gases. the properties of gases only 4 quantities are needed to define the state of a gas:

GASESGASES

The Properties of GasesThe Properties of Gases

Only 4 quantities are needed to Only 4 quantities are needed to

define the define the state of a gasstate of a gas: :

a) the a) the quantityquantity of the gas, n (in moles) of the gas, n (in moles)

b) the b) the temperaturetemperature of the gas, T (in of the gas, T (in

KELVIN) KELVIN)

c) the c) the volumevolume of the gas, V (in liters) of the gas, V (in liters)

d) the d) the pressurepressure of the gas, P (in of the gas, P (in

atmospheres)atmospheres)

A gas uniformly fills any container, A gas uniformly fills any container,

is easily compressed & mixes is easily compressed & mixes

completely with any other gas.completely with any other gas.

Gas PressureGas Pressure

A measure of the force that a gas A measure of the force that a gas

exerts on its container. exerts on its container.

Force is the physical quantity that Force is the physical quantity that

interferes with inertia. Gravity is interferes with inertia. Gravity is the the

force responsible for weight.force responsible for weight.

Newton’s 2nd LawNewton’s 2nd Law

Force = mass x acceleration Force = mass x acceleration

N = kg x m/sN = kg x m/s22

PressurePressure

Force / unit areaForce / unit area

N / mN / m22

Barometer -Barometer -

invented by invented by

Evangelista Torricelli Evangelista Torricelli

in 1643; uses the in 1643; uses the

height of a column height of a column

of mercury to of mercury to

measure gas measure gas

pressure (especially pressure (especially

atmospheric)atmospheric)

1 mm of Hg = 1 torr1 mm of Hg = 1 torr

760.00 mm Hg 760.00 mm Hg

= =

760.00 torr 760.00 torr

==

1.00 atm 1.00 atm

= =

101.325 kPa ≈ 10101.325 kPa ≈ 1055 Pa Pa

At sea level, all of the previous At sea level, all of the previous define STANDARD PRESSURE. define STANDARD PRESSURE.

The SI unit of pressure is the The SI unit of pressure is the Pascal Pascal

(Blaise Pascal).(Blaise Pascal).

1 Pa = 1 N / m1 Pa = 1 N / m22

The ManometerThe Manometer

a device for measuring the pressure a device for measuring the pressure

of a gas in a container of a gas in a container

The pressure of the gas is given by h The pressure of the gas is given by h

[the difference in mercury levels] in [the difference in mercury levels] in

units of torr (equivalent to mm Hg).units of torr (equivalent to mm Hg).

a) Gas pressure = atmospheric pressure a) Gas pressure = atmospheric pressure – h– h

b) Gas pressure = atmospheric pressure b) Gas pressure = atmospheric pressure + h+ h

Exercise 1Exercise 1 Pressure Conversions Pressure Conversions

The pressure of a gas is The pressure of a gas is measured measured

as 49 torr. as 49 torr.

Represent this pressure in both Represent this pressure in both

atmospheres and pascals.atmospheres and pascals.

SolutionSolution

6.4 x 106.4 x 10-2-2 atm atm

6.5 x 106.5 x 1033 Pa Pa

ExerciseExercise

Rank the following pressures in Rank the following pressures in

decreasing order of magnitude decreasing order of magnitude

(largest first, smallest last) (largest first, smallest last)

75 kPa 300. torr 75 kPa 300. torr

0.60 atm 350. mm Hg.0.60 atm 350. mm Hg.

GAS LAWSGAS LAWS

THE EXPERIMENTAL BASISTHE EXPERIMENTAL BASIS

BOYLE’S LAW – BOYLE’S LAW – “father of chemistry”“father of chemistry”

The volume of a confined gas is The volume of a confined gas is inversely inversely

proportional to the pressure exerted proportional to the pressure exerted on on

the gas. the gas.

ALL GASES BEHAVE IN THIS MANNER!ALL GASES BEHAVE IN THIS MANNER!

Robert Boyle was Robert Boyle was

an Irish chemist. an Irish chemist.

He studied P V He studied P V

relationships relationships using using

a J-tube set up in a J-tube set up in

the multi-story the multi-story

entryway of his entryway of his

home.home.

