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Chapter 5Chapter 5

The Gas LawsThe Gas Laws

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Gas molecules fill containerGas molecules fill container Molecules move around and hit sides.Molecules move around and hit sides. Collisions are force.Collisions are force. Container has area.Container has area. Measured with a barometer.Measured with a barometer.

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First Barometer Invented by TorricelliFirst Barometer Invented by Torricelli

The pressure of the The pressure of the atmosphere at atmosphere at sea sea level level will hold a will hold a column of mercury column of mercury 760 mm Hg.760 mm Hg.

1 atm = 760 mm Hg1 atm = 760 mm Hg

1 atm Pressure

760 mm Hg

Vacuum

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ManometerManometer

Gas

h

Greater Greater Atmospheric Atmospheric PressurePressure

PPgasgas = P = Patmatm - h - h

Gas Pressure Less Than Atmospheric Pressure

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ManometerManometer

Higher Gas PressureHigher Gas Pressure

PPgasgas = P = Patmatm + h + hh

Gas

Gas Pressure Greater Than Atmospheric Pressure

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Units of pressureUnits of pressure 1 atmosphere = 760 mm Hg = 760 torr = 1 atmosphere = 760 mm Hg = 760 torr =

101,325 Pascals = 101.325 kPa101,325 Pascals = 101.325 kPa Can make conversion factors from these.Can make conversion factors from these.

1 Atm760 mmHg101,325

1 Atm101.325 kPa

760 torr

Pascals

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What is 724 mm Hg in kPa?What is 724 mm Hg in kPa?

in torr?in torr?

in atm?in atm?

1724 mm Hg 1 atm 9.53 10 atm1 760 mmHg

x x

2724 mm Hg 760 torr torr1 760 mmHg

7.24 10x x

1724 mm Hg 101.325 kPa 9.65 10 kPa1 760 mmHg

x x

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The BoyleThe Boyle’’s Gas Lawss Gas Laws Pressure and volume are inversely Pressure and volume are inversely

related at constant temperature.related at constant temperature. PV= kPV= k As one goes up, the other goes down.As one goes up, the other goes down. PP11VV11 = P = P22 V V22 Ideal Gas is one that strictly obeys Ideal Gas is one that strictly obeys

BoyleBoyle’’s Laws Law What does it look like graphically?What does it look like graphically?

9 Pressure

Pressure vs. Volume @ Constant Temperature

10 1/Pressure

Slope = k

1/Pressure vs. Volume @ Constant Temperature

11 Pressure

CO2

O2

22.4

L a

tmNe

Ideal Gas

PV vs. P For Several Gases Below 1 Atm

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ExamplesExamples20.5 L of nitrogen at 25ºC and 742 torr is 20.5 L of nitrogen at 25ºC and 742 torr is

compressed to 9.8 atm at constant compressed to 9.8 atm at constant Temperature.Temperature.

What is the new volume?What is the new volume?Need to be in same unitsNeed to be in same units

Your turn #33 p. 219Your turn #33 p. 219

1

12

2

742 torr 1 atm 9.76 10 atm1 760 torr

9.76 10 atm x 20.5 L = 9.8 atm x V2.04 LV

x x

x

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CharleCharle’’s Laws Law Volume of a gas varies directly with Volume of a gas varies directly with

the absolute temperature at constant the absolute temperature at constant pressure.pressure.

V = kT V = kT T must be in KelvinT must be in Kelvin

GraphicallyGraphically

1 21 2

V VT T

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V (L

)

T (ºC)

He

H 2O

CH 4

H2

-273.15ºC

Volume vs. Temperature @ Constant Pressure

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ExamplesExamples What would the final volume be if 247 mL of What would the final volume be if 247 mL of

gas at 22ºC is heated to 98ºC , if the gas at 22ºC is heated to 98ºC , if the pressure is held constant?pressure is held constant?

Change Change ooC to KC to K

VV22 = 3.10 x 10 = 3.10 x 1022 mL mL

Your Turn - #34 p. 219

2

C + 273.15=295.15K; 98 C + 273.15=371.15K247 mL295.15 371.15

22o o

VK K

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Avogadro's LawAvogadro's Law At constant temperature and pressure,

the volume of gas is directly related to the number of moles.

(n = number of moles)

Try #35 p. 220

1 2

1 2

V Vn n

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Gay- Lussac LawGay- Lussac Law At constant volume, pressure and absolute At constant volume, pressure and absolute

temperature are directly related.temperature are directly related.

What would be the final pressure of a gas at What would be the final pressure of a gas at

243 kPa and 355K if it was heated 15K 243 kPa and 355K if it was heated 15K more?more? 2243 kPa 2.53 10 kPa

355 K 370 kx x

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Combined Gas LawCombined Gas Law If the moles of gas remains constant, If the moles of gas remains constant,

use this formula and cancel out the use this formula and cancel out the other things that donother things that don’’t change.t change.

