Chapter 10: Thermal Physics Temperature and the Zeroth Law of Thermodynamics Terminology Two...

Chapter 10: Thermal PhysicsTemperature and the Zeroth Law of

Thermodynamics Terminology

• Two objects are in thermal contact if energy can be exchanged between them.

• Two objects are in thermal equilibrium if they are in thermal contact and there is no net exchange of energy.

• The exchange of energy between two objects because of differences in their temperature is called heat. But what is temperature!

Zeroth law of thermodynamics

If objects A and B are separately in thermal equilibrium with a thirdobject C, then A and B are in thermal equilibrium with each other.

Two objects in thermal equilibrium with each other are at the sametemperature.

Thermometers and Temperature Scales

Thermometer

• Thermometers are devices used to measure the temperature of an object or a system.

• When a thermometer is in thermal contact with a system, energy is exchanged until the thermometer and the system are in thermal equilibrium with each other.

• All the thermometers use some physical properties that depend on the temperature. Some of these properties are: 1) the volume of a fluid 2) the length of a solid 3) the pressure of a gas held at constant volume 4) the volume of a gas held at constant pressure 5) electric resistance of a conductor 6) the color of very hot object.


Thermometer (cont’d)• One common thermometer consists of a mass of liquid: mercury or alcohol. The fluid expands into a glass capillary tube when its temperature rises.

• When the cross-sectional area of the tube is constant, the change in volume of the liquid varies linearly with its length along the tube.

• The thermometer can be calibrated by placing it in thermal contact with environments that remain at constant temp.

• Two of such environments are: 1) a mixture of water and ice in thermal equilibrium at atmospheric pressure. 2) a mixture of water and steam in thermal equilibrium at atmospheric pressure.

0oC (Celsius) 100oC

Freezingpoint Boiling

point

Thermometers and Temperature Scales Constant-volume gas thermometer and the Kelvin scale

• A constant-volume gas thermometer measures the pressure of the gas contained in the flask immersed in the bath. The volume of the gas in the flask is kept constant by raising or lowering reservoir B to keep the mercury level constant in reservoir A.


Constant-volume gas thermometer and the Kelvin scale

• It has been experimentally observed that the pressure varies linearly with temperature of a fixed volume of gas, which does not depend on what gas is used.

• It has been experimentally observed that these straight lines merge at a single point at temp. -273.15oC at pressure = 0. This temperature is called absolute zero, which is the base of the Kelvin temperature scale T=TC-273.15 measured in kelvin (K) where TC is temperature in Celsius..

0 K = -273.15oC


Temperature scales

• The common temperature scale in US is Fahrenheit:

)32(9

5;32

5

9 FCCF TTTT

CF TT 5

9

Thermal Expansion of Solids and Liquids

Thermal (linear) expansion

• Thermal expansion : As temperature of a substance increases, its volume in general increases. This phenomenon is called thermal expansion.

• The overall thermal expansion of an object is a consequence of the change in the average separation between its constituent atoms or molecules.

• If the thermal expansion of an object is sufficiently small compared with the object’s initial dimensions, the change in any dimension is, to a good approximation, proportional to the first power of the temp. change.

)( 0000 TTLLLTLL

L0 is the initial length, L is the final length, and is calledthe coefficient of linear expansion for a given material (unit: ( oC) -1 ).

Thermal Expansion of Solids and Liquids Thermal area expansion

• Because the linear dimensions of an object change due to variations in temperature, it follows that the surface area and volume of the object also change.

• Consider a square of material having an initial length L0 on a side and therefore initial area A0=L0

2. As the temperature increases, the length of each side increase to:

The new area is then:

)( 000 TTLLL

tLL

TLTLL

TLLTLLLA

20

20

220

220

20

00002

2

)(2

))((

TAAAATAAA 0000 2 : coefficient of area expansion


Thermal volume expansion

• In a similar fashion we can show that the increase in volume of an object accompanying a change in temperature is:

TVV 0 : coefficient of area expansion

• Note that if the coefficient of linear expansion of the object is the same in all directions.


Examples

• Some applications of thermal expansion

Thermal expansion joint A extreme hot day


Examples (cont’d)

• Bimetal

Thermal Expansion of Solids and Liquids Examples (cont’d)

• Example 10.3 : Expansion of a railroad track(a) A steal rail road has a length of 30.000 m when the temperature is 0oC. What is the length on a hot day when the temperature is 40.0oC?

m 0.013C)m)(40.0 000.30]()(1011[ 160 CTLL

m 013.300 LLL

(a) Find the stress if the track cannot expand.

