lecture note 1(2011 spring) -...

Thermodynamics 1

Lecture Note 1Lecture Note 1

March 02, 2011

Kwang KimKwang Kim

Yonsei University

[email protected]

39 8 5334739

Y88.91

8

O16.00

53

I126.9

34

Se78.96

7

N14.01

교수 소개

김광범 (Kwang‐Bum Kim)Email : [email protected]공과대학 B 관 325호

전공연구분야 : 재료 화학Materials Chemistry

재료전기화학Materials Electrochemistry재료전기화학Materials Electrochemistry

전기에너지 저장 장기 (리튬이차전지, 초고용량 커패시터)

대학원생연구실 : GS 칼텍스 연구동 411호

에너지 저장소재 국가지정연구National Research Laboratory (NRL) of E C i d St M t i lEnergy Conversion and Storage Materials

과목 소개 : Physical Chemistry of Materials, 필수 과목강의 시간 : 월 6, 7교시, 수 1, 2교시, 16주 면담 시간 : 수요일 3-4교시

강의실 : 공대 B관 042호

주교재 : Principle of Physical Chemistry (Lionel M. Raff)

성적 평가 시험 4회 1차 2차 3차 시험 각 20% 4차 시험 30%성적 평가 : 시험 4회 : 1차, 2차, 3차 시험 각 20%, 4차 시험 : 30%출석 및 강의 참여 10%

출석 평가 전자출결확인시 템 이상 결석 시 성적 여출석 평가 : 전자출결확인시스템 (1/3 이상 결석 시 성적 부여 불능)매 강의 출석 및 참여는 필수 사항입니다

출석: 강의시작 10분 전 - 강의시작 5분(연강의 경우 첫 강의 시간만 확인)

지각 : 강의시간 5분 후 20분지각 : 강의시간 5분 후 - 20분결석 : 강의시간 20분 후

강의 조교 : 윤희창 박사과정 대학원생 (02-365-7745)

GS 칼텍스 연구동 411호

주 기간 수업내용 교재범위,과제물 비고

12011-03-02

2011-03-07

Properties of Matters

Equation of State, Virial EquationChapter 1 수강신청 확인 및 변경

2011 03 08PVT behavior of matters

22011-03-08

2011-03-14 The 1st law of Thermodynamics

Heat and work

Chapter 1 수강신청 확인 및 변경

2011 03 15 The 1st law of Thermodynamics3

2011-03-15

2011-03-21

The 1st law of Thermodynamics

Internal energy, enthalpyChapter 2

42011-03-22 The 1st law of Thermodynamics Chapter 2

Exam 142011-03-28 Internal energy, enthalpy Chapter 3

Exam. 1

52011-03-29

2011 04 04Thermochemistry Chapter 3 (3.30 ~ 4.1) 수강철회

2011-04-04

62011-04-05

2011-04-11

The 2nd law of Thermodynamics

Heat engine, entropyChapter 4 (4.7) 학기 1/3선

2011 04 12 The 2nd law of Thermodynamics7

2011-04-12

2011-04-18

The 2nd law of Thermodynamics

Heat engine, entropyChapter 4

82011-04-19

2011 04 25중 간 고 사

(4.20 ~ 4.25) 중간시험2011-04-25 중간시험

주 기간 수업내용 교재범위,과제물 비고

92011-04-26 The 2nd law of Thermodynamics

Chapter 492011-05-02 Free energy

Chapter 4

102011-05-03 Chemical Equilibrium in Pure Substances

Chapter 5 (5.5) 어린이날102011-05-09 Chemical potential

Chapter 5 (5.5) 어린이날

2011-05-10 Chemical Equilibrium in Pure Substances

(5 10) 석가탄신일11

2011 05 10

2011-05-16 Equilibria involving vapors

Reversible process, Equilibrium constant

Chapter 5(5.10) 석가탄신일(5.16) 학기 2/3 선

2011 05 17 Phase Equilibrium12

2011-05-17

2011-05-23

Phase Equilibrium

Clausius-Clapeyron equationChapter 5 Exam. 3

2011-05-24 Equilibrium in Pure Substances13

2011 05 24

2011-05-30

Equilibrium in Pure Substances

Phase diagramChapter 6

142011-05-31 Solutions, Partial molar quantities,

Chapter 8 (6 6) 현충일142011-06-06

qGibbs phase rule

Chapter 8 (6.6) 현충일

152011-06-07

2011 06 13

Solutions

Colligative propertiesChapter 8

2011-06-13 Colligative properties

162011-06-14

2011-06-20 학기말 시험 기말시험

It’s all about the blue marble

2004 6.5 Billion People2050 ~ 10 Billion People

Humanity’s Top Ten Problemsfor next 50 years

1. ?2. ?3. ?4. ? 5. ?6. ?7. ?8. ?9. ?10. ? 2004 6.5 Billion People

2050 ~ 10 Billion Peoplep

Prof. Richard E. Smalley1996 Nobel Prize Winner in Chemistry

Humanity’s Top Ten Problemsfor next 50 yearsDISEASEDEMOCRACY

ENVIRONMENT

WATER

TERRORISM & WAR

POVERTYEDUCATION

POPULATION

2004 6.5 Billion People2050 ~ 10 Billion PeopleENERGY

POVERTYFOOD

p


Humanity’s Top Ten Problemsfor next 50 years

1. ENERGY2. WATER3. FOOD4. ENVIRONMENT 5. POVERTY6. TERRORISM & WAR7. DISEASE8. EDUCATION9. DEMOCRACY10. POPULATION 2004 6.5 Billion People

2050 ~ 10 Billion Peoplep


World EnergyMillions of Barrels per Day (Oil Equivalent)

300

200

100

01860 1900 1940 1980 2020 2060 2100

Source: John F. Bookout (President of Shell USA) ,“Two Centuries of Fossil Fuel Energy” International Geological Congress, Washington DC; July 10,1985. Episodes, vol 12, 257-262 (1989).


PRIMARY ENERGY SOURCESAlternatives to Oil

• Conservation / Efficiency not enough• Conservation / Efficiency -- not enough• Hydroelectric -- not enough• Biomass -- not enough• Wind -- not enough• Wind -- not enough• Wave & Tide -- not enough

• Natural Gas -- sequestration? cost?Natural Gas sequestration?, cost?• Clean Coal -- sequestration?, cost?

• Nuclear Fission -- radioactive waste?, terrorism?, cost?uc ea ss o ad oac e as e?, e o s ?, cos ?• Nuclear Fusion -- too difficult?, cost?