P:1/VP:1/Vplot = straight lineplot = straight line

Pressure and volume are Pressure and volume are inverselyinversely

proportional.proportional.

Volume ↑ pressure Volume ↑ pressure (at constant temperature)(at constant temperature)

The converse is also true.The converse is also true.

For a given quantity of a gas at For a given quantity of a gas at

constant temperature, the constant temperature, the product product

of pressure and volume is a of pressure and volume is a

constant. constant.

PV = kPV = k

Therefore, Therefore,

which is the equation for a straight which is the equation for a straight

line of the type y = mx + b line of the type y = mx + b

where m = slope, and b is the where m = slope, and b is the

y-intercept.y-intercept.

Pk

P

kV

1

In this case, In this case,

y = V, x = 1/P y = V, x = 1/P

and b = 0. and b = 0.

Check out the plot Check out the plot

on the left (b). on the left (b).

The data Boyle The data Boyle

collected is collected is graphed graphed

on (a) above.on (a) above.

PP11VV11 = P = P22VV22

is the easiest form of Boyle’s law tois the easiest form of Boyle’s law to

MEMORIZE !MEMORIZE !

Boyle’s Law has been tested for Boyle’s Law has been tested for over three centuries. It holds true over three centuries. It holds true only at low pressuresonly at low pressures. .

A plot of PV A plot of PV

versus P for versus P for

several gases at several gases at

pressures below pressures below

1 atm is pictured 1 atm is pictured to the left.to the left.

An An idealideal gas is gas is

expected to expected to

have a constant have a constant

value of PV, as value of PV, as

shown by the shown by the

dotted line.dotted line.

COCO22 shows the shows the

largest change largest change in in

PV, and this PV, and this

change is change is

actually quite actually quite

small.small.

PV changes PV changes

from about from about

22.39 L·atm 22.39 L·atm at at

0.25 atm to 0.25 atm to

22.26 L·atm 22.26 L·atm at at

1.00 atm.1.00 atm.

Thus, Boyle’s Law is a good Thus, Boyle’s Law is a good

approximation at these approximation at these relatively relatively

low pressures.low pressures.

Exercise 2Exercise 2 Boyle’s Boyle’s Law ILaw I

Sulfur dioxide (SOSulfur dioxide (SO22), a gas that ), a gas that

plays a central role in the plays a central role in the formation formation

of acid rain, is found in the of acid rain, is found in the exhaust exhaust

of automobiles and power of automobiles and power plants.plants.

Consider a 1.53- L sample of Consider a 1.53- L sample of

gaseous SOgaseous SO22 at a pressure of 5.6 x at a pressure of 5.6 x

101033 Pa. Pa.

If the pressure is changed to 1.5 x If the pressure is changed to 1.5 x

101044 Pa at a constant temperature, Pa at a constant temperature,

what will be the new volume of the what will be the new volume of the

gas ?gas ?

SolutionSolution

0.57 L0.57 L

Exercise 3Exercise 3 Boyle’s Boyle’s Law IILaw II

In a study to see how closely In a study to see how closely

gaseous ammonia obeys Boyle’s gaseous ammonia obeys Boyle’s

law, several volume law, several volume measurements measurements

were made at various pressures, were made at various pressures,

using 1.0 mol NHusing 1.0 mol NH33 gas at a gas at a

temperature of 0º C.temperature of 0º C.

Using the results listed below, calculate Using the results listed below, calculate the the

Boyle’s law constant for NHBoyle’s law constant for NH33 at the various at the various

pressures.pressures.

Experiment Experiment Pressure (atm) Pressure (atm) Volume (L)Volume (L)

11 0.1300 0.1300 172.1 172.1

22 0.2500 0.2500 89.28 89.28

33 0.3000 0.3000 74.35 74.35

44 0.5000 0.5000 44.49 44.49

55 0.7500 0.7500 29.55 29.55

66 1.000 1.000 22.08 22.08

SolutionsSolutions

experiment 1 is 22.37experiment 1 is 22.37

experiment 2 is 22.32experiment 2 is 22.32

experiment 3 is 22.31experiment 3 is 22.31

experiment 4 is 22.25experiment 4 is 22.25

experiment 5 is 22.16experiment 5 is 22.16

experiment 6 is 22.08experiment 6 is 22.08

PLOT the values of PLOT the values of

PV for the previous PV for the previous

six experiments. six experiments.

