Thermodynamics The Ideal Gas Equation - De Anza...

ThermodynamicsThe Ideal Gas Equation

Lana Sheridan

De Anza College

April 21, 2020

Last time

• temperature scales

• thermal expansion

Overview

• wrap up thermal expansion

• the ideal gas equation

• moles and molecules

• applying the ideal gas equation

Volume Thermal Expansion

We can model volume expansion in a similar way:

∆V = βVi ∆T

β is the average coefficient of volume expansion.

If the material is isotropic (the same in all directions, symmetrywrt rotations of coordinate systems) then:

β = 3α

using the fact that α∆T � 1.

(This entire model is only approximately true over a restrictedrange of temperatures.)

Volume Thermal ExpansionSuppose Vi = `wh

∆V ≈ (∆`)wh + `(∆w)h + `w(∆h)

= (α`∆T )wh + `(αw ∆T )h + `w(αh∆T )

= (α∆T )(3`wh)

= (3α)Vi ∆T

= βVi ∆T

So, β ≈ 3α.

Thermal Expansion QuestionThe figure here shows four rectangular metal plates, with sides ofL, 2L, or 3L. They are all made of the same material, and theirtemperature is to be increased by the same amount. Rank theplates according to the expected increase in

(a) their vertical heights greatest first.

48318-7 TE M PE RATU R E AN D H EATPART 2

HALLIDAY REVISED

This behavior of water is the reason lakes freeze from the top down rather thanfrom the bottom up. As water on the surface is cooled from, say, 10°C toward thefreezing point, it becomes denser (“heavier”) than lower water and sinks to thebottom. Below 4°C, however, further cooling makes the water then on the surfaceless dense (“lighter”) than the lower water, so it stays on the surface until it freezes.Thus the surface freezes while the lower water is still liquid. If lakes froze from thebottom up, the ice so formed would tend not to melt completely during the sum-mer, because it would be insulated by the water above. After a few years, manybodies of open water in the temperate zones of Earth would be frozen solid all yearround—and aquatic life could not exist.

CHECKPOINT 2

The figure here shows four rectangularmetal plates, with sides of L, 2L, or 3L.Theyare all made of the same material, and theirtemperature is to be increased by the sameamount. Rank the plates according to the ex-pected increase in (a) their vertical heightsand (b) their areas,greatest first.

Sample Problem

volume of the fuel did also, as given by Eq. 18-10 (!V "Vb !T).

Calculations: We find!V" (37 000 L)(9.50 # 10$4/C°)($23.0 K) " $808 L.

Thus, the amount delivered was

Vdel " V % !V " 37 000 L $ 808 L" 36 190 L. (Answer)

Note that the thermal expansion of the steel tank has nothing to do with the problem. Question: Who paid for the“missing” diesel fuel?

Thermal expansion of a volume

On a hot day in Las Vegas, an oil trucker loaded 37 000 L ofdiesel fuel. He encountered cold weather on the way toPayson, Utah, where the temperature was 23.0 K lower thanin Las Vegas, and where he delivered his entire load. Howmany liters did he deliver? The coefficient of volume expan-sion for diesel fuel is 9.50 # 10$4/C°, and the coefficient oflinear expansion for his steel truck tank is 11 # 10$6/C°.

The volume of the diesel fuel depends directly on the tem-perature. Thus, because the temperature decreased, the

KEY I DEA

Additional examples, video, and practice available at WileyPLUS

(1) (2) (3) (4)

18-7 Temperature and HeatIf you take a can of cola from the refrigerator and leave it on the kitchen table, itstemperature will rise—rapidly at first but then more slowly—until the tempera-ture of the cola equals that of the room (the two are then in thermal equilibrium).In the same way, the temperature of a cup of hot coffee, left sitting on the table,will fall until it also reaches room temperature.

In generalizing this situation, we describe the cola or the coffee as a system(with temperature TS) and the relevant parts of the kitchen as the environment(with temperature TE) of that system. Our observation is that if TS is not equal toTE, then TS will change (TE can also change some) until the two temperatures areequal and thus thermal equilibrium is reached.