• Geothermal HDR -- cost ?• Solar terrestrial -- cost ?• Solar power satellites -- cost ?• Lunar Solar Power -- cost ?


Enabling Nanotech Revolutionsh t lt i l ti t d t b 0 t 00 f ld• Photovoltaics -- a revolution to drop cost by 10 to100 fold.

• H2 storage -- a revolution in light weight materials for pressure tanks, and/or a new light weight, easily reversible hydrogen chemisorption system

• Fuel cells -- a revolution to drop the cost by nearly 10 to 100 fold

• Batteries and supercapacitors -- revolution to improve by 10-100x for automotive d di t ib t d ti li tiand distributed generation applications.

• Photocatalytic reduction of CO2 to produce a liquid fuel such as methanol.

S t li ht i ht t i l t d t t LEO GEO d l t th • Super-strong, light weight materials to drop cost to LEO, GEO, and later the moon by > 100 x, and to enable huge but low cost light harvesting structures in space.

• Robotics with AI to enable construction/maintenance of solar structures in space and on the moon; and to enable nuclear reactor maintenance and fuel and on the moon; and to enable nuclear reactor maintenance and fuel reprocessing. (nanoelectronics, and nanomaterials enable smart robots)

• Actinide separation nanotechnologies both for revolutionizing fission fuel reprocessing and for mining uranium from sea waterreprocessing, and for mining uranium from sea water

• Alloy nanotechnologies to improve performance under intense neutron irradiation (critical for all of the GEN IV advanced reactor designs, and for fusion).

• Thermoelectrics or some other way of eliminating compressors in refrigeration.


Enabling Nanotech Revolutionsh t lt i l ti t d t b 0 t 00 f ld• Photovoltaics -- a revolution to drop cost by 10 to100 fold.

• H2 storage -- a revolution in light weight materials for pressure tanks, and/or a new light weight, easily reversible hydrogen chemisorption system

• Fuel cells -- a revolution to drop the cost by nearly 10 to 100 fold

• Batteries and supercapacitors -- revolution to improve by 10-100x for automotive d di t ib t d ti li tiand distributed generation applications.

• Photocatalytic reduction of CO2 to produce a liquid fuel such as methanol.

Nano Materials + Materials Electrochemistry

The road to success is paved

with advanced materials.

Imagine driving it

Imagine driving it without the need of this

The Future of Transportation is Electric

The Future of Transportation is Electric

Two huge industries are transforming

and a new one is emerging...and a new one is emerging...

B tt I d t

Electricity Transportation

Battery Industry

First Hybrid Electric Vehicle (HEV) in late 1800sThis car is a front wheel drive

electric‐gasoline hybrid car and

has power steering A gasolinehas power steering. A gasoline

engine supplements the battery

pack. Between 1890 and 1910,

there were many hybrid electric

cars and four wheel drive electric

cars. Electric cars were more

expensive than gasoline cars and

electrics were considered moreelectrics were considered more

reliable and safer. With the

development of the starter motor

for gasoline cars and increased

range of gasoline cars, most

people public interest switchedwww.didik.com

people public interest switched

from electrics to gasoline by 1915.

relevant enabling technology : Batteries

a cell that converts chemical

energy into electrical energy gy gy

by reversible chemical

reactions and that may be

recharged by passing a

current through it in the

di ti it t th t fdirection opposite to that of

its discharge

Electric Vehicle (EV)

relevant enabling technology : Batteries

• Battery issues:

longevity, cost, energy

voltage, & weight!

Battery technology is not mature yet

Storage technologyStorage technology Energy densityEnergy densityg gyg gy gy ygy y

LeadLead--acid batteriesacid batteries 100 kJ/kg (30 W100 kJ/kg (30 W--h/kg)h/kg)

LithiumLithium--ion batteriesion batteries 600 kJ/kg600 kJ/kg

Compressed air 10 Compressed air 10 MPaMPa 80 kJ/kg (not including tank)80 kJ/kg (not including tank)Compressed air, 10 Compressed air, 10 MPaMPa 80 kJ/kg (not including tank)80 kJ/kg (not including tank)

Conventional capacitorsConventional capacitors 0.2 kJ/kg0.2 kJ/kg

UltracapacitorsUltracapacitors 20 kJ/kg20 kJ/kg

FlywheelsFlywheels 100 kJ/kg100 kJ/kgFlywheelsFlywheels 100 kJ/kg100 kJ/kg

GasolineGasoline 43000 kJ/kg43000 kJ/kg

Material Science and Engineering

Materials Science and Engineering (MSE) grew out of the

disciplines of metallurgy and ceramics and now includes

polymers, semiconductors, magnetic materials, photonic

materials, energy materials and biological materials.

The field of MSE researches all classes of materials with an

emphasis on the connections between 1) the underlying

structure, 2) processing, 3) properties, and 4) performance

of the material.

Material Science and Engineering (MSE)The field of MSE researches all classes of materials with anThe field of MSE researches all classes of materials with an emphasis on the connections between 1) the underlying structure 2) processing 3) properties and 4) performancestructure, 2) processing, 3) properties, and 4) performance of the material.

"Tetrahedron of Materials Science and Engineering"

Performance

Properties

Synthesis/Composition/Structure

Synthesis/Processing

Four elements of materials and strong interrelationship

Material Science and Engineering (MSE)Four elements of materials and strong interrelationship

among them define a field of Materials Science and

Engineering. Materials Science and Engineering rooted in

the classical description of physics and chemistryp p y y

Performance

Properties


y /Processing

Material Science and Engineering (MSE)

Final materials must perform Performance

Properties and performance : l d i i da given task in an economical

and societally acceptable mannerrelated to composition and structure

Properties

Composition/Synthesis/Processing Composition/

Structureresult of synthesis and processing

Processing

Material Science and Engineering (MSE)Properties and Performance

Properties : descriptor that defines the functional attributes and

utility of materials

Diamond

Properties Performance

brilliance and transparency gem tone, optical coating

hardness and thermal conductivity cutting tools

Properties : collective response of materials to external stimuli

→ results of the structure and composition of the synthesized

or processed substance


Structure and Composition

→ Which atoms are present? How are these atoms arranged?→ c ato s a e p ese t o a e t ese ato s a a ged

Advance in Materials Science and Engineering

→ What You See is What You Get→ What You See is What You Get

→ Development of tools for probing structures and composition

(XRD, XAS, TEM, SEM, STM, AFM)(XRD, XAS, TEM, SEM, STM, AFM)

→ Models to explain the origin of materials properties from

their structure and compositiontheir structure and composition

Increased understanding of the relationship among propertiesIncreased understanding of the relationship among properties,

structure and composition of materials leads to design of new

materials with desirable combination of propertiesmaterials with desirable combination of properties.