Extrapolate it back Extrapolate it back

to see what PV to see what PV

equals at 0.00 atm equals at 0.00 atm

pressure.pressure.

Compare it to Compare it to the PV vs. P the PV vs. P graph at the graph at the right.right.

What is the y-intercept for all What is the y-intercept for all of these gases?of these gases?

Remember, gases behave most Remember, gases behave most

ideally at low pressures. ideally at low pressures.

You can’t get a pressure lower You can’t get a pressure lower than than

0.00 atm!0.00 atm!

Charles’ LawCharles’ Law

If a given quantity of gas is held If a given quantity of gas is held at at

a constant pressure, then its a constant pressure, then its

volume is directly proportional to volume is directly proportional to

the absolute temperature. the absolute temperature.

Must use KELVIN !Must use KELVIN !

Jacques Charles was a French Jacques Charles was a French

physicist and the first person to physicist and the first person to fill a fill a

hot “air” balloon with hydrogen hot “air” balloon with hydrogen gas gas

and made the first solo balloon and made the first solo balloon

flight!flight!

V:T plot = straight lineV:T plot = straight line

VV11TT22 = V = V22TT11

Temperature ↑ Volume ↑ Temperature ↑ Volume ↑

at constant pressureat constant pressure

This figure shows the plots of V vs. T This figure shows the plots of V vs. T (Celcius) for several gases. (Celcius) for several gases.

The solid lines The solid lines represent represent experimental experimental measurements on measurements on gases. The gases. The

dashed dashed lines represent lines represent extrapolation of extrapolation of the data into the data into regions where regions where these gases these gases

would would become liquids or become liquids or solids.solids.

Note that the samples Note that the samples of of

the various gases the various gases

contain different contain different

numbers of moles.numbers of moles.

What is the What is the

temperature when temperature when

the volume the volume extrapolates to zeroextrapolates to zero??

Exercise 4Exercise 4 Charles’s Charles’s LawLaw

A sample of gas at 15º C and 1 A sample of gas at 15º C and 1 atm atm

has a volume of 2.58 L. has a volume of 2.58 L.

What volume will this gas What volume will this gas occupy at occupy at

38º C and 1 atm?38º C and 1 atm?

SolutionSolution

2.79 L2.79 L

Gay-Lussac’S Law Gay-Lussac’S Law of Combining of Combining VolumesVolumesVolumes of gases always Volumes of gases always

combine combine

with one another in the ratio of with one another in the ratio of

small whole numbers, as long as small whole numbers, as long as

volumes are measured at the volumes are measured at the same same

T and P.T and P.

PP11TT22 = P = P22TT11

Avogadro’s HypothesisAvogadro’s Hypothesis

Equal volumes of gases under Equal volumes of gases under the the

same conditions of temperature same conditions of temperature and and

pressure contain equal pressure contain equal numbers of numbers of

molecules.molecules.

Avogadro’s LawAvogadro’s Law

The volume of a gas, at a given The volume of a gas, at a given

temperature and pressure, is temperature and pressure, is

directly proportional to the quantity directly proportional to the quantity

of gas.of gas.

V:nV:n

n n ↑↑ Volume Volume ↑↑

at constant T & Pat constant T & P

Here’s an easy way to Here’s an easy way to MEMORIZE all this….MEMORIZE all this….

Start with the combined gas Start with the combined gas law: law:

PP11VV11TT22 = P = P22VV22TT11

Memorize it.Memorize it.

Next,Next,

put the guy’s names in put the guy’s names in alphabetical alphabetical

order. order.

Boyle’s uses the first 2 variables, Boyle’s uses the first 2 variables,

Charles’ the second 2 variables & Charles’ the second 2 variables &

Gay-Lussac’s the remaining Gay-Lussac’s the remaining

combination of variables. What combination of variables. What

ever doesn’t appear in the ever doesn’t appear in the formula, formula,

is being held CONSTANT!is being held CONSTANT!