Such a change in temperature is due to a change in the thermal energy of thesystem because of a transfer of energy between the system and the system’senvironment. (Recall that thermal energy is an internal energy that consists of the

halliday_c18_476-506v2.qxd 22-10-2009 12:03 Page 483

(A) 1, 2, 3, 4

(B) (2 and 3), 1, 4

(C) 3, 2, (1 and 4)

(D) all the same1Halliday, Resnick, Walker, page 483.

Thermal Expansion QuestionThe figure here shows four rectangular metal plates, with sides ofL, 2L, or 3L. They are all made of the same material, and theirtemperature is to be increased by the same amount. Rank theplates according to the expected increase in

(b) their areas greatest first.

48318-7 TE M PE RATU R E AN D H EATPART 2

HALLIDAY REVISED

This behavior of water is the reason lakes freeze from the top down rather thanfrom the bottom up. As water on the surface is cooled from, say, 10°C toward thefreezing point, it becomes denser (“heavier”) than lower water and sinks to thebottom. Below 4°C, however, further cooling makes the water then on the surfaceless dense (“lighter”) than the lower water, so it stays on the surface until it freezes.Thus the surface freezes while the lower water is still liquid. If lakes froze from thebottom up, the ice so formed would tend not to melt completely during the sum-mer, because it would be insulated by the water above. After a few years, manybodies of open water in the temperate zones of Earth would be frozen solid all yearround—and aquatic life could not exist.

CHECKPOINT 2

The figure here shows four rectangularmetal plates, with sides of L, 2L, or 3L.Theyare all made of the same material, and theirtemperature is to be increased by the sameamount. Rank the plates according to the ex-pected increase in (a) their vertical heightsand (b) their areas,greatest first.

Sample Problem

volume of the fuel did also, as given by Eq. 18-10 (!V "Vb !T).

Calculations: We find!V" (37 000 L)(9.50 # 10$4/C°)($23.0 K) " $808 L.

Thus, the amount delivered was

Vdel " V % !V " 37 000 L $ 808 L" 36 190 L. (Answer)

Note that the thermal expansion of the steel tank has nothing to do with the problem. Question: Who paid for the“missing” diesel fuel?

Thermal expansion of a volume

On a hot day in Las Vegas, an oil trucker loaded 37 000 L ofdiesel fuel. He encountered cold weather on the way toPayson, Utah, where the temperature was 23.0 K lower thanin Las Vegas, and where he delivered his entire load. Howmany liters did he deliver? The coefficient of volume expan-sion for diesel fuel is 9.50 # 10$4/C°, and the coefficient oflinear expansion for his steel truck tank is 11 # 10$6/C°.

The volume of the diesel fuel depends directly on the tem-perature. Thus, because the temperature decreased, the

KEY I DEA

Additional examples, video, and practice available at WileyPLUS

(1) (2) (3) (4)

18-7 Temperature and HeatIf you take a can of cola from the refrigerator and leave it on the kitchen table, itstemperature will rise—rapidly at first but then more slowly—until the tempera-ture of the cola equals that of the room (the two are then in thermal equilibrium).In the same way, the temperature of a cup of hot coffee, left sitting on the table,will fall until it also reaches room temperature.

In generalizing this situation, we describe the cola or the coffee as a system(with temperature TS) and the relevant parts of the kitchen as the environment(with temperature TE) of that system. Our observation is that if TS is not equal toTE, then TS will change (TE can also change some) until the two temperatures areequal and thus thermal equilibrium is reached.

Such a change in temperature is due to a change in the thermal energy of thesystem because of a transfer of energy between the system and the system’senvironment. (Recall that thermal energy is an internal energy that consists of the

halliday_c18_476-506v2.qxd 22-10-2009 12:03 Page 483

(A) 1, 2, 3, 4

(B) (2 and 3), 1, 4

(C) 3, 2, (1 and 4)

(D) all the same1Halliday, Resnick, Walker, page 483.