Synthesis and Processing : applied to make a given material

→ comprehensive array of techniques for building of new

arrangement of atoms and molecules

Synthesis : physical and chemical means by which atoms

and molecules are assembled (atomic scale)

Processing : materials manufacturing (large scale)

solidification, sintering, welding, machining forging


"Tetrahedron of Materials Science and Engineering"

Performance

Properties


Synthesis/Processing

Physical Chemistry

Chemistry is the study of Matter and the Changes it undergoes.

Change in Matters accompanied by Changes in Energy

AB + CD → AC + BD

Matter and Energy are what chemistry is all about.

Physical Chemistry : can be studied from a macroscopic view pointy y p p

→ Thermodynamics

→ Macroscopic Science (T, P, V)→ p ( , , )

or can be studied from a microscopic viewpoint

→Molecular concept→ Molecular concept

→ Quantum Chemistry

Physical Chemistry : more abstract and more mathematical than other chemistry courses

Physical Chemistry : can be studied from a macroscopic view point

→ Thermodynamics

→ Macroscopic Science (T, P, V)

→ forget about the existence of atoms and moleclues

or can be studied from a microscopic viewpointp p

→ Molecular concept

→ Quantum ChemistryQ y

Four branches of Physical Chemistry :

Thermodynamicsy

Kinetics

Statistical MechanicsStatistical Mechanics

Quantum Chemistry

Chemical reaction

Will it be possible produce AC from a mixture of AB and CD?p p

AB + CD → AC + BD

1) C i ? Di i f h i ?1) Can it occur? Direction of the reaction?

2) Will it occur spontaneously?

What determines the reaction direction?

3) How to control reaction variables to drive the reaction to form AC?3) How to control reaction variables to drive the reaction to form AC?

4) How fast can we produce AC?

Chemical reaction

Will it be possible produce AC from a mixture of AB and CD?p p

AB + CD → AC + BD

1) C i ? Di i f h i ?1) Can it occur? Direction of the reaction?

2) Will it occur spontaneously?

What determines the reaction direction?

3) How to control reaction variables to drive the reaction to form AC?3) How to control reaction variables to drive the reaction to form AC?

4) How fast can we produce AC?

Thermodynamics: 1), 2) and 3); Kinetics: 4)

h l hHow to get an A in Physical Chemistry

‐Working problems is essential to learning Physical Chemistry.

Work with a pencil and a calculator.

‐ Learn mathematics while learning science.g

Mathematical expressions and equations :

meant to be understood not to be memorizedmeant to be understood, not to be memorized

E ti t t t f l ti b t h i l titiEquations : statement of a relation between physical quantities

Learning Curve

Central theme of Physical Chemistry

‐ System

관찰 대상의 정의

‐ State

관찰 대상의 상태에 대한 정량적 표현관찰 대상의 상태에 대한 정량적 표현

‐ Processes

조작을 통한 관찰 대상의 상태 변화

Scientific Thinking

‐ System under investigation

‐ Description or Behavior of a system

‐ Variables of a systemVariables of a system

‐ Correlation between behavior and variables of a system

‐Modeling of a system

‐ Comparison of a model with a systemp y

‐ Revision of a model

Scientific Thinking

‐ System under investigationSystem under investigation


Scientific Thinking



성별, 이름, 생년월일, 거주지, 출생국가, 인종

학력, 직업, 가족관계, 종교, 지능지수, 감성지수학력, 직업, 가족관계, 종 , 지능지수, 감성지수

키, 몸무게, 혈압, 혈액형, 혈당지수, 체성분, 건강상태

인생관 성격 자세 태도 습관 첫인상인생관, 성격, 자세, 태도, 습관, 첫인상

취미, 좋아하는 음식, 은행잔고, 대인관계, 외모, 음성

대화법

Scientific Thinking



Gas in a balloon

Scientific Thinking



Which gas?

Ideal gas or non-ideal gas?

Amount of a gas?

Pressure?Gas in a balloon

Volume?

Temperature?p

Density?

Price?Price?

Chapter 1 : Properties of gases

‐ Ideal gas : PV = nRT

‐ Real gas : molecular interaction

Non ideality of real gas‐ Non‐ideality of real gas

‐ Deviation of real gas from ideal gas

‐ Quantifying non‐ideality

‐ Van der Walls equation

‐ Check how well Van der Walls equation describe l b h ia real gas behavior

Cause and Effecty = F(x) ; y is a function of x, y ( ) ; y ,x : controlled variable, y : observed variable

System ‐ a part of the universe of interest to you‐ surrounded by the boundary to separate from the other part ofthe universe the surroundingsthe universe, the surroundings

‐What interactions are there between the system and the surroundings?

‐What is exchanged between thegsystem and the surroundings?

‐What changes in the system areWhat changes in the system areobserved?

‐ How do we describe the system?‐ How do we describe the system?

System

System / Boundary / Surroundings

Heat and work : means to change the energy of a system

Open System

Closed System

Isolated System

Description of the System

How can we describe a system that consists of a pure gas? Properties (variables or descriptors) of a pure gas system : Properties (variables or descriptors) of a pure gas system :

‐ physical description: macroscopic properties of a gas system

pressure P volume V and temperature Tpressure P, volume V, and temperature T, chemical composition C, number of atoms or molecules, n

‐ chemical description: chemical potential)‐ chemical description: chemical potential)

i t i i bl T P h i l t ti l)intensive variables : T, P, chemical potential)

extensive variables : V, n

If numerical values given to those descriptors, we know everything we need to know about the properties of the system.

‐ the state of the system specified

Properties of a gas : amount of gas n, temperature T, volume, V and pressure P

Description of a gas at a certain state : a gas at equilibrium

system at equilibrium :system at equilibrium :

Thermal equilibrium

Mechanical equilibrium

Chemical equilibriumChemical equilibrium

State of a gas: defined with numerical values given to amount of gas n temperature T volume V and pressure Pamount of gas n, temperature T, volume V, and pressure P

Equation of state of a gas : relationship among amount of t t l dgas, temperature, volume, and pressure

P = f(T, V, n)

pV = nRT for an ideal gas : Equation of state of an ideal gas

Equilibrium of a system :

The word equilibrium means a state of balance. qIn an equilibrium state, there are no unbalanced potentials (or driving forces) within the system.