These balloons These balloons

each hold 1.0 L of each hold 1.0 L of

gas at 25gas at 25°°C and 1 C and 1

atm. Each balloon atm. Each balloon

contains 0.041 mol contains 0.041 mol

of gas, or 2.5 x 10of gas, or 2.5 x 102222

molecules.molecules.

Exercise 5Exercise 5 Avogadro’s Law Avogadro’s Law

Suppose we have a 12.2-L sample Suppose we have a 12.2-L sample

containing 0.50 mol oxygen gas containing 0.50 mol oxygen gas

(O(O22) at a pressure of 1 atm and a ) at a pressure of 1 atm and a

temperature of 25º C. temperature of 25º C.

If all this OIf all this O22 were converted to were converted to

ozone (Oozone (O33) at the same ) at the same temperature temperature

and pressure, what would be and pressure, what would be the the

volume of the ozone ?volume of the ozone ?

SolutionSolution

8.1 L 8.1 L

The Ideal Gas LawThe Ideal Gas Law

Four quantities describe the Four quantities describe the state of state of

a gas: a gas:

pressure, volume, temperature, pressure, volume, temperature, and and

# of moles (quantity).# of moles (quantity).

Combine all 3 laws…Combine all 3 laws…

V : V : nT nT

PP

Replace the : with a constant, R, Replace the : with a constant, R, and and

you get:you get:

PV = nRTPV = nRT

The Ideal Gas Law!The Ideal Gas Law!It is an Equation of State.It is an Equation of State.

R = 0.8206 L R = 0.8206 L •• atm/mol atm/mol •• K K

Useful only at low pressures and Useful only at low pressures and high high

temperatures! Guaranteed points temperatures! Guaranteed points on on

the AP Exam!the AP Exam!

These next exercises can all be These next exercises can all be

solved with the ideal gas law. solved with the ideal gas law.

BUT, you can use another if you BUT, you can use another if you

like!like!

Exercise 6Exercise 6 Ideal Gas Law I Ideal Gas Law I

A sample of hydrogen gas (HA sample of hydrogen gas (H22) has ) has

a volume of 8.56 L at a temperature a volume of 8.56 L at a temperature

of 0º C and a pressure of 1.5 atm. of 0º C and a pressure of 1.5 atm.

Calculate the moles of HCalculate the moles of H22 molecules molecules

present in this gas sample.present in this gas sample.

SolutionSolution

0.57 mol0.57 mol

Exercise 7Exercise 7 Ideal Gas Law II Ideal Gas Law II

Suppose we have a sample of Suppose we have a sample of ammonia ammonia

gas with a volume of 3.5 L at a gas with a volume of 3.5 L at a pressure of 1.68 atm. The gas is pressure of 1.68 atm. The gas is compressed to a volume of 1.35 L at a compressed to a volume of 1.35 L at a constant temperature. constant temperature.

Use the ideal gas law to calculate the Use the ideal gas law to calculate the final pressure.final pressure.

SolutionSolution

4.4 atm4.4 atm

Exercise 8Exercise 8 Ideal Gas Law III Ideal Gas Law III

A sample of methane gas that A sample of methane gas that has a has a

volume of 3.8 L at 5º C is heated volume of 3.8 L at 5º C is heated to to

86º C at constant pressure. 86º C at constant pressure.

Calculate its new volume.Calculate its new volume.

SolutionSolution

4.9 L4.9 L

Exercise 9Exercise 9 Ideal Gas Law IV Ideal Gas Law IV

A sample of diborane gas (BA sample of diborane gas (B22HH66), ), a a

substance that bursts into flame substance that bursts into flame

when exposed to air, has a when exposed to air, has a pressure pressure

of 345 torr at a temperature of of 345 torr at a temperature of

––15º C and a volume of 3.48 L.15º C and a volume of 3.48 L.

If conditions are changed so If conditions are changed so that that

the temperature is 36º C and the temperature is 36º C and the the

pressure is 468 torr, what will pressure is 468 torr, what will be be

the volume of the sample ?the volume of the sample ?

