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Chapter 5

Gases and the Kinetic-Molecular Theory

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Gases and the Kinetic Molecular Theory

5.1 An Overview of the Physical States of Matter

5.2 Gas Pressure and Its Measurement

5.3 The Gas Laws and Their Experimental Foundations

5.4 Further Applications of the Ideal Gas Law

5.5 The Ideal Gas Law and Reaction Stoichiometry

5.6 The Kinetic-Molecular Theory: A Model for Gas Behavior

5.7 Real Gases: Deviations from Ideal Behavior

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Table 5.1 Some Important Industrial Gases

Methane (CH4)

Ammonia (NH3)

Chlorine (Cl2)

Oxygen (O2)

Ethylene (C2H4)

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

Name (Formula) Origin and Use

Atmosphere-Biosphere Redox Interconnections

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An Overview of the Physical States of Matter

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.

4. Most gases have relatively low densities under normal conditions.

5. Gases are miscible.

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Figure 5.1 The three states of matter.

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Figure 5.2 Effect of atmospheric pressure on objects at the Earth’s surface.

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Definition of Pressure

•Gas molecules in constant motion.

•Collide with each other.

•Collide with container walls → exert a force on container walls.

•Pressure = force per unit area:

AFP =Nm–2

N ≡ kg ms–2

m2

Pa

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Figure 5.3 A mercury barometer.

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Figure 5.4

Two types of manometer

closed-end

open-end

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Table 5.2 Common Units of Pressure

Atmospheric PressureUnit Scientific Field

chemistryatmosphere(atm) 1 atm*

pascal(Pa); kilopascal(kPa)

1.01325x105Pa; 101.325 kPa

SI unit; physics, chemistry

millimeters of mercury(Hg)

760 mm Hg* chemistry, medicine, biology

torr 760 torr* chemistry

pounds per square inch (psi or lb/in2)

14.7lb/in2 engineering

bar 1.01325 bar meteorology, chemistry, physics

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

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Sample Problem 5.1 Converting Units of Pressure

PROBLEM: On a certain day, the barometer in a laboratory indicates that the atmospheric pressure is 764.7 torr. A sample of gas is placed in a vessel attached to an open-end mercury manometer. A meter stick is used to measure the height of the mercury above the bottom of the manometer. The level of mercury in the open-end arm of the manometer has a measured height of 136.4 mm and that in the arm that is in contact with the gas has a height of 103.8 mm.

What is the pressure of the gas (a) in atmospheres; (b) in kPa?

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Figure 5.5 The relationship between the volume and pressure of a gas.

Boyle’s Law

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Sample Problem 5.2 Applying the Volume-Pressure Relationship

PROBLEM: A gas cylinder of volume 50.0 L is pressurized at 21.5 atm. When a glass vessel is hooked up to the cylinder and the valve opened, the pressure gauge reads 1.55 atm. What is the volume of the glass vessel (neglect the volume of any tube attaching the cylinder to the vessel)

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Figure 5.6

The relationship between the volume and temperature of a

gas.

Charles’s Law

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Boyle’s Law n and T are fixedV α1

P

Charles’s Law V α T P and n are fixed

V

T= constant V = constant x T

Amontons’s Law P α T V and n are fixed

P

T= constant P = constant x T

combined gas law V αT

PV = constant x

T

P

PV

T= constant

V x P = constant V = constant / P

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Figure 5.7 An experiment to study the relationship between the volume and amount of a gas.

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Figure 5.8 Standard molar volume.

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Figure 5.9 The volume of 1 mol of an ideal gas compared with some familiar objects.

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THE IDEAL GAS LAW

PV = nRT

IDEAL GAS LAW

nRT

PPV = nRT or V =

Boyle’s Law

V =constant

P

R = PVnT

= 1atm x 22.414L1mol x 273.15K

= 0.0821atm*L

mol*K

V = V =

Charles’s Law

constant X T

Avogadro’s Law

constant X n

fixed n and T fixed n and P fixed P and T

Figure 5.10

R is the universal gas constant

3 significant figures

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Sample Problem 5.3 Using the Ideal Gas Equation Properly

PROBLEM: Calcium carbonate, CaCO3(s), decomposes upon heating to give CaO(s) and CO2(g). A sample of CaCO3 is decomposed, and the carbon dioxide is collected in a 250-mL flask. After the decomposition is complete, the gas has a pressure of 1.3 atm at a temperature of 31 ºC. How many moles of CO2 gas were generated?