Thermal Expansion of Rings 19.4 Thermal Expansion of Solids and Liquids 575

temperature changes to Ti 1 DT, its volume changes to Vi 1 DV, where each dimen-sion changes according to Equation 19.4. Therefore,

Vi 1 DV 5 (, 1 D,)(w 1 Dw)(h 1 Dh)

5 (, 1 a, DT )(w 1 aw DT )(h 1 ah DT )

5 ,wh(1 1 a DT)3

5 Vi[1 1 3a DT 1 3(a DT )2 1 (a DT )3]

Dividing both sides by Vi and isolating the term DV/Vi , we obtain the fractional change in volume:

DVVi

5 3a DT 1 3 1a DT 22 1 1a DT 23

Because a DT ,, 1 for typical values of DT (, , 1008C), we can neglect the terms 3(a DT)2 and (a DT)3. Upon making this approximation, we see that

DVVi

5 3a DT S DV 5 13a 2Vi DT

Comparing this expression to Equation 19.6 shows that

b 5 3a

In a similar way, you can show that the change in area of a rectangular plate is given by DA 5 2aAi DT (see Problem 61). A simple mechanism called a bimetallic strip, found in practical devices such as mechanical thermostats, uses the difference in coefficients of expansion for differ-ent materials. It consists of two thin strips of dissimilar metals bonded together. As the temperature of the strip increases, the two metals expand by different amounts and the strip bends as shown in Figure 19.9.

Q uick Quiz 19.3 If you are asked to make a very sensitive glass thermometer, which of the following working liquids would you choose? (a) mercury (b) alco-hol (c) gasoline (d) glycerin

Q uick Quiz 19.4 Two spheres are made of the same metal and have the same radius, but one is hollow and the other is solid. The spheres are taken through the same temperature increase. Which sphere expands more? (a) The solid sphere expands more. (b) The hollow sphere expands more. (c) They expand by the same amount. (d) There is not enough information to say.

Table 19.1 Average Expansion Coefficients for Some Materials Near Room Temperature Average Linear Average Volume Expansion ExpansionMaterial Coefficient Material Coefficient(Solids) (a)(°C)21 (Liquids and Gases) (b)(°C)21

Aluminum 24 3 1026 Acetone 1.5 3 1024

Brass and bronze 19 3 1026 Alcohol, ethyl 1.12 3 1024

Concrete 12 3 1026 Benzene 1.24 3 1024

Copper 17 3 1026 Gasoline 9.6 3 1024

Glass (ordinary) 9 3 1026 Glycerin 4.85 3 1024

Glass (Pyrex) 3.2 3 1026 Mercury 1.82 3 1024

Invar (Ni–Fe alloy) 0.9 3 1026 Turpentine 9.0 3 1024

Lead 29 3 1026 Aira at 08C 3.67 3 1023

Steel 11 3 1026 Heliuma 3.665 3 1023

aGases do not have a specific value for the volume expansion coefficient because the amount of expansion depends on the type of process through which the gas is taken. The values given here assume the gas undergoes an expansion at constant pressure.

Steel

BrassRoomtemperature

Highertemperature

Bimetallicstrip

Off 30!COn 25!C

a

b

Figure 19.9 (a) A bimetallic strip bends as the temperature changes because the two metals have different expansion coeffi-cients. (b) A bimetallic strip used in a thermostat to break or make electrical contact.

Figure 19.8 Thermal expansion of a homogeneous metal washer. (The expansion is exaggerated in this figure.)

a

b

b " #b

a " #a

Ti " #T

Ti

As the washer is heated, all dimensions increase, including the radius of the hole.

Question

Quick Quiz 19.41 Two spheres are made of the same metal andhave the same radius, but one is hollow and the other is solid. Thespheres are taken through the same temperature increase. Whichsphere expands more?

(A) The solid sphere expands more.

(B) The hollow sphere expands more.

(C) They expand by the same amount.

(D) There is not enough information to say.

1Serway & Jewett, pg575.

Thermal Expansion and Water

Water has a strange behavior with temperature change.

Ice is less dense than water, but even in its liquid phase, waterexpands as it cools between 4◦C and 0◦C.

Water forms crystals as it freezes. Those crystals have a greatervolume than liquid water.

Between 4◦C and 0◦C some small crystals begin to form,expanding the volume of the water.

