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2009년도 제1학기 화 학 1 담당교수: 신국조 Textbook: P. Atkins / L. Jones, Chemical Principles, 4th ed., Freeman (2008) Chapter 4

CHAPTER 4 THE PROPERTIES OF GASES

Fig. 4.1 Film of earth’s atmosphere Fig. 4.2 11 gas elements

THE NATURE OF GASES

4.1 Observing Gases ◈ Gas : Bulk matter, Compressible, Rapidly fill the available space

Molecules of gases widely separated, and in ceaseless random motion

4.2 Pressure

◈ Pressure of a gas, P

exerted on a surface of area A

forcePressurearea

= FPA

=

▶ SI unit of pressure: pascal, Pa 1 21 Pa 1 kg m s− −= ⋅ ⋅

◆ Origin of pressure

Collision of gas molecules with

the walls of the container

molecular storm

Fig. 4.3 Pressure of gas molecules

Fig. 4.4 Barometer Evangelista Torricelli (17c)

pressure of mercury column = atmospheric pressure

height of mercury column, h ∝ atmospheric pressure, P

2009년도 제1학기 화 학 1 담당교수: 신국조 Textbook: P. Atkins / L. Jones, Chemical Principles, 4th ed., Freeman (2008) Chapter 4 ◈ Relation between P and h :

Volume of mercury in the column, V hA=Mass, m, of this volume of mercury, m d h= ⋅

d : density of mercury

Force exerted by the mercury column at its base, F mg=

g : acceleration due to gravity

Pressure exerted by the mercury column at its base

m

F mg dhA gP dhgA A A

= = = =

◆ Manometer

Fig. 4.5 (a) An open-tube manometer. (b) A closed-tube manometer.

Ex. 4.2 Calculating the pressure inside an apparatus using a manometer.

Atmospheric pressure : 756 mmHg at 15oC

Height of mercury column in the system side :

10 mm higher than that in the open tube side

system pressure is lower than atmospheric pressure

System pressure

756 mm – 10 mm = 746 mmHg

Or ( ) ( ) ( )3 2

4 1 2 4

13,595 kg m 0.746 m 9.806 m s

9.95 10 kg m s 9.95 10 Pa

P dhg − −

− −

= = ⋅ × × ⋅

= × ⋅ ⋅ = ×


4.3 Alternative Units of Pressure

Balloonist Charles, 1783

THE GAS LAWS

Robert Boyle Jacques Charles Joseph-Louis Gay-Lussac Amedeo Avogadro

(英,1627-1691) (佛,1746-1823) (佛,1778-1850) (伊,1776-1856) 4.4 The Experimental Observations

(1) Boyle’s law (1662):

For a fixed amount of gas at constant temperature,

volume is inversely proportional to pressure.

constant (at constant and )PV n T=

Fig. 4.7 Boyle’s experiment Fig. 4.8 Boyle’s law Fig. 4.9 A plot of pressure against 1/volume


(2) Charles’s law

For a fixed amount of gas under constant pressure,

the volume varies linearly with the temperature

constant (at constant and )V T n P= ×

Fig. 4.10 Volume vs. temperature. Fig. 4.11 V vs. T for several gases Fig. 4.12 Pressure vs. temperature

▶ Extrapolations of the straight lines of V vs. T plot all reach zero volume at –273.15oC

◆ Alternative version of Charles’s law:

constant (at constant and )P T n= × V

(3) Avogadro’s principle

Under the same conditions of temperature and

pressure, a given number of gas molecules occupy

the same volume regardless of their chemical identity

m (at constant and )V nV T P=

Molar volume, Vm ≈ 22 L·mol–1 at 0oC and 1 atm

Fig. 4.13 Molar volumes of gases at 0oC and 1 atm.

2009년도 제1학기 화 학 1 담당교수: 신국조 Textbook: P. Atkins / L. Jones, Chemical Principles, 4th ed., Freeman (2008) Chapter 4 ◈ Consistency of properties of a gas with the molecular model of gas

▶ Boyle’s law

Compression increases the number of molecules in a given region of the sample

increases the number of collisions increases the pressure

Fig 4.14 Pressure of a gas arises from impact on the walls of the container.

▶ Charles’s law

▷ Effect of temperature on the pressure of a gas at constant volume (Fig. 4.12)

Increase in temperature increases the average speed of the molecules

increase in the number and strength of collisions

increases in pressure

▷ Effect of temperature on the volume of a gas at constant pressure (Fig. 4.10)

Increase in temperature increase in volume so as not to increase the number of collisions

keeping the pressure constant

▶ Avogardo’s principle

Increase in number of molecules increase in volume so as not to increase the number of collisions

keeping the pressure constant

(1) + (2) + (3)

Ideal gas law : PV nRT=

Equation of state

Limiting law for real gases as P 0

Universal gas constant, R

R = 8.314 J·K–1·mol–1

= 8.206 x 10–2 L·atm·K–1

4.5 Applications of the Ideal Gas Law

◈ Combined gas law : 1 1 2 2

1 1 2 2

PV PVn T n T

=

Ex. 4.3 Calculating the pressure of a given sample

Ex. 4.4 Using the combined gas law when one variable is changed

Ex. 4.5 Using the combined gas law when two variable are changed


▶ STP (Standard Temperature and Pressure) 0oC, 1 atm (=101.325 kPa)

▶ SATP (Standard Ambient Temperature and Pressure) 25oC, 1 bar (=100 kPa)

◈ Molar volume

m/V nRT P RTV

n n= = =

P

Molar volume of an ideal gas : 22.41 L·mol–1 at STP (24.79 L·mol–1 at SATP)

Fig. 4.15 Molar volume of ideal gase at SATP.

4.6 Gas Density

▶ /Molar concentration n pV RT P

V V R= = =

T

Same molar concentration for all gases for a given P and T

Same amount of gas molecules in equal volume at the same P and T for all gases

▶ Density of a gas, ( )/PV RT Mm nM MPd

V V V RT= = = =

☺ density of air ~ 1.6 g·L–1 at SATP Ex. 4.6 Calculating the molar mass of a gas from its density Geraniol – volatile organic compound, component of rose oil

d = 0.480 g·L–1 at 260oC and 103 Torr M ?

(Solution)

d = 0.480 g·L–1, T = (273.15+260)K = 533 K, P = 103 Torr

( ) ( ) ( )1 1 1

10.480 g L 62.364 L Torr K mol 533 K155 g mol

103 TorrdRTM

P

− − −−

⋅ × ⋅ ⋅ ⋅ ×= = = ⋅


4.7 The Stoichiometry of Reacting Gases

Ex. 4.7 Calculating the mass of reagent needed to react with a specified volume of gas

☺ Removing CO2 from the air in a closed-system breathing environment: submarine, spacecraft

2 2 324 2 CO (g) 2 K CO (s) 3 O ( )K s) gO ( + ⎯⎯→ + 2 Potassium superoxide

Calculate the mass of KO2 needed to react with 50 L of CO2 at 25oC and 1 atm.

(Solution)

Molar volume of CO2: Vm = 24.47 L·mol–1

1 mol CO2 2 mol KO2

Molar mass of KO2

39.10 + 2(16.00) g·mol–1 = 71.10 g·mol–1

Convert from volume of CO2 to mass of KO2

22

1 mol COMass of KO (50 L)24.47 L

⎛ ⎞= × ⎜ ⎟⎝ ⎠

22

2 2

2 mol KO 71.1 g 2.9 10 g1 mol CO 1 mol KO

⎛ ⎞ ⎛ ⎞× × = ×⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Fig. 4.16 2 CO2 + 4 KO2 2 K2CO3 + 3 O2

2009년도 제1학기 화 학 1 담당교수: 신국조 Textbook: P. Atkins / L. Jones, Chemical Principles, 4th ed., Freeman (2008) Chapter 4 ◈ Explosive formation of gases ▶ Solid fuel: Formation of CO and CO2

▶ Lead azide, Pb(N3)2

Detonator for explosives

( )3 22Pb N (s) Pb(s) 3 N (g)⎯⎯→ +

when it is struck

▶ Sodium azide, NaN3

Air bags in automobiles

explosive release of N2

electrically detonated

Fig. 4.17 Explosion by ignition of coal dus

Fig. 4.18 Rapid decomposition of sodium azide, NaN3, forming a large volume of N2 gas in an air bag. 4.8 Mixtures of Gases

▶ Partial pressure (Pi) of the i-th gas

in a mixture of gases

pressure that the i-th gas would exert

if it occupied the container alone

◈ Dalton’s Law of Partial Pressures

The total pressure of a mixture of gases is the

sum of the partial pressures of its component.

A B ii

P P P P= + + ⋅⋅⋅ =∑

Fig. 4.19 Law of partial pressures all gases must be ideal gases

2009년도 제1학기 화 학 1 담당교수: 신국조 Textbook: P. Atkins / L. Jo cal Principles, 4th ed., Freeman (2008) Chapter 4 nes, Chemi ▶ Mole fraction of A component, Ax

AA

A B

nxn n

=+ + ⋅⋅⋅

A B 1x x+ = for a binary mixture

Fig. 4.20 Mole fraction of red molecules in a binary mixture, REDx

A

An RTP

V=

( )A BnRT RTP n nV V

= = + + ⋅⋅⋅ A B

RT PV n n

=+ + ⋅⋅⋅

AA A

A B

n PP xn n

= =+ + ⋅⋅⋅

P

∴ A AP x P=

MOLECULAR MOTION 4.9 Diffusion and Effusion ▶ Diffusion : gradual dispersal of one substance through another substance

Fig. 4.21 In diffusion, the molecules of one substance spread into the region occupied by molecules of another in a series of random steps, undergoing collisions as they move. ▶ Effusion : escape of a gas through a small hole into a vacuum or a region of low pressure

◈ Graham’s Law of Effusion:

Rate of effusion ∝ 1M

B

A

Rate of effusion of A moleculesRate of effusion of B molecules

MM

=

Fig. 4.22 Effusion

Thomas Graham (蘇格蘭,1805-1869)


▷1Average speedM

∝ A

B

Time for A to effuseTime for B to effuse

MM

=

Rate of effusion T∝ 2 2

1 1

Rate of effusion at Rate of effusion at

T TT T

=

▷ speAverage ed T∝ Nature of temperature !

▷▷ Average speed TM

∝

4.10 The Kinetic Model of Gases ◈ Kinetic theory of gases (or Kinetic model of gases)

1. A gas consists of a collection of molecules in continuous random motion.

2. Gas molecules are infinitesimally small points.

3. The molecules move in straight lines until they collide.

4. The molecules do not influence one another except during collisions.

Fig. 4.23 Kinetic model of gases Fig. 4.24 Kinetic model of gas

◈ Derivation of Boyle’s law

(1) Momentum of a particle moving right along the x-axis toward the wall : xmv(2) Momentum of a particle moving left along the x-axis from the wall : xm− v

(3) Magnitude of change in Momentum during the collision: 2 2x x xm m m m− − = − =v v v xv

(4) All the molecules within a distance x t∆v of the wall and traveling toward it will strike the wall

during the interval . t∆ (5) All the particles in the volume x t∆Av will reach the wall. (A: area of wall)

(6) Overall, N molecules in V. Number of molecules in the volume x t∆Av is xA t NV∆

×v

.

(3) (4) (5) (6)


(7) Average number of collisions Half the number of molecules in x t∆Av 12

xA t NV∆

× ×v

(8) Total momentum change during : t∆2

22

x xx

NA t NmA tmV V∆ ∆

× =v vv

(9) Force = Rate of momentum change = (Total momentum change) / t∆ = 2xNmA

Vv

(10) Pressure = Force / Area : 2xNm

Vv

(7) (9) (10)

(11) Introduce average value, 2xv

2xNm

PV

=v

(12) but 2 2x y= + +2v v v v2

z2z

2 2x y= =v v v 2 1

3x = 2v v

(13) Root mean square speed, 1/ 2

rms =2v v

(13) ∴ 2 2rms A rms A rms

3Nm Nm nN m N MP

V= = = =

2v v v3V 3V 3V

2v

(14) Boyle’s law: 21rms3 PV nM= v = constant (at constant T)

(15) Ideal gas law: PV nRT=

(16) 1/ 23RT

M⎛ ⎞= ⎜ ⎟⎝ ⎠

rmsv or 2rms

3RMvT =

Fig. 4.25 ’s of gases at 25rmsv oC 4.11 The Maxwell-Boltzmann Distribution of Speeds

James Clerk Maxwell Ludwig Boltzmann (蘇格蘭,1831-1879) (墺,1844-1906)

◈ Maxwell-Boltzmann distribution of speeds

( )f∆∆

N = vN

v with 3/ 2

/ 2( ) 42

M RTMf eRT

ππ

−⎛ ⎞= ⎜ ⎟⎝ ⎠

22 vv v


Or B

3/ 2/ 2

B

( ) 4 2

m k Tmf ek T

ππ

−⎛ ⎞= ⎜ ⎟

⎝ ⎠

2v2v v

where 23 1

0/ 1.38066 10 J KBk R N − −= = × : Boltzmann constant

Fig. 4.26 Range of molecular speeds of three gases Fig. 4.27 Temperature dependence of the distribution of

at the same temperature, 300K. molecular speeds of the same gas.

* Most probable speed

mp2 2Bk T RT

m= =v

M , mp

( )dfd =

⎡ ⎤=⎢ ⎥⎣ ⎦v v

vv

0

* Average speed

0

8 8( ) Bk T RTf dm Mπ π

∞

= = =∫v v v v

* Mean square speed

2 2

0

3 3( ) Bk T RTf dm M

∞

= = =∫v v v v

* Root-mean-square speed

2rms

3 3Bk T RTm M

= = =v v

mp rms 1.000 :1.128 :1.225< < =v v v

From Oxtoby


Box 4.1 HOW DO WE KNOW . . . THE DISTRIBUTION OF MOLECULAR SPEEDS? ◈ Measuring speeds of molecules

Molecular beam

Rotating discs with slits oriented askew one another.

Rotate the discs with different rotational rates.

Measure the intensity of beams arriving at the detector at various rotational rates.

THE IMPACT ON MATERIALS: REAL GASES 4.12 Deviations from Ideality

Condensation of gases into liquids Strong attraction (Phase transition)

Liquids are very difficult to compress Strong repulsion

◈ Compression (or Compressibility) factor, Z

m m

idealm /V V PZ

V RT P R= = = mV

T

For an ideal gas, 1Z = .

Real gases deviate from 1Z = as P ↑ .

Presence of intermolecular forces

Z < 1 attractive forces Z > 1 repulsive force

Fig. 4.29 Variation on the potential energy of a molecule as it Fig. 4.28 Compression factor against pressure

approaches another molecule.

2009년도 제1학기 화 학 1 담당교수: 신국조 Textbook: P. Atkins / L. Jones, Chemical Principles, 4th ed., Freeman (2008) Chapter 4 The Liquefaction of Gases ◈ Chlorine, Cl2

Condenses to a liquid

at 1 atm and below –35oC.

Upper flask contains a “cold finger”

a smaller tube filled with dry ice in

acetone at –78oC

Fig. 4.30 Liquefaction of chorine

◈ Joule-Thomson effect Temperature vs. molecular speeds

Expansion of real gas against attraction between gas molecules

Slowing the molecules (Lowering of average speeds)

Cooling the gas

▶ Linde refrigerator

Compression of a gas expansion through a throttle

gas cools as it expands circulates past incoming gas which enters the same cycle

liquefies gas is distilled to separate components

Used for producing O2, N2, Ne, Ar,…. in air

Fig. 4.31 Cooling by the Joule-Thomson effect Fig. 4.32 Linde refrigerator for liquefying gases. 4.13 Equations of State of Real Gases ◈ Virial equation of state

2 32

m m

( ) ( )1 B T B TPV nRTV V

⎛ ⎞= + + + ⋅⎜ ⎟

⎝ ⎠⋅ ⋅

2 ( )B T : second virial coefficient, repulsion >0, attraction <0

2009년도 제1학기 화 학 1 담당교수: 신국조 Textbook: P. Atkins / L. Jones, Chemical Principles, 4th ed., Freeman (2008) Chapter 4 ◈ van der Waals equation of state (1873) ▶ Corrections to the ideal equation of state

Attraction at long distance:

Reduction in collision frequency /n V∝

Reduction in intensity of collision /n V∝ 2

2idealnP P aV

= +

Johannes Diderik van der Waals

(和,1837-1923) Nobel Prize ’10 Physics

"for his work on the equation of state for gases and liquids"

Repulsions at short distance :

No overlap of molecules excluded volume effect

Reduction in free volume n∝

idealV V b= − n

( )2

2 nP a V nb nRTV

⎛ ⎞+ − =⎜ ⎟

⎝ ⎠

2

2

nRT nP aV nb V

= −−

can also predict the phase transition ! Hint: cubic equation in V

▶ Compressibility factor

11 /

PV V a n a nZnRT V nb RT V bn V RT V

= = − = −− −

1 11 /

nbbn V V

+ + ⋅⋅⋅−

21 1 ( )a n nZ bV

BV

TRT

⎛ ⎞+ − + ⋅ ⋅ ⋅ = + + ⋅ ⋅ ⋅⎜ ⎟⎝ ⎠

cf. virial eq. of state

B(T)=0 at the Boyle temperature, BaT

Rb= for a van der Waals gas.

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