Thermal equilibrium : Ts (thermal potential) are the same in every partof a system

Mechanical equilibrium : Ps (mechanical potential) are the same in every part of a system

Chemical equilibrium : s (chemical potential) are the same in everyChemical equilibrium : s (chemical potential) are the same in everypart of a system

Gas A in a container


The word equilibrium means a state of balance. In an equilibrium q qstate, there are no unbalanced potentials (or driving forces) within the system.

Thermal equilibrium : Ts (thermal potential) are the same in every partof a system

Mechanical equilibrium : Ps (mechanical potential) are the same in every part of a system

Chemical equilibrium : s (chemical potential) are the same in everyChemical equilibrium : s (chemical potential) are the same in everypart of a system

T1 T2 T3 pV = nRT for an ideal gas

.Ti Tj

pV = nRT for an ideal gas

Gas A in a container P

VTzTz‐1


Thermal equilibrium

M h i l ilib iMechanical equilibrium

Chemical equilibrium

State of a gas: defined with single numerical values given toState of a gas: defined with single numerical values given to amount of gas n, temperature T, volume V, and pressure P

Eq ation of state of a as relationship amon amo nt ofEquation of state of a gas : relationship among amount of gas, temperature, volume, and pressure

P f(T V )P = f(T, V, n)

pV = nRT for an ideal gas : Equation of state of an ideal gas

The Zeroth Law of Thermodynamics

C id t t A d B i hi h th t tConsider two systems, A and B, in which the temperature of A is greater than the temperature of B. ‐ Each is a closed system.

‐ No material transfer, but heat and work transfer across the boundary

What happens to the temperatureWhat happens to the temperature when A and B are brought together? ‐ Heat flux from A to B due toHeat flux from A to B due totemperature difference‐ Thermal energy transfer, or heat Thermal energy transfer, or heattransfer from A to B till TA = TB‐ Two systems at thermal equilibriumy q


B, C : Mercury Thermometer

TA, TB, TC = ?

If TA = TB, TB = TC , then TA = TC


If t t f i i th l ilib iIf two systems of any size are in thermal equilibrium with each other and a third system is in thermal

ilib i ith f th th it i i th lequilibrium with one of them, then it is in thermal equilibrium with the other, too.

If TA = TB and TB = TC, then TA = TC

If TA = TB If TA = TB

Then T = TThen, TB = TC

The state of a system: dictated not by what they were, or how they got there,y y , y g ,

but by what the state variables are.

Process 1 Process 2

Final State

Initial State

Process 3

Final State

State Variables : State functions

A state variable is a precisely measurable physical property that characterizes the state of a system independently ofthat characterizes the state of a system, independently of how the system was brought to that state. (path‐independent)

‐must be inherently single‐valued to characterize a state.

Common examples of state variables : pressure P volume VCommon examples of state variables : pressure P, volume V, and temperature T, internal energy U, enthalpy H, Helmholtz free energy F chemical potential Gibbs free energy G andfree energy F, chemical potential , Gibbs free energy G and entropy S

In the ideal gas law, the state of n moles of gas is precisely determined by these three state variables of P, V and T.

PV = nRT





Thermal equilibriumThermal equilibrium

Mechanical equilibrium

Chemical equilibrium




Common examples of state variables : pressure P volume VCommon examples of state variables : pressure P, volume V, and temperature T, internal energy U, enthalpy H, Helmholtz free energy F chemical potential Gibbs free energy G andfree energy F, chemical potential , Gibbs free energy G and entropy S

In the ideal gas law, the state of n moles of gas is precisely determined by these three state variables of P, V and T.

PV = nRT

Equation of State of a gas

Ph l i l th d i b d i tPhenomenological thermodynamics : based on experiment

For any fixed amount of a pure gas ( n = fixed), consider two y p g ( ),state variables P and V.‐ Each can be controlled independently from each other.p y‐ Another state variable T cannot be changed independentlyfrom P or V.

Experience shows that if a certain P, V, and T are specified f ti l l f t ilib i th ll thfor a particular sample of gas at equilibrium, then all the measurable, macroscopic properties of that sample have

t i ifi lcertain specific values.

Equation of State of a gas

E i h th t if t i P V d T ifi dExperience shows that if a certain P, V, and T are specified for a particular sample of gas at equilibrium, then all

bl i ti f th t l hmeasurable, macroscopic properties of that sample have certain specific values.

Arbitrary values for all four variables n, P, V, and T are not possible simultaneously.p y‐ For any fixed amount of a pure gas, only two of the three state variables P, V, and T are truly independent., , y p

Mathematical equation with which we can calculate the thi d t t i bl f th t k t t i blthird state variable from the two known state variables : equation of state

Ideal Gas Law

A id lAn ideal gas :

‐ all collisions between atoms or molecules are perfectlyp yelastic‐ there are no intermolecular attractive forces‐ a collection of perfectly hard spheres which collide butwhich otherwise do not interact with each other‐ no potential energy‐ all the internal energy is in the form of kinetic energygy gyand any change in internal energy is accompanied by achange in temperature. g p

An ideal gas : characterized by three state variables: b l t (P) l (V) d b l tabsolute pressure (P), volume (V), and absolute temperature (T).

Ideal Gas Law with Constraints

All the possible states of an ideal gas can be represented by a PVT surface as illustrated in the left.

The behavior whenThe behavior when any one of the three state variables isstate variables is held constant is also shownshown.