1Image from its.caltech.edu.

Thermal Expansion and Water

1Figure from Chemwiki ucdavis.edu.

Ideal Gases

An ideal gas is a gas

• at low pressure

• at a temperature much higher than its condensation point

• at a low density

Ideal Gas Equation

The equation of state for an ideal gas:

PV = nRT

where

• P is pressure

• V is volume

• n is the number of moles (amount of gas)

• R = 8.314 J mol−1 K−1 is the universal gas constant

• T is temperature measured in Kelvin

The LHS and RHS of this equation both have units of Joules(energy).

Moles?

1 mole or 1 mol. of a substance is NA = 6.022× 1023 moleculesof that substance. (NA is Avogadro’s number.)

1Photo from http://thelazybfarm.com.

Moles?

1 mole or 1 mol. of a substance is NA = 6.022× 1023 moleculesof that substance. (NA is Avogadro’s number.)

1Photo from http://thelazybfarm.com.

X

Moles

Why is this even a unit?

Not such a long time ago scientists really did not have any idea ofhow much mass an individual molecule would have, or how manymolecules would be present in a cubic meter of gas. Even theexistence of atoms and molecules was controversial.

1 mole was a macroscopic unit they could work with.

1 mole of a substance is the amount that has the same number ofmolecules as there are atoms in 12 grams of a pure Carbon-12sample. (That is NA = 6.022× 1023.)

Some History of Atoms and Molecules (Skipping)

Many ancient Indian, Greek, and Roman philosophers argued for abasic unit of matter: the atom.

In Europe, most philosophers thought matter was insteadcontinuous (Aristotle).

Nevertheless, ideas about “corpuscles” (small particles) wereimportant for Newton and his contemporaries.

A chemist, Robert Boyle (1627-1692) speculated that if atoms /corpuscles made up matter, that would resolve many problemsarising in chemistry.

He proved correct.

Understanding from Chemical Reactions (Skipping)

John Dalton, a physicist and chemist in ∼1803 started trying tounderstand the patterns of chemical reactions.

Electrolysis can dissociate water

water −→ oxygen + hydrogen

and always the same proportions are produced: twice as muchhydrogen as oxygen (by volume, at equal pressure, temperature).

This lead him to suppose that

• matter was composed of atoms

• chemical substances that could not be dissociated wereelements

• chemical substances that could be dissociated were formedfrom combinations of atoms


Rules of Dalton’s theory2:

• Elements are made of extremely small particles called atoms.

• Atoms of a given element are identical in size, mass, andother properties; atoms of different elements differ in size,mass, and other properties.

• Atoms cannot be subdivided, created, or destroyed.

• Atoms of different elements combine in simple whole-numberratios to form chemical compounds.

• In chemical reactions, atoms are combined, separated, orrearranged.

Which of these turned out to be incorrect?

1Wikipedia, Dalton, “A New System of Chemical Philosophy” (1808)


1Images from Dalton, “A New System of Chemical Philosophy” (1808) andWikimedia.

Additional Evidence for atoms: Brownian Motion(Skipping)

In 1827 Robert Brown, a botanist, observed pollen grainssuspended in water through a microscope.

He expected to see them suspended at rest, but did not.

Instead the grains of pollen seemed to jump and wiggle about forno discernible reason.

He wondered if it was something peculiar that pollen did so hetried again with dust and soot – and saw the same thing!

This motion is called Brownian Motion.

Brownian Motion (Skipping)

Brownian motion remained a mystery until 1905.

Einstein, building on tools he had just developed in his doctoralthesis, developed a theory describing Brownian motion.

It is the result of fast-moving water molecules (too small to see)colliding with the pollen molecules, and jostling them.