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Sample Problem 5.4 Applying the Temperature-Pressure Relationship

PROBLEM: The gas pressure in an aerosol can is 1.5 atm at 25 ºC. Assuming that the gas inside obeys the ideal-gas equation, what would the pressure be if the can were heated to 450 ºC?

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Sample Problem 5.5 Multiple Physical Changes

PROBLEM: An inflated balloon has a volume of 6.0 L at sea level (1.0 atm) and is allowed to ascend in altitude until the pressure is 0.45 atm. During the ascent the temperature of the gas falls from 22 ºC to –21 ºC. Calculate the volume of the balloon at its final altitude.

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Sample Problem 5.6 Using Gas Laws to Determine a Balanced Equation

PROBLEM: The cylinders below depict a gaseous reaction carried out at constant temperature and volume. All reactants and products behave as ideal gases. Before the reaction, the pressure is 90 kPa; when it is complete, the pressure is 60 kPa.

Which of the following balanced equations describes the reaction?

(1) A2 + B2 2AB (2) 2AB + B2 2AB2

(4) 2AB2 A2 + 2B2(3) A + B2 AB2

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density = m/V

n = m/M

The Density of a Gas

PV = nRT PV = (m/M)RT

m/V = M x P/ RT

•The density of a gas is directly proportional to its molar mass.

•The density of a gas is inversely proportional to the temperature.

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Sample Problem 5.7 Calculating Gas Density

PROBLEM: What is the density of carbon tetrachloride vapor at 714 torr and 125 ºC?

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The Molar Mass of a Gas

n =mass

M=

PV

RT

M =

M = d RT

P

m RT

VPd =

m

V

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Sample Problem 5.8 Finding the Molar Mass of a Gas

PROBLEM: A series of measurements are made in order to determine the molar mass of an unknown gas:

(1) A large flask is evacuated and found to weigh 134.567 g.

(2) It is then filled with the gas to a pressure of 735 torr at 31 ºC and reweighed – its mass is now 137.456 g.

(3) Then the flask is filled with water at 31 ºC and found to weigh 1067.9 g (density of water at this temperature = 0.997 g/cm3)

Assuming that the ideal gas equation applies, calculate the molar mass of the unknown gas.

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Figure 5.11

Determining the molar mass of an unknown

volatile liquid.

based on the method of J.B.A. Dumas (1800-1884)

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Dalton’s Law of Partial Pressures

Ptotal = P1 + P2 + P3 + ...

P1= χ1 x Ptotal where χ1 is the mole fraction

χ1 = n1

n1 + n2 + n3 +...=

n1

ntotal

Mixtures of Gases•Gases mix homogeneously in any proportions.

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

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Sample Problem 5.9 Applying Dalton’s Law of Partial Pressures

PROBLEM: A gaseous mixture made from 6.00 g O2 and 9.00 g CH4 is placed in a 15.0-L vessel at 0 ºC. What is the partial pressure of each gas, and what is the total pressure of the vessel?

Sample Problem 5.10 Applying Dalton’s Law of Partial Pressures

PROBLEM: A study of the effects of certain gases on plant growth requires a synthetic atmosphere composed of 1.5 mol % CO2, 18.0 mol % O2, and 80.5 mol % Ar.

(a) Calculate the partial pressure of O2 in the mixture if the total pressure of the atmosphere is to be 745 torr.

(b) If this atmosphere is to be held in a 120-L space at 295 K, how many moles of O2 are needed?

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P,V,T

of gas A

amount (mol)

of gas A

amount (mol)

of gas B

P,V,T

of gas B

ideal gas law

ideal gas law

molar ratio from balanced equation

Figure 15.13

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).