Equation of StateBoyle’s lawBoyle s law

Charles’s law

Avogadro’s law

fi dat fixed n, T

at fixed n pat fixed n, p

at fixed p Tat fixed p, T

Boyle’s law

Charles’s law

Avogadro’s law

Ideal Gas Law The relationship between T P and V : deduced from kineticThe relationship between T, P and V : deduced from kinetic theory of gases and is called the

n = number of moles

R = universal gas constant = 8.3145 J/mol K

N = number of molecules

k = Boltzmann constant = 1.38066 x 10‐23 J/K = 8.617385 x 10‐5 eV/K

k = R/NA

NA = Avogadro's number = 6.0221 x 1023 /mol

one mole of an ideal gas at STP occupies 22.4 liters.STP is used widely as a standard reference point for expression of y p pthe properties and processes of ideal gases. ‐ standard temperature : freezing point of water, 0°C = 273.15 K‐ standard pressure : 1 atmosphere = 760 mmHg = 101.3 kPa‐ standard volume of 1 mole of an ideal gas at STP : 22.4 liters

Ideal Gas Law pV = nRTpV = nRT

SI units:

V (m3), P (Pa), T (K)

R 8 314 J K 1 l 1R = 8.314 J K‐1 mol‐1

1 J = 1 Nm, 1 N = 1 kg m s‐2

1 Pa = 1 N/m2 = 1 kg m s‐2/m2 = 1 kg m‐1 s‐2

alternative units:alternative units:

V (L), (1 L = 1 dm3 = 10‐3 m3), P (atm)

R = 8.206 x 10‐2 L atm K‐1 mol‐1

1 Pa = 1 N/m2 = 10−5 bar = 9 8692×10−6 atm1 Pa = 1 N/m = 10 bar = 9.8692×10 atm

= 7.5006×10−3 torr = 145.04×10−6 psi

How does a bubble jet printer work?

Factors needed to consider in the ejection of ink jet from a capillary of a printing head of a bubble jet p y p g jprinter?

Kinetic Theory of gases

‐ study of the microscopic behavior of molecules and the interactions

which lead to macroscopic relationships like the ideal gas lawwhich lead to macroscopic relationships like the ideal gas law

‐ Pressure can be viewed as arising from the kinetic pressure

of gas molecules colliding with the walls of a container inof gas molecules colliding with the walls of a container in

accordance with Newton's laws.

temperature is taken to be proportional to this average‐ temperature is taken to be proportional to this average

kinetic energy


F = ma = m (du/dt) = d(mu)/dt : force acting on the particle is equal to the change in momentum per unit time.

Change in momentum between before collision (mu1) and after collision (‐mu1) : (‐mu1) – (mu1) = ‐2mu1Time between collisions : 2 l /u1. Number of collisions per unit time : u1/2 l. Change in momentum per second = (‐2mu1) x (u1/2 l) Force acting on the particle = ‐mu12/ l. Force acting on the wall = +mu12/ l. Pressure P’ = +mu12/(l A) = +mu12/V per one particle1 /( ) 1 / p p

Kinetic Theory of gasesPressure P’ = +mu12/(l A) = +mu12/V per one particle1 /( ) 1 / p p

Total pressure P = m (u12 + u22 + u32 + ‐‐‐ + uj2 + ‐‐‐‐‐‐ )/V

<u2> = (u 2 + u 2 + u 2 + ‐‐‐ + u 2 + ‐‐‐‐‐‐ )/N<u > = (u1 + u2 + u3 + ‐‐‐ + uj + ‐‐‐‐‐‐ )/N

P = Nm <u2>/V for one dimensional gas

F th di i l 2 2 + 2 + 2For three dimensional gas, c2 = u2 + v2 + w2. <c2> = <u2> + <v2> + <w2>, <u2> = <v2> = <w2> = (1/3) <c2>

P = Nm <c2>/(3V)P = Nm <c >/(3V)

Kinetic energy of any molecule = mc2/2

< < 2 /2<>= m <c2>/2

PV = (2/3) N <>

PV = nRT = (2/3) N <>

N = N/NA, RT = (2/3) NA <>

Total kinetic energy U = NA <> = (3/2)RT

Ideal Gas Law with Constraints : pV = nRT

All the possible states of an ideal gas can be represented by a PVT surface as illustrated in the left.

The behavior whenThe behavior when any one of the three state variables isstate variables is held constant is also shownshown.

Partial Derivatives and Ideal Gas Law

How is one state variable affected when another state variable changes?

Partial Derivatives and Ideal Gas Law

How is one state variable affected when another state variable changes? Change in F expressed by dF caused by change in x by dx, dy, dz and so on? Total derivatives of function of multiple variables F (x,y,z,….)

the derivative of the function F taken w.r.t. one variable i i h h h i bl h ldat a time with the other variables held constant

the derivative of the function F taken w.r.t. x only withthe derivative of the function F taken w.r.t. x only with y, z and so on treated as constants : partial derivatives

Partial Derivative

The ordinary derivative of a function of one variable can be

carried out because everything else in the function is a

constant and does not affect the process of differentiationconstant and does not affect the process of differentiation.

When there is more than one variable in a function, it is

often useful to examine the variation of the function withoften useful to examine the variation of the function with

respect to one of the variables with all the other variables

constrained to stay constant.

This is the purpose of a partial derivative. p p p

Partial Derivative

the partial derivative with respect to x :

the partial derivative with respect to y :the partial derivative with respect to y :

Partial Derivatives and Ideal Gas LawHow is one state variable affected when another stateHow is one state variable affected when another state variable changes?

V = f(T,P)( , )

Change in V with T and P with constraints?with constraints?

PV

VT

Partial Derivatives and Ideal Gas LawHow does the P varies w.r.t. T, assuming constant n and V?How does the P varies w.r.t. T, assuming constant n and V?

Non‐ ideal Gas

‐ Ideal gas : PV= nRT

‐ Real gas : l l i t timolecular interaction,

molecular size

‐ Non‐ideality of real gas

‐ Deviation of real gas f id lfrom ideal gas

Q tif i id lit‐ Quantifying non‐ideality

Non‐ ideal Gas

‐ Ideal gas : PV= nRT n, T : fixed

‐ Real gas : l l i t timolecular interaction,

molecular size

‐ Non‐ideality of real gas

‐ Deviation of real gas f id lfrom ideal gas

Q tif i id lit‐ Quantifying non‐ideality

Intermolecular Forces

Coulomb's Law : Like charges repel, unlike charges attract.The electric force acting on a point charge q1 as a result ofThe electric force acting on a point charge q1 as a result of the presence of a second point charge q2 is given by Coulomb's Law:Coulomb s Law:

where ε0 = permittivity of space0 p y p

Ideal gas law : l l f i t ti l‐molecules of a gas as point particles

with perfectly elastic collisionslid f dil t‐ valid for dilute gases,

but gas molecules are not point masses

van der Waals equation of state ;

‐ A modification of the ideal gas lawproposed by Johannes D van der Waals in 1873 to takeproposed by Johannes D. van der Waals in 1873 to takeinto account molecular size and molecular interaction forces.

‐ Constants a and b : positive values and characteristic of the pindividual gas : gas‐specific properties,

different values for different gasesTh d W l i f h h id l l‐ The van der Waals equation of state approaches the ideal gas law PV=nRT as the values of these constants approach zero.

van der Waals Equation of State

Constant a : correction for the intermolecular forces Constant b : correction for finite molecular size

‐ its value is the volume of one mole of the atoms or molecules

‐ could be used to estimate the radius of an atom or molecule, modeled as a sphere.

Fishbane et al. give the value of b for nitrogen gas as 39.4 x10‐6 m3/mol. This leads to the following estimate of radius:

atomic radius of 0.075 nm for nitrogen

Non‐ ideal Gas : how to express non‐ideality?

C ibili f ZCompressibility factor Z for nmoles of non‐ideal gas = Vm/Vidat constant T and Pat constant T and P

Z = 1 for an ideal gas

Non‐ ideal Gas Ideal GasI t l l f ?Intermolecular force?

No liquefaction of an ideal gasLiquefaction of a real gas

Requirements for liquefaction ?Requirements for liquefaction ?

Phase Equilibria in waterq

Phase Equilibria in water

temperature increase

pressure increase

Non‐ ideal Gas

D D’

a discontinuity in the isothermE D’’

a discontinuity in the isotherm

At point C, CO2 condensation starts.

Th l d i ifi tl hil th i h d

Equilibrium vapor pressure of pure liquid : Function of temperature

The volume drops significantly, while the pressure remains unchanged.At points E and F, CO2 is in the liquid state. Since the liquid is nearlyincompressible, a huge pressure increase is required to decrease thel b li h l h i lvolume by a slight amount. Consequently, the P‐V curve is nearly

vertical from point E to point F.

Non‐ ideal Gas

phase transition: condensation, gas to liquid

iti l i tcritical point: critical temperature = maximum temperature at which a gas can be liquefiedliquefied

Non‐ ideal Gas

As the temperature is increased a higherAs the temperature is increased, a higher

pressure is required to condense CO2.

A lt th d i lAs a result, the decrease in volume upon

condensation becomes smaller at higher

temperature.

Finally, a temperature is reached above which the gas cannot be

liquefied regardless of the pressure applied. The highest temperature

at which liquefaction is still possible is called the critical temperature

Tc . At this temperature, the discontinuity in the isotherm at the point

of condensation will have been reduced to a single point, which will

be an inflection point in the curve. The asterisk marks the inflection

point. The pressure and volume at the inflection point of the critical

isotherm are termed the critical pressure Pc and critical volume Vc

Non‐ ideal Gas : Intermolecular force vs. critical temperature

The P V and T values of a real gas depend primarily uponThe Pc , Vc , and Tc values of a real gas depend primarily upon

magnitude of the intermolecular forces.

G ith l i t l l f hibit hi h iti l t tGases with large intermolecular forces exhibit high critical temperatures.

Gases with near-zero large intermolecular forces have very low values of

Tc . For example, helium, which exhibits nearly ideal gas type behavior,

has a critical temperature of-267.9°C.

The normal melting point of

materials, T°m , also depends upon

the magnitude of the g

intermolecular forces holding the

molecules in the solid state.o ecu es t e so d state.

Cryophorus 실험

본 실험은 진공 상태에 위치한 water drolpet의 상변화 과정본 실험은 진공 상태에 위치한 ate d o pet의 상변화 과정을 물의 상태도를 고려하여 예측하고 , 이를 실제 관찰할 수있도록 진행함.

가정 : 진공상태에 위치한 water droplet은 주변과 단열상태를 유지함.

A droplet of water is placed in a bottle connected to a vacuum pump. Explain what you would observe in a water droplet p p p y pwith time.

VacuumVacuum

Water droplet in a bottle under vacuumunder vacuum

Cryophorus 실험

가정 : 진공상태에 위치한 water droplet은 주변과 단열상태가정 진공상태에 위치한 ate d op et은 주변과 단열상태를 유지함.

A droplet of water is placed in a bottle connected to a vacuumA droplet of water is placed in a bottle connected to a vacuum pump. Explain what you would observe in a water droplet with time.

*

Vacuum

W d l iWater droplet in a bottle under vacuum

Cryophorus 실험

1) 진공상태에 위치한 water droplet의 evaporation1) 진공상태에 위치한 water droplet의 evaporation

2) 흡열 반응인 evaporation 진행,

자체의 온도 감소

*

자체의 온도 감소

liquid water/vapor 평형상태 공존선 도달

3) 지속적 온도 감소, 3중점 도달

(ice/liquid water/vapor 평형공존점)

4) 온도, 압력 저하

liquid water/vapor 평형상태 공존선에서

solid/vapor 평형 공존선으로 변화

5) liquid water droplet가 solid ice 로 변화5) liquid water droplet가 solid ice 로 변화

6) solid ice는 sublimation에 의해 vapor 상으로 승화

궁극적으로는 solid ice는 모두 sublimation 하여 사라짐궁극적으로는 solid ice는 모두 sublimation 하여 사라짐

Intermolecular Forces and Potentials

The interaction force F between two such particles with charges q1 and

q2 separated by a distance r in vacuum is given by Coulombs law of

electrostatic interaction, where ε0 , the permittivity of the vacuum.

The charges are in coulombs (C), the distance is in meters, and the

force is in newtons Thus the force between two charged particlesforce is in newtons. Thus, the force between two charged particles

decreases as the inverse square of the distance between them. Note

that if q and q have the same sign F will be positivethat if q1 and q2 have the same sign, F will be positive.

Repulsive forces are associated with positive values of the interaction

force, while attractive forces have negative values of F.


Potential Energy : energy a system possesses by virtue of its position inPotential Energy : energy a system possesses by virtue of its position in

force field

Where Fz is the force in the z direction and V(z) is the potential energy

of the system. The force is, therefore, the negative of the slope of they , , g p

potential energy function. Physically, this means that the force always

acts in a direction so as to reduce the potential of the system.acts in a direction so as to reduce the potential of the system.

where the variable z is replaced with r. Integration of Eq.1.47 gives

where c is a constant of integration. If we take V(r=∞)=0 as our

reference point, then c=0, and Eq.1.48 becomes

Intermolecular Forces and PotentialsWhen two molecules, such as CO2 and benzene, do not possess a

permanent dipole moment, there will still be a net attractive force

between them at a large separation. This force is called a Londong p

dispersion force and arises because the interaction of the electron clouds

surrounding the molecules distorts the charge density and induces asurrounding the molecules distorts the charge density and induces a

temporary dipole. When the atoms are near each other, the mutual

repulsion felt by each charge cloud distorts the spherical symmetry ofrepulsion felt by each charge cloud distorts the spherical symmetry of

the cloud, with the result that a temporary dipole is produced.