SolutionSolution

3.07 L3.07 L

Exercise 10Exercise 10 Ideal Gas Law V Ideal Gas Law V

A sample containing 0.35 mol argon A sample containing 0.35 mol argon

gas at a temperature of 13º C and a gas at a temperature of 13º C and a

pressure of 568 torr is heated to pressure of 568 torr is heated to

56º C and a pressure of 897 torr. 56º C and a pressure of 897 torr.

Calculate the change in volume that Calculate the change in volume that

occurs.occurs.

SolutionSolution

decreases by 3 Ldecreases by 3 L

Gas StoichiometryGas Stoichiometry

Use:Use:

PV = nRT PV = nRT

to solve for the volume of one to solve for the volume of one mole mole

of gas at STP.of gas at STP.

Look familiar?Look familiar?

This is the This is the molar volumemolar volume of a gas of a gas at at

STP. Work stoichiometry problems STP. Work stoichiometry problems

using your favorite method, using your favorite method,

dimensional analysis, mole map, the dimensional analysis, mole map, the

table way…just work FAST! Use the table way…just work FAST! Use the

ideal gas law to convert quantities ideal gas law to convert quantities

that are NOT at STP.that are NOT at STP.

Exercise 11Exercise 11 Gas Stoichiometry I Gas Stoichiometry I

A sample of nitrogen gas has a A sample of nitrogen gas has a

volume of 1.75 L at STP. volume of 1.75 L at STP.

How many moles of NHow many moles of N22 are are present ?present ?

SolutionSolution

7.81 x 107.81 x 10-2-2 mol N mol N22

Exercise 12Exercise 12 Gas Stoichiometry II Gas Stoichiometry II

Quicklime (CaO) is produced by Quicklime (CaO) is produced by the the

thermal decomposition of thermal decomposition of calcium calcium

carbonate (CaCOcarbonate (CaCO33). ).

Calculate the volume of COCalculate the volume of CO22 at at STP STP

produced from the decomposition produced from the decomposition of of

152 g CaCO152 g CaCO33 by the reaction: by the reaction:

CaCOCaCO33((ss) → CaO() → CaO(ss) + CO) + CO22((gg))

SolutionSolution

34.1 L CO34.1 L CO22 at STP at STP

Exercise 13Exercise 13 Gas Stoichiometry III Gas Stoichiometry III

A sample of methane gas having a A sample of methane gas having a volume of 2.80 L at 25º C and 1.65 volume of 2.80 L at 25º C and 1.65 atm was mixed with a sample of atm was mixed with a sample of oxygen gas having a volume of oxygen gas having a volume of

35.0 L 35.0 L at 31º C and 1.25 atm. The mixture at 31º C and 1.25 atm. The mixture was then ignited to form carbon was then ignited to form carbon dioxide and water. dioxide and water.

Calculate the volume of COCalculate the volume of CO22 formed formed

at a pressure of 2.50 atm and a at a pressure of 2.50 atm and a

temperature of 125º C.temperature of 125º C.

SolutionSolution

2.47 L2.47 L

The Density of Gases The Density of Gases

d = d = m m = = P(FW) P(FW) {for ONE {for ONE V RT mole of gas}V RT mole of gas} = = FW FW 22.4 L22.4 LAND…AND… Molar Mass = FW = Molar Mass = FW = dRT dRT

PP

““Molecular Mass Kitty Molecular Mass Kitty Cat”Cat”

All good cats put dirt [dRT] over All good cats put dirt [dRT] over

their pee [P]. their pee [P].

Corny, but you’ll thank me Corny, but you’ll thank me later! later!

Just remember that densities of Just remember that densities of

gases are reported in g/L gases are reported in g/L NOTNOT

g/mL.g/mL.

What is the approximate molar What is the approximate molar mass mass

of air? _________of air? _________

The density of air is approximately The density of air is approximately _______ g/L. _______ g/L.

List 3 gases that float in air:List 3 gases that float in air:

List 3 gases that sink in air:List 3 gases that sink in air:

Exercise 14Exercise 14 Gas Density/Molar Gas Density/Molar MassMassThe density of a gas was measured The density of a gas was measured

at 1.50 atm and 27º C and found to at 1.50 atm and 27º C and found to

be 1.95 g/L. be 1.95 g/L.

