PHY1039 - PowerPoint Presentation

Slide1

PHY1039

Properties of Matter

Heat Capacity

of

Ideal Gases (

C

P

and

C

V

) and

Adiabatic Expansion of Ideal Gas

(

See Finn’s

Thermal Physics, Ch. 4)

March 12 and 15, 2012

Lectures 11 and 12Slide2

From next week:

Lectures on Monday at 3 pm will meet in Lecture Theatre E.

Week 7 (next week) only:

Lecture on Monday, March 19 at 4 pm in 35AC04 (instead of tutorial)

Tutorial will be held on Thursday, March 22 at 9 am in the Austin Pearce Building, Lab2 (AP Lab2)Slide3

V

P

Two Types of Heat Capacity

Isochoric Process:

C

V

Isobaric Process,

C

P

T

1

T2

V

P

T

1

T

2

(

V1, P1)

(V1, P2)

P

1

P

2

V

1

V

2

T2 > T1

T2 > T1

V

1

P

1

(V1, P1)

(

V

2

,

P

1

)Slide4

Internal Energy,

U,

of Monoatomic Gas

All of the

k

inetic energy of a monoatomic gas is contained in

translational

motion with a velocity n

.Monoatomic gases have one atom per molecule: e.g. He, Ne, Xe

, and Kr.There are three degrees of freedom.Each

d.o.f. has ½ kT

in thermal energy.k = 1.38 x 10

-23 J/K

The total energy of

each molecule (ignoring potential energy) is

.

n

y

n

zn

KE =

=

n

x

Translational energy

Ideal GasSlide5

Internal Energy,

U,

of Diatomic Gas

Diatomic gases have two atoms per molecule:

e.g.

H

2, O

2, N2, CO, Cl2.

Diatomic molecules have translational, rotational and vibrational energy:Three

translational degrees of freedom: nx,

ny, nz

Three rotational degrees of freedom: about x, y and z axes.

Figure from “Understanding Properties of Matter” by M. de

Podesta

But the energy of rotation about the molecule’s axis is not

accessible

at lower temperatures.Slide6

Internal Energy,

U,

of Diatomic Gas

N

N

Diatomic molecules can also have vibrational energy, but it is not

accessible

at lower

T

.

(Kinetic

energy and potential energy of vibration

each

contribute one

d.o.f

.)

There are

five

degrees of freedom accessible at lower temperatures

.

The total energy of each

molecule (ignoring potential energy) is 5(1/2)

kT = (5/2)

kT at lower T, but it will increase with T

.r

PE = ½K (

r – ro

)2

Inaccessible states

A

ccessible stateSlide7

Internal Energy,

U,

of Triatomic Gas

Linear

triatomic gases have three atoms per molecule that all lie along the same axis:

e.g.

CO

2.

O=C=OLike a diatomic molecule, a linear triatomic molecule has three translational and only

two accessible rotational degrees of freedom.

Symmetric &

anti-symmetric stretching vibration

Bending vibration

A linear molecule with

N

atoms has 3

N

-5 modes of vibration.

There are two

degrees of freedom for vibrational energy accessible at lower temperatures.At “lower” temperatures, in total there are seven accessible degrees of freedom.

The total energy of each molecule is

7(1/2)kT

= (7/2)kTSlide8

Internal Energy,

U,

of Triatomic Gas

Non-linear

triatomic gases have three atoms per molecule that do

not

lie along the same axis:

e.g. N

2O; SO2; H2O

Figure from

P. Atkin’s

The Elements of Physical Chemistry

For a non-linear triatomic molecule, there are rotations about

three

axes at lower T

: three rotational degrees of freedom.

A non-linear molecule with N atoms has 3

N-6 modes of vibration. A non-linear triatomic molecule has three

degrees of freedom.In total, there are 9 possible degrees of freedom (3 translational, 3 rotational, and three vibrational), but only

7 are accessible at lower temperatures.The total energy of each molecule is 7(1/2)kT = (7/2)

kTSlide9

http://jcwinnie.biz/wordpress/?p=2235

http://www.dailymail.co.uk/sciencetech/article-483191/Arctic-ice-cap-melts-smallest-size.html

Greenhouse Effect: A Problem of Thermodynamics

Earth can be treated as a thermodynamic system

.Slide10

http://en.wikipedia.org/wiki/File:Atmosfaerisk_spredning.gif

Most Intense Thermal

Radiation

from Earth

Why Do Water

and Carbon Dioxide Block Thermal Radiation from

Earth?

Wavelength (

m

m)

Thermal radiation transmitted through atmosphere

to

Earth from Sun

Thermal radiation

from

colder Earth transmitted through atmosphere

en.wikipedia.org/wiki/User:Dragons_flight/Images

The resonant frequency of molecular vibrations is in the same frequency as infrared radiation.Slide11

Heat Capacity of Ideal Gases

Type of Gas

Accessible

d.o.f

. at lower

T

Internal Energy, U

Cv

Monoatomic 3 3/2 nRT 3/2

nR Diatomic 5 5/2 nRT 5/2

nR Triatomic 7

7/2 nRT 7/2 nR

U for ideal gases (ignoring potential energy) depends only on T.

Cv depends only on the amount of gas (through n

).Slide12

Important Conclusions

Isothermal processes:

U

of an ideal gas is a function of

T only. If the temperature is constant (

DT = 0), then internal energy is constant: DU

= 0. Changes in P and V will

not affect U.First Law tells us:

DU = 0 = Q + W

So, W = - Q. If there is isothermal work on an ideal gas (W is positive), then heat must go out of the gas.

Adiabatic processes:

In an adiabatic process, Q = 0.First Law tells us:

DU = 0

+ W.

If work is done on an ideal gas, such as by compression, W is positive, and hence

DU is positive. As

DU is

proportional to nRD

T, the temperature of the gas will also

increase.Slide13

Molar Heat Capacity,

C

P

, of Monoatomic Gas

Figure from “Understanding Properties of Matter” by M. de

Podesta

At higher temperatures, more degrees of freedom are

not

excited. There is no T dependence of CV

or CP.CP

= 3/2 nR + nR

= 5/2 nRSlide14

Figure from “Understanding Properties of Matter” by M. de

Podesta

C

P

= 5/2

nR

+

nR

= 7/2 nR

Molar Heat Capacity, CP, of Diatomic Gas

As temperature is increased from room temperature, more degrees of freedom are accessible (rotational and vibrational)Slide15

From Tipler’s

Physics

Molar

Heat Capacities

of

Various Gases

at 25 °CSlide16

V

P

V

2

V

1

P

2

Adiabatic

versus

Isothermal Expansions (or Compressions) of an Ideal Gas

P

1

Adiabatic compression

:

Isothermal compression

: