Chapter 5

Gases and the Kinetic-Molecular Theory Name (Formula) Methane (CH4) Ammonia (NH3) Chlorine (Cl2)

Oxygen (O2) Ethylene (C2H4)

Origin and Use natural deposits; domestic fuel from N2+H2; fertilizers, explosives

electrolysis of seawater; bleaching and disinfecting liquefied air; steelmaking high-temperature decomposition of natural gas; plastics

Organisms reduce CO2 in the atmosphere , animals breath O2 microbes take in N2 and produce ammonia etc.

The three states of matter.

An Overview of the Physical States of Gases Note The Distinction of Gases from Liquids and Solids

1. Gas volume changes greatly with pressure. 2. Gas volume changes greatly with temperature. 3. Gases have relatively low viscosity (resistance to flow).

4. Most gases have relatively low densities under normal conditions. 5. Gases are miscible. 6. Gases are compressible.

7. Gas particles have negligible attraction for each other (ideal gases).

Effect of atmospheric pressure on objects at the Earth’s surface.

Variations in pressure, temperature, and composition of the Earth’s atmosphere.

A mercury barometer.

closed-end

Two types of manometer

open-end

Closed end Vacuum

Evacuated flask

A

Closed end Vacuum

Evacuated flask

A

Mercury levels equal

Closed end Vacuum

Evacuated flask

A

Mercury levels equal Pgas

h

B

Patm Closed end Open end

Vacuum

Evacuated flask

A

Mercury levels equal Pgas

h Pgas

B

C

Patm Closed end Open end

Vacuum

Evacuated flask

A

Mercury levels equal Pgas

h Pgas

B

C Pgas = Patm

Patm

Patm

Closed end Open end

Vacuum

Evacuated flask

A

Mercury levels equal Pgas

h

h Pgas

B

Pgas

C Pgas = Patm D Pgas < Patm Pgas = Patm - h

Patm

Patm

Patm

h

h

Closed end Open end

Vacuum

Evacuated flask

A

Mercury levels equal Pgas

h Pgas

B

Pgas

C Pgas = Patm D Pgas < Patm Pgas = Patm - h

Pgas

E Pgas > Patm Pgas = Patm + h

Common Units of Pressure Unit

Atmospheric Pressure

Scientific Field

pascal(Pa); kilopascal(kPa)

1.01325x105Pa; 101.325 kPa

SI unit; physics, chemistry

atmosphere(atm)

1 atm*

chemistry

millimeters of mercury(Hg)

760 mm Hg*

chemistry, medicine, biology

torr

760 torr*

chemistry

14.7lb/in2

engineering

1.01325 bar

meteorology, chemistry, physics

pounds per square inch (psi or lb/in2) bar

*This is an exact quantity; in calculations, we use as many significant figures as necessary.

PROBLEM SOLVING A geochemist heats a limestone (CaCO3) sample and collects the CO2 released in an evacuated flask attached to a closed-end manometer. After the system comes to room temperature, h = 291.4 mm Hg. Calculate the CO2 pressure in torrs, atmospheres, and kilopascals.

Exercise # 8: Do problems 1 & 2.

1

Boyle’s Law

V

VxP

= constant

Charles’s Law

V

V

Amontons’s Law

T

T

= constant

T

P

P

P

T

= constant

n and T are fixed V = constant / P P and n are fixed

V = constant x T

V and n are fixed P = constant x T

T combined gas law

V

P

V = constant x

T

PV

P

T

= constant

Basic Behavior of a Gas Boyle’s Law: P1V1 = P2V2 Where T and n are held constant.

The relationship between the volume and pressure of a gas.

Boyle’s Law

Problem Solving Do #’s 3 & 4 of Exercise 8

Relationship of Pressure and Temperature Charles’ Law: V1 = V2 T1 T2 Where P and n are held constant.

**** Temperature must be in Kelvin****

The relationship between the volume and temperature of a gas.

Charles’s Law

Problem Solving Do Problem #’s 5 & 6 of Exercise 8

Relationship Between Temperature and Pressure Amonton’s Law:

P1 T1

=

P2 T1

Where V and n are held constant. **** Temperature must be in Kelvin****

Relationship Between Volume, Temperature, and Pressure The Combined Gas Law:

P1V1 = T1

Where n is held constant. Do problem #’s 7 & 8 of Exercise 8

P2V2 T2

The Relationship Between the Volume and Amount of a Gas.

Avogadro’s Law Equal volumes of a gas will contain the same number of particles if P & T are held constant. At standard temperature, 273.15 K and pressure, 1.00 atm (STP) 1 mole of gas = 22.4 L

Standard molar volume.

