention for Q

:

Q>0  system gains heat from environment

iclicker

question:

Assuming the system does not change phase, what can you say about T

F

versus T

I

for the system?

T

F

>T

ITF<TISlide5

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

5

From

Webassign

(Assignment #19)

A 2.20-mol sample of helium gas initially at 300 K, and 0.400

atm is compressed isothermally to 1.80 atm. Note that the helium behaves as an ideal gas.

(

a) Find the final volume of the gas

.

(b) Find the work done on the gas.

(c) Find the energy transferred by heat

.Slide6

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

6

From

Webassign

(Assignment #19)

A 2.20-mol sample of helium gas initially at 300 K, and 0.400

atm is compressed isothermally to 1.80 atm. Note that the helium behaves as an ideal gas.

(

a) Find the final volume of the gas

.Slide7

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

7

From

Webassign

(Assignment #19)

A 2.20-mol sample of helium gas initially at 300 K, and 0.400

atm is compressed isothermally to 1.80 atm. Note that the helium behaves as an ideal gas.

(

b) Find the work done on the gas.

(

c) Find the energy transferred by heat

.Slide8

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

8

From

Webassign

(Assignment #19)

One

mole of an ideal gas does 2 900 J of work on its surroundings as it expands isothermally to a final pressure of 1.00 atm and volume of 28.0 L. (a) Determine the initial volume of the gas.

(b) Determine the temperature of the gas.Slide9

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

9

From

Webassign

(Assignment #19)

One

mole of an ideal gas does 2 900 J of work on its surroundings as it expands isothermally to a final pressure of 1.00 atm and volume of 28.0 L. Determine the initial volume of the gas.

Determine

the temperature of the gas.Slide10

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

10

From

Webassign

(Assignment #19)

In

the figure, the change in internal energy of a gas

that

is taken from

A

to

C

along the blue path is +795 J. The work done on the gas along the red path ABC is -530 J. (a) How much energy must be added to the system by heat as it goes from

A through B to C?(b) If the pressure at point A is five times that of point C

, what is the work done on the system in going from C to

D?(c) What is the energy exchanged with the surroundings by heat as the gas goes from C to A along the green path?(d) If the change in internal energy in going from point D to point

A is +495 J, how much energy must be added to the system by heat as it goes from point C

to point D?Slide11

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

11

Review:

Consider the process described by A

BCA

iclicker

exercise:

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

-12000 J

12000 J

0Slide12

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

12

Equation of “state” for ideal gas

(from experiment)

pressure in

Pascals

volume in m

3

# of moles

temperature in K

8.314 J/(

mol

K)Slide13

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

13

Ideal gas -- continued

Note that at this point, the above equation for

E

int

is completely unjustified…Slide14

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

14

From The New Yorker Magazine, November 2003Slide15

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

15

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.Slide16

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

16

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:

d

x

v

ix

-v

ix

number of atoms

volumeSlide17

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

17Slide18

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

18

iclicker

question:

What should we call ?

Average kinetic energy of atom.

We cannot use our macroscopic equations at the atomic scale -- so this quantity will go unnamed.

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

Very boring.Slide19

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

19

for mono atomic ideal gasSlide20

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

20

Average atomic velocities:

(note <v

i

>=0)

Relationship between average atomic velocities with TSlide21

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

21

Periodic

table:

http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpgSlide22

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

22

Periodic

table:

http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg

Molecular massSlide23

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

23

Periodic

table:

http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg

Molecular massSlide24

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

24

For monoatomic ideal gas:

General form for ideal gas (including mono-, di-, poly-atomic ideal gases):Slide25

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

25

Macroscopic

Microscopic

8.314 J/mole

o

K

1.38 x 10

-23

J/molecule

o

KSlide26

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

26

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

N

2

7/5

1.41

H

2

O

4/3

1.30

Big leap!Slide27

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

27

Comment on “big leap” – case of diatomic molecule

v

CM

w

Note: We are assuming that molecular vibrations are not taking much energySlide28

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

28

Comment on “big leap” – continued

Internal energy of an ideal gas:

derived for monoatomic ideal gas

more general relation for polyatomic ideal gas

Big leap!

can be measured for each gaseous system

Note:

g

= C

P

/C

VSlide29

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

29

Determination of Q for various processes in an ideal gas:

Example: Isovolumetric process – (V=constant

 W=0)

In terms of “heat capacity”: Slide30

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

30

Example: Isobaric process (P=constant):

In terms of “heat capacity”:

Note:

g

= C

P

/C

VSlide31

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

31

SummarySlide32

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

32

iclicker

question:

The previous discussion

Made me appreciate the

g

factor in thermo analyses

Made me want to scream

Put me to sleep

No problem – as long as this is not on the testSlide33

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

33

More examples:

Isothermal process (T=0)

D

T=0 

D

E

int

= 0  Q

=-WSlide34

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

34

Even more examples:

Adiabatic process (Q=0)Slide35

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

35Slide36

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

36

iclicker

question:

Suppose that an ideal gas expands adiabatically. Does the temperature

(A) Increase (B) Decrease (C) Remain the sameSlide37

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

37

Review of results from ideal gas analysis in terms of the specific heat ratio

g

º

C

P

/C

V

:

For an isothermal process,

D

Eint = 0  Q=-W

For an adiabatic process, Q = 0Slide38

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

38

Note:

It can be shown

that the work done by an ideal gas which has an initial pressure P

i and initial volume Vi when it expands adiabatically to a volume V

f

is given by

:Slide39

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

39

P (1.013 x 10

5

) Pa

V

i

V

f

P

i

P

f

A

B

C

D

Examples process by an ideal gas:

A

®

B

B

®

C

C

®

D

D

®

A

Q

W

0

-P

f

(

V

f

-V

i

)

0

P

i

(

V

f

-V

i

)

D

E

int

Efficiency as an engine:

e =

|

W

net

/

|/

Q

inputSlide40

11/14/2013

PHY 113 C Fall 2013 -- Lecture 22

40

From

Webassign

(#19)

An ideal gas initially at

P

i

,

V

i

, and

T

i is taken through a cycle as shown below. (Let the factor n = 2.6.)

(a) Find the net work done on the gas per cycle for 2.60

mol

of gas initially at 0°C.

(b) What is the net energy added by heat to the system per cycle?