Periodic TableA P P E N D I X

C Periodic Table of the Elements

*Lanthanide series

**Actinide series

Atomic numberSymbol

Electron configuration

20CaAtomic mass†

58

90

57

89

3

11

19

37

55

87

20

38

56

88

21

39

57–71*

89–103**

22

40

72

104

23

41

73

105

24

42

74

106

25

43

75

107

26

44

76

108

27

45

77

109

4

12

59 60 61 62

94939291

1

Li

Na

K

Rb

Cs

Fr

Ca

Sr

Ba

Ra

Sc

Y

Ti

Zr

Hf

Rf

V

Nb

Ta

Db

Cr

Mo

W

Sg

Mn

Tc

Re

Bh

Fe

Ru

Os

Hs

Co

Rh

Ir

Mt

Be

Mg

Ce Pr Nd Pm Sm

PuNpUPaTh

H

La

Ac

4s2

5f 67s25f 46d 17s25f 36d17s25f 26d17s26d 27s26d17s2

4f 66s24f 56s24f 46s24f 36s25d14f 16s25d16s2

6d 37s26d 27s27s27s1

5d 76s25d 66s25d 56s25d 46s25d 36s25d 26s26s26s1

4d 85s14d 75s14d 55s24d 55s14d 45s14d 25s24d15s25s25s1


3s23s1

2s22s1

1s

(261) (262) (266) (264) (277) (268)

6.941 9.0122

1.007 9

22.990

39.098

85.468

132.91

(223)

40.078

87.62

137.33

(226)

44.956

88.906

47.867

91.224

178.49

50.942

92.906

180.95

51.996

95.94

183.84

54.938

(98)

186.21

55.845

101.07

190.23

58.933

102.91

192.2

24.305

140.12 140.91 144.24 (145) 150.36

(244)(237)238.03231.04232.04

40.078

138.91

(227)

GroupI

GroupII Transition elements

Note : Atomic mass values given are averaged over isotopes in the percentages in which they exist in nature.For an unstable element, mass number of the most stable known isotope is given in parentheses.†

Periodic Table of the Elements A-23

1.007 9

26.982 28.086 30.974 32.066 35.453 39.948

58.693

106.42

195.08

63.546

107.87

196.97

65.41

112.41

200.59

114.82

204.38

118.71

207.2

121.76

208.98

127.60

(209)

126.90

(210)

131.29

(222)

162.50 164.93 167.26 168.93 173.04

(259)(258)(257)(252)(251)

158.93

(247)

157.25

(247)

151.96

(243)

69.723 72.64 74.922 78.96 79.904 83.80

10.811 12.011 14.007 15.999 18.998 20.180

4.002 6

174.97

(262)

1

13 14 15 16 17 18

28

46

78

29

47

79

30

48

80

49

81 82 83

52

84

53

85

54

86

66 67 68 69 70

1021011009998

65

97

64

96

63

95

31 33 34 35 36

5 6 7 8 9 10

2

50 51

32

71

103

In

Ga

H

Al Si P S Cl Ar

Ni

Pd

Pt

Cu

Ag

Au

Zn

Cd

Hg Tl Pb Bi

Te

Po

I

At

Xe

Rn

Dy Ho Er Tm Yb

NoMdFmEsCf

Tb

Bk

Gd

Cm

Eu

Am

As Se Br Kr

B C N O F Ne

He

Sn Sb

Ge

Lu

Lr

GroupIII

GroupIV

GroupV

GroupVI

GroupVII

Group0

5f146d17s25f147s25f 137s25f 127s25f 107s25f 86d17s25f 76d17s25f 77s2

4f 146s24f 136s24f 126s24f 116s24f 106s25d 14f 7 6s24f 76s2

6p66p56p46p36p26p15d106s25d106s15d 96s1

5p65p55p45p35p25p14d105s24d105s14d10

4p64p54p44p34p24p13d104s23d104s13d 84s2

3p63p53p43p33p23p1

2p62p52p42p32p22p1

1s21s1

5f 117s2

(271) (272) (285)

110 111 112

(289)

114

(293)

116Ds Rg Cn(284) (288)

115††113†† Fl Lv(294) (294)

5d 14f 8 6s2 5d 14f 14 6s2

Elements 113, 115, 117, and 118 have not yet been officially named. Only small numbers of atoms of these elements have been observed.Note : For a description of the atomic data, visit physics.nist.gov/PhysRefData/Elements/per_text.html.

††

117†† 118††

2Serway & Jewett, Appendix C.