Pa 1067.8)30.0

m 0.013Pa)( 1000.2( 711

L

LY

A

F


• Example 10.4 : Rings and rods(a) A circular copper ring at 20.0oC has a hole with an area of 9.98 cm2. What minimum temperature must it have so that it can be slipped onto a steel metal rod having a cross-sectional area of 10.0 cm2?

20 cm 02.0 TAA

CTTTCT 9.789.58 0

(b) If the ring and the rod are heated simultaneously, what change in temperature of both will allow the ring to be slipped onto the end of the rod?

TAATAAAAAA SSSCCCSSCC

CAA

AAT

SSCC

CS

168


• Example 10.5 : Global warming and coastal flooding(a) Estimate the fractional change in the volume of Earth’s oceans due to an average temperature change of 1oC?

TVV 0

4

0

102

T

V

V

(b) Use the fact that the average depth of the ocean is 4.00x103 m to estimate the change in depth. water =2.07x10-4 (oC)-1.

m 3.03 00 TLTLL

Macroscopic Description of an Ideal Gas Ideal gas

• An ideal gas is a collection of atoms or molecules that move randomly and exert no long-range forces on each other. Each particle of the ideal gas is individually point-like, occupying a negligible volume.

• Low-density/low-pressure gases behave like ideal gases.

• Most gases at room temperature and atmospheric pressure can be approximately treated as ideal gases.

Equation of state

• The pressure P, volume V, temperature T and amount n of gas in a container are related to each other by an equation of state.

• In general, equation of state is complex but for an ideal gas it is simple.

Macroscopic Description of an Ideal Gas

Avogadro’s number • The same number of particles is found in a mole of a substance.• This number is called Avogadro’s number NA=6.02x1023 particles/mole.• The mass in grams of one Avogadro’s number of an element is numerically the same as the mass of one atom of the element, expressed in atomic mass u.• Atomic mass of hydrogen 1H is 1 u, and that of carbon 12C is 12 u. 12 g of 12C consists of exactly NA atoms of 12C. The molecular mass of molecular hydrogen H2 is 2u, and NA molecules are in 2 g of H2 gas.

Molar mass of a substance • The molar mass of a substance is defined as the mass of one mole of that substance, usually expressed in grams per mole.

Number of moles

• The number of moles of a substances n is:

massmolar

mn m : mass of the substance


Definition of a mole • One mole (mol) of any substance is that amount of the substance that contains as many particles (atoms, or other particles) as there are atoms in 12 g of the isotope carbon-12 12C.

• One atomic mass unit is equal to 1.66x10-24 g.

• The mass m of an Avogadro’s number of carbon-12 atoms is :

g 0.12u

g 1066.1)u 12(1002.6)u 12(

2423

ANm

• The mass per atom for a given element is:

Aatom N

mmolar

g/atom1064.6atoms/mol 1002.6

g/mol 00.4 2423

Hem

1.66x10-24=1/6.02x1023


Ideal gas law (Equation of state for ideal gas) • Boyle’s law

When a gas is kept at a constant temperature, its pressure isinversely proportional to its volume.

• Charles’s law

When the pressure of a gas is kept constant, its volume isdirectly proportional to the temperature.

• Gay-Lussac’s lawWhen the volume of a gas is kept constant, its pressure isdirectly proportional to the temperature.

Ideal gas law:

nRTPV

P : pressure, V : volume, T : temperature in KR : universal gas constant 8.31 J/(mole K) 0.0821 L atm/(mol K) 1 L (litre) = 103 cm3 = 10-3 m3

The volume occupied by 1 mol of an ideal gas at atmosphericpressure and at 0oC is 22.4 L


Ideal gas law (Equation of state for ideal gas) (cont’d)

Ideal gas law:

nRTPV

P : pressure, V : volume, T : temperature in KR : universal gas constant 8.31 J/(mole K) 0.0821 L atm/(mol K) 1 L (litre) = 103 cm3 = 10-3 m3

The volume occupied by 1 mol of an ideal gas at atmosphericpressure and at 0oC is 22.4 L

RTN

NPV

N

Nn

AA

Defining kB=R/NA=1.38x10-23 J/K (Boltzmann’s constant),

TNkPV B


Examples • Example 10.7 : Message in a bottleA corked bottle was found with a message in air inside at atmosphericpressure and at T=30.0oC. The cork has a cross-sectional area of 2.30cm2. The finder placed the bottle over a fire to eject the cork, whichhappened at T=99oC. (a) What was the pressure just before the corkpopped out from the bottle?

Pa 1024.1K 303

K 372Pa) 1001.1( 55

i

fif

i

f

ii

ff

T

TPP

nRT

nRT

VP

VP

(b) What force of friction held the cork in place? Neglect any change in in volume of the bottle.

00 frictionoutin FAPAPF

N 29.5)m 1030.2(Pa] 10)01.124.1[()( 245 APPF outinfriction


Examples • Example 10.8 : Submerging a balloonA balloon with volume 0.500 m3 is attached to 2.50x102 kg iron weightand tossed overboard into a freshwater. The air in the balloon initiallyat atmospheric pressure. The system fails to sink and there are nomore weights, so a skin diver decides to drag it deep enough so thatthe balloon will remain submerged. Ignore the weight of the balloonmaterial. (a) Find the volume of the balloon at the point where thesystem will remain submerged in equilibrium?

kg 65.0m 0318.0/ 3 balairbalFeFeFe VmmV

0 balbalFeFe wBwB

0 gmgVgmgV balbalwatFeFewat

3m 219.0balVSolve for Vbal.


Examples • Example 10.8 : Submerging a balloon (cont’d)(b) What is the balloon’s pressure at that point?

Pa 1031.21 5 if

if

i

f

ii

ff PV

VP

nRT

nRT

VP

VP Vi=0.500 m3

Vf=0.219 m3

(c) To what minimum depth must the balloon be dragged?

gdPP atmf

m 3.13

g

PPd

water

atmf


Kinetic theory of gases • Assumptions The number of molecules in the gas is large, and the average separation between them is large compared with their dimension.

A large number of molecules behave statistically in a stable fashion.Large separation between molecules allows us to neglect thevolume occupied by a molecule.

The molecules obey Newton’s laws of motion, but as a whole they move randomly.

Randomness guarantees that any molecule move in any directionwith equal probability. From this randomness emerges “regularity”.

The molecules interact only through short-range forces during elastic collisions.

Lack of long-range force is consistent with the ideal gas model. The molecules make elastic collisions with the walls.

All molecules in the gas are identical.


Kinetic theory of gases • Molecular model for the pressure of an ideal gasConsider the collision of a molecule movingwith a velocity –vx in x-direction toward theleft hand wall. After colliding elastically, themolecule moves in the x-direction with avelocity vx. The change in momentum is:

xxxx mvmvmvp 2)(

The magnitude of the average force exertedby a molecule on the wall in time t is:

t

mv

t

pF xx

2

1

The time interval t between two collisionswith the same wall is :

xv

dt

2


Kinetic theory of gases • Molecular model for the pressure of an ideal gas (cont’d)

The force imparted to the wall in a time t by a single molecule is:

d

mv

vd

mv

t

mv

t

pF x

x

xxx2

1 /2

22

The total force exerted by all the molecules on the wall is then:

N

vvv

d

Nmv

d

mF ix

xxix

2222

22222222222

3

13 vvvvvvvvvvv xxzyxzyx

randomness

22

3v

Nv

d

NmF x

222

32 2

1

3

2

3

1

3

1vm

V

NPvm

V

Nvm

d

N

d

F

A

FP

average translationalkinetic energy


Kinetic theory of gases • Molecular interpretation of temperature

Ideal gas law + pressure in terms of average kinetic energy :

TNkvmNPV B

2

2

1

3

2

2

2

1

3

2vm

kT

B

The temperature of a gas is a direct measure ofthe average molecular kinetic energy of the gas.

Tkvm B2

3

2

1 2 The average translational kinetic energy per moleculeis (3/2) of kBT.

nRTTNkvmNKE Btotal 2

3

2

3)

2

1( 2

The total translational kinetic energyof N molecules of gas


Kinetic theory of gases • Internal energy, root-mean-square speed

For a monatomic gas, the translational kinetic energy is the onlytype of energy that molecules can have. Therefore the internal energyU of a monatomic gas is :

nRTU2

3

For diatomic and polyatomic molecules, additional types of energysources are available from the vibration and rotation of molecules.

The square root of is called the root-mean-square (rms) speed ofmolecules.

2v

M

RT

m

Tkvv B

rms

332 O2 : M =32x10-2 kg/mol vrms =1.0x103 m/s

Chapter 10: Thermal Physics Temperature and the Zeroth Law of Thermodynamics Terminology Two...

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Transcript of Chapter 10: Thermal Physics Temperature and the Zeroth Law of Thermodynamics Terminology Two...