Problem Solving Do problems 9 & 10 of Exercise 8

THE IDEAL GAS LAW

PV = nRT 3 significant figures

R=

PV

nT

=

1atm x 22.414L 1mol x 273.15K

=

0.0821atm*L

mol*K

R is the universal gas constant

IDEAL GAS LAW nRT PV = nRT or V = fixed n and T Boyle’s Law V=

constant

P

P

fixed n and P

fixed P and T

Charles’s Law

Avogadro’s Law

V=

constant X T

V=

constant X n

Problem Solving Do problem #’s 11 & 12 of Exercise 8

Determination of Density of a Gas Think about what units you will need. Consider: PV = nRT, then

P/RT = n/ V where, n(molar mass)/ V = density Do Problem #15 of Exercise 8

? Grams ? Volume

Determination of The Molar Mass of a Gas

n=

mass

MW

=

PV

RT

MW =

m RT VP

d=

MW =

m V

d RT

P

Determination of the Molar Mass of a Gas Think about what units you will need to determine. ? Grams ? Moles Do Problem #’s 13 and 17 of Exercise 8

Determining the molar mass of an unknown volatile liquid. based on the method of J.B.A. Dumas (1800-1884)

Problem Solving

A chemical engineer isolates a colorless liquid from a petroleum sample. She uses the Dumas method and obtains the following data: Volume of flask = 213 mL

T = 100.00C

Mass of flask + gas = 78.416 g

Mass of flask = 77.834 g

Calculate the molar mass of the liquid.

P = 754 torr

Mixtures of Gases •Gases mix homogeneously in any proportions.

•Each gas in a mixture behaves as if it were the only gas present.

Dalton’s Law of Partial Pressures Ptotal = P1 + P2 + P3 + ...

P1=

1

x Ptotal

where

1

=

1 is

the mole fraction n1

n1 + n2 + n3 +...

=

n1 ntotal

Problem Solving Do problem # 21 of Exercise 8

Collecting a water-insoluble gaseous reaction product and determining its pressure (Week 8 Lab).

1 Water-insoluble gaseous product bubbles through water into collection vessel Ptotal Patm

1 Water-insoluble gaseous product bubbles through water into collection vessel

2 Pgas adds to vapor pressure of water (PH2O) to give PtotalAs shown Ptotal < Patm

Ptotal Patm

1 Water-insoluble gaseous product bubbles through water into collection vessel

2 Pgas adds to vapor pressure of water (PH2O) to give PtotalAs shown Ptotal < Patm

Ptotal Patm

Ptotal

Patm

3 Ptotal is made equal to Patm by adjusting height of vessel until water level equals that in beaker

1 Water-insoluble gaseous product bubbles through water into collection vessel

2 Pgas adds to vapor pressure of water (PH2O) to give PtotalAs shown Ptotal < Patm

3 Ptotal is made equal to Patm by adjusting height of vessel until water level equals that in beaker

Ptotal

Pgas Patm

Ptotal

Patm

Ptotal

=

P H2 O

1 Water-insoluble gaseous product bubbles through water into collection vessel

2 Pgas adds to vapor pressure of water (PH2O) to give PtotalAs shown Ptotal < Patm

3 Ptotal is made equal to Patm by adjusting height of vessel until water level equals that in beaker

Ptotal

Pgas Patm

Ptotal

Patm

Ptotal

=

P H2 O

4 Ptotal equals Pgas plus PH2O at temperature of experiment. Therefore, Pgas = Ptotal – PH2O

Vapor Pressure of Water (P ) at Different Temperatures H2 O

T(0C) 0 5 10 11 12 13 14 15 16 18 20 22 24 26 28

P (torr)

T(0C)

P (torr)

4.6 6.5 9.2 9.8 10.5 11.2 12.0 12.8 13.6 15.5 17.5 19.8 22.4 25.2 28.3

30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

31.8 42.2 55.3 71.9 92.5 118.0 149.4 187.5 233.7 289.1 355.1 433.6 525.8 633.9 760.0

Collecting a water-insoluble gaseous reaction product and determining its pressure (Week 8 Lab).

Draw Diagram of Lab Apparatus PT = PH2O + Pgas PT = barometric pressure PH2O = vapor pressure of H2O at a particular temperature

Problem Solving Do Problem # 24 of Exercise 8

Summary of the stoichiometric relationships among the amount (mol,n) of gaseous reactant or product and the gas variables pressure (P), volume (V), and temperature (T).

P,V,T

of gas A ideal gas law

amount (mol)

amount (mol)

P,V,T

of gas A

of gas B

of gas B

molar ratio from balanced equation

ideal gas law

Gas Stoichiometry Problem Solving Do problem #’s 18, 13, 19, 20, and 25 of Exercise 8

Postulates of the Kinetic-Molecular Theory Postulate 1: Particle Volume Because the volume of an individual gas particle is so small compared to the volume of its container, the gas particles are considered to have mass, but no volume. Postulate 2: Particle Motion Gas particles are in constant, random, straight-line motion except when they collide with each other or with the container walls.