Quantitative treatment of this effect shows that the resulting potentialQuantitative treatment of this effect shows that the resulting potential

energy is approximately proportional to the inverse sixth power of the

i h i C 6 h C i i iseparation; that is, V(r)=-C/r6 , where C is a positive constant.


The preceding interactions all produce attractive forces HoweverThe preceding interactions all produce attractive forces. However,

when molecules approach close enough, their electronic charge

l d b i t l d th lti l i f d i tclouds begin to overlap, and the resulting repulsive forces dominate

the interaction. These repulsive interactions usually vary with inverse

powers of the separation between 10 and 13. That is, we usually

observe that Vrep(r)=K/rn , with n=10 to13 and K appositive constant.

The total potential can be written as the sum of the attractive and

repulsive interactions. For a molecule exhibiting only London

dispersion interactions, we would expect to have a total potential of

the form


The Lennard-Jones potential is a typical example; ε and α are

parameters that determine the attractive well depth and the position

of the potential minimum, respectively.

Non‐ ideal Gas

As the pressure approaches zero the assumptions of noAs the pressure approaches zero, the assumptions of nointermolecular forces and a zero volume for the gasmolecules become increasingly good approximationsmolecules become increasingly good approximations.

The reason is that the molecular volume becomesinsignificant as the total volume of the gas becomes verylarge.

As V increases and P decreases, the average distancebetween molecules <r> increases Since intermolecularbetween molecules, <r>, increases. Since intermolecularforce varies approximately with <r‐7>, the forces ofattraction become vanishingly small at low pressures.attraction become vanishingly small at low pressures.


repulsive interactions: hard spheres excluded volumerepulsive interactions: hard spheres excluded volume,

V (V – nb)

where b is a material constant, equal to the volume of 1 mol ofwhere b is a material constant, equal to the volume of 1 mol of

densely packed gas particles

P(V - nb) = nRT or P = nRT /(V – nb)


repulsive interactions: hard spheres excluded volumerepulsive interactions: hard spheres excluded volume,

P = nRT /(V – nb)

attractive forces: diminish pressure; pressure is the result of collisionsattractive forces: diminish pressure; pressure is the result of collisions

of the gas particles with the walls;

as a particle is about to hit the container wall, it is “held back”, and itsp

impact is diminished, by the attractive forces from surrounding gas

particles; this is a pair effect ~ number of pairs of particles ~ (n/V )2

P = nRT/(V – nb) – a(n/V)2


The van der Waals equation is a two-parameter equation of state. The

d b diff f b h htwo parameters, a and b, differ from gas to gas, but they are the same

at all values of P, V, and T for a given gas. Division by the first factor in

Eq. 1.57, followed by the addition of b to both sides, produces

If we now take the limit as P → ∞, the first term on the right-hand side

of Eq. 1.58 becomes negligible relative to b, and we haveq g g ,

When the gas is compressed with a very large pressure, the molar

l t d b l th l f b Th b t tvolume cannot decrease below the value of b. Thus, b must represent

the volume occupied by the molecules themselves. As a result, the van

der Waals equation of state compensates for inaccuracies produced by

the ideal-gas-law assumption of zero volume for the gas molecules.


That is, the real-gas pressure is pressure the gas would exert if were

ideal min s a q antit that depends pon the in e se si th po e ofideal, minus a quantity that depends upon the inverse sixth power of

the distance variable. This observation suggests that the parameter a is

f h i f i b h l l ha measure of the attractive forces acting between the molecules. Such

attractive forces would serve to reduce the pressure exerted on the walls

of the container, since they would have the same effect as that expected

if there were a spring between pairs of molecules.


Note in the table that the smaller molecules or atoms, such as He and Ne, have the

ll b l h l l l l h d l dsmallest b values, while larger molecules, such as CO2 and acetylene, are associated

with larger values of b. Figure 1.13 illustrates the correlation between the a values

determined from fitting pressure, volume, and temperature data and the measured

melting points of the compounds. The correlation is obvious and suggests that our

conclusion that a is associated with the magnitude of the attractive interactions

between molecules is reasonable.

Other Equations of State

virial equation of state good for quantitative work, B(T ), C(T ), . . .are tabulated for gases, but it does not provide understanding of

the above phenomena of real gases

Law of Corresponding StatesPV = nRT

Law of Corresponding States

Where X* is some specific value of X that usually has a special

i ifi Si h d d i f 1 6 hsignificance. Since the numerator and denominator of Eq.1.67 have

identical units, the reduced variable will always be unitless. In the case of

pressure, volume, and temperature, we define their reduced form by

Since there exists a linear relationship between pressure, volume, and

temperature and the reduced variables it is clear that any equation oftemperature and the reduced variables, it is clear that any equation of

state can be expressed in terms of the reduced variables rather than P, V,

and T When this transformation is made it is often observed that theand T. When this transformation is made, it is often observed that the

equation of state becomes identical for every gas. This is the law of

di Wh d i f d d i bl hcorresponding states. When expressed in terms of reduced variables, the

equation of state becomes identical for all gases.

Law of Corresponding StatesConsider a real gas at the inflection point of its critical isotherm At thisConsider a real gas at the inflection point of its critical isotherm. At this

point, where P=Pc , V=Vc, and T=Tc, the itself must be satisfied by the

critical variables Then at an inflection point we must also havecritical variables. Then, at an inflection point, we must also have

The first and second partial derivatives of the pressure with respect to VThe first and second partial derivatives of the pressure with respect to V

are, respectively,

Law of Corresponding States : Universality


The critical isotherm with TR=1.0 exhibits the expected inflection point,

hi h i k d i h i l i h fi A hi i hwhich is marked with an open circle in the figure. At this point, we have

PR=VR=TR=1.0. The isotherms with TR≥1.0 closely resemble those seen in

Figure 1.8 for CO2. Those with TR<1.0 qualitatively reflect the measured

data above the condensation point and after the gas has condensed.

Law of Corresponding StatesR = 0.082 L atm mol‐1 K‐1

The average absolute percent error in R is about 24%. This reflects the

fact that real gases do not obey the van der Waals equation of state atfact that real gases do not obey the van der Waals equation of state at

the critical point. Nevertheless, the law of corresponding states is a

f l t th t t i ifi t i t th id luseful concept that represents a significant improvement over the ideal-

gas equation of state.

Residual VolumeIn the ideal-gas regime at low pressure, where the volume occupied by

the molecules becomes negligible and the intermolecular forces

approach zero, we expect the ideal-gas equation of state to hold. That is,

we expect to observe that

For an idea-gas, we also expect to have

for all values of P, V, and T. However, we do not expect to see Eq. 1.88

hold for a real gas even when it is in the ideal-gas region where Eq. 1.87

is satisfied.

Residual VolumeAlth h th i t t t t b t di ti it iAlthough the previous statement may seem to be a contradiction, it is

not: It is quite common for the ratio of two functions to approach a

limiting value of unity, but their difference not to approach a limiting

value of zero. As an example, consider two simple functions

I h li i W l l hIn the limit as x →∞ , We clearly have

However, we do not have

Since, f(x)-g(x)=(x+5)-(x+3)=2. Thus, a limiting ratio may be unity, while

th li iti diff i Thi i tl th it ti th t i tthe limiting difference is nonzero. This is exactly the situation that exists

for real gases: The ratio of Vm to RT/P is unity as the pressure

h b h diff b h i i iapproaches zero, but the difference between these quantities is not zero.

Residual VolumeThe nonzero difference between Vm and RT/P in the limit of zerom /

pressure or infinite volume is defined to be the residual volume

As an example, let us consider a van der Waals gas whose equation of

state is given by Eq. 1.57. To determine the residual volume of this gas,

We need an analytic expression for the difference [Vm-RT/P]. This can be

easily obtained by expanding the left side of Eq. 1.57. Such an expansiony y p g q p

produces

Residual VolumefTaking limits of both sides as P→0, we obtain

From Eq. 1.87, we know that PVm approaches RT in the limit of lowFrom Eq. 1.87, we know that PVm approaches RT in the limit of low

pressure so that Eq.1.91, may be rewritten as

Since Vm becomes infinite as the pressure approaches zero, the third

term on the right vanishes in the limit and we obtainterm on the right vanishes in the limit, and we obtain

Residual Volume

The residual volume is the difference between the actual volume

occupied by the gas and that which the gas would have occupied if it

had been ideal. There are two contributions, b and –a/RT. The first of

these is due to the volume occupied by the molecules of the gas

themselves. This is zero for an ideal-gas, but is given by the parameter b

for a van der Waals gas. Consequently, this effect makes a positive

contribution to Vres. The second contribution arises due to theres

intermolecular attractive forces is determined by the parameter a, which

causes the molecules to be pulled closer together. The resulting volumecauses the molecules to be pulled closer together. The resulting volume

is less than would be observed for an ideal gas, and the contribution to

V is negative which shows that the effect of the intermolecular forcesVres is negative, which shows that the effect of the intermolecular forces

decreases with increasing temperature.

Compression FactorThe residual volume is an experimental measure of the deviation of realp

gases from ideal behavior in the limit of low pressure. In this limit, the

ratio (PVm)/(RT) approaches unity even though the difference [Vm – RT/P]ratio (PVm)/(RT) approaches unity even though the difference [Vm RT/P]

does not. At higher pressures, (PVm)/(RT) can deviate significantly from

unity This ratio which is called the compression factor or compressibilityunity. This ratio, which is called the compression factor or compressibility,

is frequently used as an experimental measure of the nonideality of real

gases In most texts the compression factor is given the symbol Z(T P)gases. In most texts, the compression factor is given the symbol Z(T,P),

where the arguments emphasize the dependence of Z upon temperature

d Wi h hi i hand pressure. With this notation, we have

Since P / RT is the reciprocal of the molar ideal-gas volume and Vm/(RT)

i th i l f th id l th i f tis the reciprocal of the ideal-gas pressure, the compression factor can

also be written in the form

Non- ideal Gas

Compression FactorThe residual volume is an experimental measure of the deviation of realp

gases from ideal behavior in the limit of low pressure. In this limit, the

ratio (PVm)/(RT) approaches unity even though the difference [Vm – RT/P]ratio (PVm)/(RT) approaches unity even though the difference [Vm RT/P]

does not. At higher pressures, (PVm)/(RT) can deviate significantly from

unity This ratio which is called the compression factor or compressibilityunity. This ratio, which is called the compression factor or compressibility,

is frequently used as an experimental measure of the nonideality of real

gases In most texts the compression factor is given the symbol Z(T P)gases. In most texts, the compression factor is given the symbol Z(T,P),

where the arguments emphasize the dependence of Z upon temperature

d Wi h hi i hand pressure. With this notation, we have

Compression FactorIn most texts, the compression factor is given the symbol Z(T,P), wherep g y

the arguments emphasize the dependence of Z upon temperature and

pressure. With this notation, we havepressure. With this notation, we have

Since P/RT is the reciprocal of the molar ideal gas volume and V /(RT) isSince P/RT is the reciprocal of the molar ideal-gas volume and Vm/(RT) is

the reciprocal of the ideal-gas pressure, the compression factor can also

b i i h fbe written in the form

Compression Factor

For the same reasons that the residual volume can be either positive or

i i f b i h l h inegative, compression factors can be either greater or less than unity.

When Z(T,P )<1, the actual molar volume of the real gas is less than that

of an ideal gas under similar conditions. Compression factors less than

unity are usually observed in gas molecules with relatively large

intermolecular forces that contract the molecules, thereby producing a

volume less than that which would be observed for an ideal gas.

Compression factors greater than unity are the result of the nonzero

volume of the molecules. When this factor becomes important,p

compression factors larger than unity are observed.

Non- ideal Gas

Virial Equation of StateThe pressure, volume, and temperature behavior of real gases can be

represented by

where V is the molar volume of the gas The expansion coefficientswhere Vm is the molar volume of the gas. The expansion coefficients

C2(T),C3(T),…….,Cn(T) are termed the second virial coefficient the third virial

coefficient and so on to the nth virial coefficientcoefficient and so on to the nth virial coefficient.

As Vm ∞, all real gases must approach ideal behavior, since the

interaction forces approach zero and the volume occupied by theinteraction forces approach zero and the volume occupied by the

molecules becomes negligible relative to the volume of the gas. Hence,

l th fi t t i th i i i th li it V It ionly the first term in the expansion remains in the limit as Vm ∞. It is

important to note that each of the virial coefficients is a function of the

temperature. Consequently, the value of Cn is different at each

temperature, and there are effectively an infinite number of parameters

available for adjustment to fit the measured pressure volume, and

temperature behavior of the gas.

Non- ideal Gas

lecture note 1(2011 spring) -...

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