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Ideal Gas LawIdeal Gas Law PV = nRTPV = nRT

» V in L, V in L, » P in atm, P in atm, » T in KT in K

R is the universal gas constant, there are R is the universal gas constant, there are 4 different numbers depending on units.4 different numbers depending on units.

Watch the units when setting up Ideal Gas Watch the units when setting up Ideal Gas Law – they all must matchLaw – they all must match

.0.08206 L atmRK mol

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Ideal Gas LawIdeal Gas Law An An equation of stateequation of state.. Where the state of the gas is its condition Where the state of the gas is its condition

at the at the given timegiven time described by pressure, described by pressure, volume, temperature and number of volume, temperature and number of moles. moles.

Given 3 values in equation, you can Given 3 values in equation, you can determine the fourth.determine the fourth.

An Ideal Gas is an hypothetical gas whose An Ideal Gas is an hypothetical gas whose behavior is described by the equationbehavior is described by the equation

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Ideal Gas LawIdeal Gas Law Think of it as a limit.Think of it as a limit. Gases only approach ideal behavior at:Gases only approach ideal behavior at:

» low pressure (< 1 atm)low pressure (< 1 atm) & &»high temperaturehigh temperature

Use the laws anyway, unless told to do Use the laws anyway, unless told to do otherwise.otherwise.

They give good estimates.They give good estimates.

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Practice Using Formula p187-Practice Using Formula p187-190190 Ideal Gas Law I – Calc. # mol Ex. 5.6 p.187

Solve for n

Be careful, make sure all units match Ideal Gas Law II – Calc. final press. Ex.5.7 p.

188 Through algebraic manipulation, the problem

becomes a simple Boyles problem.

.0.08206PV L atmn RRT K mol

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Practice Using Formula p187-Practice Using Formula p187-190190Ideal Gas Law III – Calc. new vol. Ex. 5.8 p. 188

Through algebraic manipulation, the problem becomes a simple Charles Law problem.

Ideal Gas Law IV – Calc. vol. Ex. 5.9 p.189Through algebraic manipulation, the problem becomes a simple Combined Gas Law problem.

Try a couple: p.220 #44, & 46

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Gases StoichiometryGases Stoichiometry Reactions happen in molesReactions happen in moles At Standard Temperature and At Standard Temperature and

Pressure (STP, 0ºC and 1 atm) 1 Pressure (STP, 0ºC and 1 atm) 1 mole of gas occuppies 22.42 L.mole of gas occuppies 22.42 L.

If not at STP, use the ideal gas law to If not at STP, use the ideal gas law to calculate moles of reactant or calculate moles of reactant or volume of product.volume of product.

What is the volume of an ideal gas at STP? Show complete setup.

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#53 p. 221

Try #54

2 2

22 3 3

2 33

1.00 atm x 70.0 L 3.12 mol N0.08206 L atm 273 K k

3.12 mol N 2 mol NaN 65.02 g NaN 1.35 10 g 1 3 mol N 1 mol NaN

NaN

NPVRT x

mol

x x x

n

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Gas Density and Molar MassGas Density and Molar Massmass of gasmolar mass molar mass

mn

Substitute above into ideal gas equations gives:

molar mass x molar mass

m xRTn RTmTV V

RV

P

mV

D Because: we can substitute D for m/V giving new equation

massRT

molarDP OR mass RD T

Pmolar

Now if know density of gas, you can find molar mass

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Think Through #61 p. 221Think Through #61 p. 221Know the formula

Know the density = 3.164g/LKnow R = 0.08206Know STP: Temperature = 273K, Pressure = 1 atmIt’s a “Plug and Chug”

mass RD TP

molar

10.08206 / 7.03.164 / 2 983 /7 101.0

mass Latm molKxg L K x g molat

xm

molar

The gas is diatomic so 7.098x10-1 / 2 = 35.47 g which is the atomic mass of Cl so the gas is Cl2

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DaltonDalton’’s Laws Law The total pressure in a container is The total pressure in a container is

the sum of the pressure each gas the sum of the pressure each gas would exert if it were alone in the would exert if it were alone in the container.container.

PPTotalTotal = P = P11 + P + P22 + P + P33 + P + P44 + P + P55 ... ...

For each P = nRT/VFor each P = nRT/V

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Dalton's LawDalton's Law PPTotalTotal = n = n11RT + nRT + n22RT + nRT + n33RT +...RT +...

V V V V V V In the same container R, T and V are In the same container R, T and V are

the same.the same.

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The mole fractionThe mole fraction Ratio of moles of the substance to Ratio of moles of the substance to

the total moles.the total moles. Read the derivation of the mole Read the derivation of the mole

fraction on p. 195fraction on p. 195

symbol is Greek letter chi symbol is Greek letter chi χχ

1 11

total total

n Pxn P

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ExamplesExamples

3.50 L O2

1.50 L N2

2.70 atm When these valves are opened, what is each When these valves are opened, what is each

partial pressure and the total pressure?partial pressure and the total pressure?

4.00 L CH4

4.58 atm 0.752 atm

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Vapor PressureVapor Pressure Water evaporates!Water evaporates! When that water evaporates, the vapor has When that water evaporates, the vapor has

a pressure.a pressure. Gases are often collected over water so the Gases are often collected over water so the

vapor pressure of water must be subtracted vapor pressure of water must be subtracted from the total pressure.from the total pressure.

It must be given.It must be given. See #71-73 p. 222See #71-73 p. 222

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Kinetic Molecular TheoryKinetic Molecular Theory Explains why ideal gases behave the way Explains why ideal gases behave the way

they do.they do. PostulatesPostulates simplify the theory, but don simplify the theory, but don’’t t

work in real gases.work in real gases. The particles are so small we can ignore The particles are so small we can ignore

their volume.their volume. The particles are in constant motion and The particles are in constant motion and

their collisions cause pressure. their collisions cause pressure.

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Kinetic Molecular TheoryKinetic Molecular Theory

The particles do not exert force on each The particles do not exert force on each other, neither attracting or repelling.other, neither attracting or repelling.

The average kinetic energy is proportional The average kinetic energy is proportional to the Kelvin temperature.to the Kelvin temperature.

Appendix 2 p. A13 shows the derivation of Appendix 2 p. A13 shows the derivation of the ideal gas law.the ideal gas law.

Kelvin temperature is the result of random Kelvin temperature is the result of random motions of particles.motions of particles.

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EffusionEffusion Passage of gas through a small hole, into a vacuum.Passage of gas through a small hole, into a vacuum. The effusion rate measures how fast this happens.The effusion rate measures how fast this happens. GrahamGraham’’s Law the rate of effusion is inversely proportional s Law the rate of effusion is inversely proportional

to the square root of the mass of its particles.to the square root of the mass of its particles.

Rate of effusion for gas 1Rate of effusion for gas 2

MM

2

1

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DiffusionDiffusion The spreading of a gas through a The spreading of a gas through a

room.room. Slow considering molecules move at Slow considering molecules move at

100100’’s of meters per second.s of meters per second. Collisions with other molecules slow Collisions with other molecules slow

down diffusions.down diffusions.

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Real GasesReal Gases Real molecules do take up space and Real molecules do take up space and

do interact with each other do interact with each other (especially polar molecules).(especially polar molecules).

Need to add correction factors to the Need to add correction factors to the ideal gas law to account for these.ideal gas law to account for these.

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Pressure correctionPressure correction Because the molecules are attracted Because the molecules are attracted

to each other, the pressure on the to each other, the pressure on the container will be less than idealcontainer will be less than ideal

depends on the number of molecules depends on the number of molecules per liter.per liter.

since two molecules interact, the since two molecules interact, the effect must be squared.effect must be squared.

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Deriving the Ideal Gas LawDeriving the Ideal Gas LawAKA Physicists Having FunAKA Physicists Having Fun

212 23

AnN muP

V

P = Pressuren = molesNA = Avogadro’s numberm = mass of particleu2 = average velocityV = Volume of container

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Volume CorrectionVolume Correction The actual volume free to move is less The actual volume free to move is less

because of particle size.because of particle size. More molecules will have more effect.More molecules will have more effect. Corrected volume VCorrected volume V’’ = V - nb = V - nb b is a constant that differs for each gas.b is a constant that differs for each gas. PP’’ = nRT = nRT

(V-nb) (V-nb)

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Pressure correctionPressure correction Because the molecules are attracted Because the molecules are attracted

to each other, the pressure on the to each other, the pressure on the container will be less than idealcontainer will be less than ideal

depends on the number of molecules depends on the number of molecules per liter.per liter.

since two molecules interact, the since two molecules interact, the effect must be squared.effect must be squared.Pobserved = P’ - a

2

( )Vn

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AltogetherAltogether PPobsobs= nRT - a n = nRT - a n 22

V-nb VV-nb V Called the Van der WallCalled the Van der Wall’’s equation if s equation if

rearrangedrearranged

Corrected Corrected Corrected Corrected Pressure Pressure Volume Volume

( )

P + a nV

x V - nb nRTobs

2

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Where does it come fromWhere does it come from a and b are determined by a and b are determined by

experiment.experiment. Different for each gas.Different for each gas. Bigger molecules have larger b.Bigger molecules have larger b. a depends on both size and polarity.a depends on both size and polarity. once given, plug and chug.once given, plug and chug.

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ExampleExample Calculate the pressure exerted by Calculate the pressure exerted by

0.5000 mol Cl0.5000 mol Cl22 in a 1.000 L container in a 1.000 L container at 25.0ºCat 25.0ºC

Using the ideal gas law.Using the ideal gas law. Van der WaalVan der Waal’’s equations equation

– a = 6.49 atm La = 6.49 atm L22 /mol /mol22 – b = 0.0562 L/molb = 0.0562 L/mol