Postulate 3: Particle Collisions Collisions are elastic therefore the total kinetic energy(Ek) of the particles is constant.

Distribution of molecular speeds at three temperatures.

A molecular description of Boyle’s Law.

A molecular description of Dalton’s law of partial pressures.

A molecular description of Charles’s Law.

A molecular description of Avogadro’s Law.

Summary: A Molecular View of the Gas Laws Particles exert a force when they collide with the walls of a container. The force is proportional to the pressure.

More particles, more collisions, more pressure Greater T, particles move faster, more collisions, more pressure.

Smaller volume, more collisions, more pressure.

Avogadro’s Law

V

Ek = 1/2 mass x speed2

n

Ek = 1/2 mass x u 2 u 2 is the root-mean-square speed

urms =

√3RT

R = 8.314Joule/mol*K

M

Graham’s Law of Effusion The rate of effusion of a gas is inversely related to the square root of its molar mass. rate of effusion

1

√M

Recall that

KE = ½ mv2

If at some temperature a heavy particle’s KE = a light particle’s KE Then the heavy gas must be moving slower than the lighter gas.

Then KEA = KEB (1/2 mv2)A = (1/2 mv2)B or (mA/mB)1/2 = vB/vA Where m = molar mass

Relative number of molecules with a given speed

Relationship Between Molar Mass and Molecular Speed

H2 (2)

Molecular speed at a given T

Relative number of molecules with a given speed

He (4)

H2 (2)

Molecular speed at a given T

Relative number of molecules with a given speed

H2O (18) He (4)

H2 (2)

Molecular speed at a given T

Relative number of molecules with a given speed

N2 (28) H2O (18) He (4)

H2 (2)

Molecular speed at a given T

Relative number of molecules with a given speed

O2 (32) N2 (28) H2O (18) He (4)

H2 (2)

Molecular speed at a given T

Effusion: Gaseous particles passing through a tiny hole into an evacuated space. Rateeffusion is proportional to 1/(MW)1/2 Thus . . . ,

RateA = RateB =

(MWB)1/2 (MWA)1/2

Do Problem # 28 of Exercise 8. What does the ratio mean? Do Problem # 29 of Exercise 8.

Diffusion: Gaseous particles passing through another gaseous particles. Ratediffusion is proportional to 1/(MW)1/2

Thus . . . ,

RateA = RateB =

(MWB)1/2 (MWA)1/2

Real Gases • At high temperatures and low pressures most gases behave “ideally”. • Consider gases at high pressure.

The effect of molecular volume on measured gas volume.

The effect of intermolecular attractions on measured gas pressure.

(Real) Non Ideal At high pressures we can no longer say the attractions between gases are negligible. We can no longer say the volume of the gaseous particles themselves is negligible. Thus our ideal gas law equation, PV = nRT has to have some correction factors, a and b. ( P + n2/ V)( V + nb ) = nRT

The behavior of several real gases with increasing external pressure.

2.0

1.5

PV 1.0 RT 0.5

0.0 0

200

400

600

Pext (atm)

800

1000

PV 1.0 RT

0

2.0

10 20 Pext (atm)

1.5

PV 1.0 RT 0.5

0.0 0

200

400

600

Pext (atm)

800

1000

PV 1.0 RT

Ideal gas

0

2.0

10 20 Pext (atm)

1.5

PV 1.0 RT

Ideal gas

0.5

0.0 0

200

400

600

Pext (atm)

800

1000

H2 He Ideal gas

PV 1.0 RT

0

2.0

10 20 Pext (atm)

H2 He

1.5

PV/RT > 1 Effect of molecular volume predominates

PV 1.0 RT 0.5

0.0 0

200

400

600

Pext (atm)

800

1000

Ideal gas

H2 He Ideal gas

PV 1.0 RT

CH4 CO2 0

2.0

CH4

10 20 Pext (atm)

H2 CO2 He

1.5

PV/RT > 1 Effect of molecular volume predominates

PV 1.0 RT

PV/RT < 1 Effect of intermolecular attractions predominates

0.5

0.0 0

200

400

600

Pext (atm)

800

1000

Ideal gas

Table 5.5 Van der Waals Constants for Some Common Gases Van der Waals equation for n moles of a real gas

(P

He Ne Ar Kr Xe H2 N2 O2 Cl2 CO2 CH4 NH3 H2O

nb)

nRT

adjusts P up adjusts V down

a Gas

n2a )(V 2 V atm*L2 mol2

0.034 0.211 1.35 2.32 4.19 0.244 1.39 1.36 6.49 3.59 2.25 4.17 5.46

b

L mol

0.0237 0.0171 0.0322 0.0398 0.0511 0.0266 0.0391 0.0318 0.0562 0.0427 0.0428 0.0371 0.0305