Periodic Table

The atomic number is the number of protons in the nucleus of anatom for that element.

A P P E N D I X

C Periodic Table of the Elements

*Lanthanide series

**Actinide series

Atomic numberSymbol

Electron configuration

20CaAtomic mass†

58

90

57

89

3

11

19

37

55

87

20

38

56

88

21

39

57–71*

89–103**

22

40

72

104

23

41

73

105

24

42

74

106

25

43

75

107

26

44

76

108

27

45

77

109

4

12

59 60 61 62

94939291

1

Li

Na

K

Rb

Cs

Fr

Ca

Sr

Ba

Ra

Sc

Y

Ti

Zr

Hf

Rf

V

Nb

Ta

Db

Cr

Mo

W

Sg

Mn

Tc

Re

Bh

Fe

Ru

Os

Hs

Co

Rh

Ir

Mt

Be

Mg

Ce Pr Nd Pm Sm

PuNpUPaTh

H

La

Ac

4s2

5f 67s25f 46d 17s25f 36d17s25f 26d17s26d 27s26d17s2

4f 66s24f 56s24f 46s24f 36s25d14f 16s25d16s2

6d 37s26d 27s27s27s1

5d 76s25d 66s25d 56s25d 46s25d 36s25d 26s26s26s1



3s23s1

2s22s1

1s

(261) (262) (266) (264) (277) (268)

6.941 9.0122

1.007 9

22.990

39.098

85.468

132.91

(223)

40.078

87.62

137.33

(226)

44.956

88.906

47.867

91.224

178.49

50.942

92.906

180.95

51.996

95.94

183.84

54.938

(98)

186.21

55.845

101.07

190.23

58.933

102.91

192.2

24.305

140.12 140.91 144.24 (145) 150.36

(244)(237)238.03231.04232.04

40.078

138.91

(227)

GroupI

GroupII Transition elements

Note : Atomic mass values given are averaged over isotopes in the percentages in which they exist in nature.For an unstable element, mass number of the most stable known isotope is given in parentheses.†

The atomic mass number is average mass of all isotopes of anelement weighted by abundance in nature.

(Isotopes of an element have different numbers of neutrons.)

Back to Moles

Now that we know the make up of various molecules in terms oftheir constituent atoms, we can determine the mass of eachmolecule.

We can then work out how many moles are in a certain mass of asubstance, m.

n =m

M

where M is the molar mass, or mass of 1 mole of the substance.

For example, water is H2O. Hydrogen has atomic mass 1, Oxygenhas atomic mass 8.

1 + 1 + 16 = 18 amu

Moles

n =m

M

where M is the molar mass, or mass of 1 mole of the substance.

For example, water is H2O. Hydrogen has atomic mass 1, Oxygenhas atomic mass 16.

1 + 1 + 16 = 18 amu

The molar mass of water is 18 g/mol.

Question

A balloon contains 6.00 g of helium.

How many moles is that?

He: 4.00 amu ⇒ M = 4.00 g/mol

n =m

M= 1.50 mol

Question



He: 4.00 amu

⇒ M = 4.00 g/mol

n =m

M= 1.50 mol

Question



He: 4.00 amu ⇒ M = 4.00 g/mol

n =m

M= 1.50 mol

Ideal Gas Equation

The equation of state for an ideal gas:

PV = nRT

Can also be written:

PV = NkBT

where

• P is pressure

• V is volume

• N is the number of molecules

• kB = 1.380649× 10−23 J K−1 is Boltzmann’s constant

• T is temperature

Question

Quick Quiz 19.63 On a winter day, you turn on your furnace andthe temperature of the air inside your home increases. Assumeyour home has the normal amount of leakage between inside airand outside air. Is the number of moles of air in your room at thehigher temperature

(A) larger than before,

(B) smaller than before, or

(C) the same as before?

3Serway & Jewett, pg 579.

Ideal Gas Equation and the Gas ThermometerIn a constant-volume gas thermometer, the pressure varies linearlywith the temperature: a consequence of the ideal gas equation!

1000T (!C)

P

P =

(nR

V

)T

Summary

• thermal expansion

• atoms

• the ideal gas equation

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