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Sample Problem 5.11 Using Gas Variables to Find Amount of Reactants and Products

PROBLEM: The safety airbags in automobiles are inflated by nitrogen gas generated by the rapid decomposition of sodium azide, NaN3:

2NaN3(s) → 2Na(s) + 3N2(g)

If an airbag has a volume of 36 L and is to be filled with nitrogen gas at a pressure of 1.15 atm at a temperature of 26.0 ºC, how many grams of NaN3 must be decomposed?

L of H2

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Table 5.3 Vapor Pressure of Water (P ) at Different TH2O

T(0C) P (torr) T(0C) P (torr)05

10111213141516182022242628

3035404550556065707580859095

100

31.842.255.371.992.5

118.0149.4187.5233.7289.1355.1433.6525.8633.9760.0

4.66.59.29.8

10.511.212.012.813.615.517.519.822.425.228.3

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Figure 5.12 Collecting a water-insoluble gaseous reaction product and determining its pressure.

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Sample Problem 5.12 Calculating the Amount of Gas Collected Over Water

PROBLEM: A sample of KClO3 is partially decomposed producing O2 gas that is collected over water:

2KClO3(s) → 2KCl(s) + 3O2(g)

The volume of gas collected is 0.250 L at 26 ºC and 765 torrtotal pressure.

(a) How many moles of O2 are collected?

(b) How many grams of KClO3 were decomposed?

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Postulates of the Kinetic-Molecular Theory

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.

Gas particles are in constant, random, straight-line motion except when they collide with each other or with the container walls.

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

Postulate 1: Particle Volume

Postulate 2: Particle Motion

Postulate 3: Particle Collisions

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Figure 5.14 Distribution of molecular speeds at three temperatures.

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Figure 5.15 A molecular description of Boyle’s Law.

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Figure 5.16 A molecular description of Dalton’s law of partial pressures.

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Figure 5.17 A molecular description of Charles’s Law.

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Avogadro’s Law V α n

Ek = 1/2 mass x speed2 Ek = 1/2 mass x u 2

u 2 is the root-mean-square speed

urms = √3RT

MR = 8.314 J/mol*K

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

rate of effusion α1

√M

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Figure 5.18 A molecular description of Avogadro’s Law.

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Figure 5.19 Relationship between molar mass and molecular speed.

Ek = 3/2 (R/NA) T

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Sample Problem 5.13 Applying Graham’s Law of Effusion

PROBLEM: Calculate the ratio of the effusion rates of N2 and O2.

Sample Problem 5.14 Applying Graham’s Law of Effusion

PROBLEM: An unknown gas composed of homonuclear diatomic molecules effuses at a rate that is only 0.355 times that of O2 at the same temperature. What is the identity of the unknown gas?

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Figure 5.20 Diffusion of a gas particle through a space filled with other particles.

distribution of molecular speeds

mean free path

collision frequency

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Table 5.4 Molar Volume of Some Common Gases at STP (00C and 1 atm)

GasMolar Volume

(L/mol)Condensation Point

(0C)

HeH2NeIdeal gasArN2O2COCl2NH3

22.43522.43222.42222.41422.39722.39622.39022.38822.18422.079

-268.9-252.8-246.1----185.9-195.8-183.0-191.5

-34.0-33.4

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Figure 5.21

The behavior of several real gases with increasing external pressure.

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Figure 5.22 The effect of intermolecular attractions on measured gas pressure.

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Figure 5.23 The effect of molecular volume on measured gas volume.

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Table 5.5 Van der Waals Constants for Some Common Gases

0.0340.2111.352.324.190.2441.391.366.493.592.254.175.46

HeNeArKrXeH2N2O2Cl2CO2CH4NH3H2O

0.02370.01710.03220.03980.05110.02660.03910.03180.05620.04270.04280.03710.0305

Gasa

atm*L2

mol2b

L

mol

(P +n2aV 2 )(V − nb) = nRTVan der Waals

equation for nmoles of a real gas adjusts P up adjusts V down