Calculate the molar mass of the Calculate the molar mass of the

gas.gas.

SolutionSolution

32.0 g/mol32.0 g/mol

Gas Mixtures and Gas Mixtures and Partial PressuresPartial Pressures

The pressure of a The pressure of a

mixture of gases mixture of gases

is the sum of the is the sum of the

pressures of the pressures of the

different components different components

of the mixture:of the mixture:

PPtotaltotal = P = P11 + P + P22 + . . . P + . . . Pnn

John Dalton’s Law of Partial John Dalton’s Law of Partial

Pressures also uses the concept Pressures also uses the concept of of

mole fraction, X.mole fraction, X.

XXAA = = moles of A moles of A _ _

moles A + moles B + moles C + . moles A + moles B + moles C + . . .. .

so now, so now,

PPAA = X = XAA P Ptotaltotal

The partial pressure of each gas in a The partial pressure of each gas in a

mixture of gases in a container mixture of gases in a container

depends on the number of moles of depends on the number of moles of

that gas. The total pressure is the that gas. The total pressure is the

SUM of the partial pressures and SUM of the partial pressures and

depends on the total moles of gas depends on the total moles of gas

particles present, no matter what particles present, no matter what

they are!they are!

Exercise 15Exercise 15 Dalton’s Law I Dalton’s Law I

Mixtures of helium and oxygen Mixtures of helium and oxygen are are

used in scuba diving tanks to used in scuba diving tanks to help help

prevent “the bends.”prevent “the bends.”

For a particular dive, 46 L He at 25º C For a particular dive, 46 L He at 25º C and 1.0 atm and 12 L Oand 1.0 atm and 12 L O22 at 25º C and at 25º C and 1.0 atm were pumped into a tank with 1.0 atm were pumped into a tank with a volume of 5.0 L. a volume of 5.0 L.

Calculate the partial pressure of each Calculate the partial pressure of each gas and the total pressure in the tank gas and the total pressure in the tank at 25º C.at 25º C.

SolutionSolution

PPHeHe = 9.3 atm = 9.3 atm

PPO2O2 = 2.4 atm = 2.4 atm

PPTOTALTOTAL = 11.7 atm = 11.7 atm

Exercise 16Exercise 16 Dalton’s Law II Dalton’s Law II

The partial pressure of oxygen was The partial pressure of oxygen was

observed to be 156 torr in air with a observed to be 156 torr in air with a

total atmospheric pressure of 743 total atmospheric pressure of 743

torr. torr.

Calculate the mole fraction of OCalculate the mole fraction of O22

present.present.

SolutionSolution

0.2100.210

Exercise 17Exercise 17 Dalton’s Law III Dalton’s Law III

The mole fraction of nitrogen in the The mole fraction of nitrogen in the

air is 0.7808. air is 0.7808.

Calculate the partial pressure of NCalculate the partial pressure of N22

in air when the atmospheric in air when the atmospheric

pressure is 760. torr.pressure is 760. torr.

SolutionSolution

593 torr593 torr

Water DisplacementWater Displacement

It is common to collect a gas by It is common to collect a gas by water water

displacement, which means some of displacement, which means some of

the pressure is due to water vapor the pressure is due to water vapor

collected as collected as

the gas was the gas was

passing passing

through! through!

You must correct for this. You must correct for this.

You look up the partial pressure You look up the partial pressure due due

to water vapor by knowing the to water vapor by knowing the

temperaturetemperature..

Exercise 8Exercise 8 Gas Collection over Gas Collection over WaterWaterA sample of solid potassium A sample of solid potassium

chlorate (KClOchlorate (KClO33) was heated in a ) was heated in a

test tube (see the figure above) and test tube (see the figure above) and

decomposed by the following decomposed by the following

reaction:reaction:

2 KClO2 KClO33((ss) → 2 KCl() → 2 KCl(ss) + 3 O) + 3 O22((gg))

The oxygen produced was collected The oxygen produced was collected

by displacement of water at 22º C by displacement of water at 22º C

at a total pressure of 754 torr. The at a total pressure of 754 torr. The

volume of the gas collected was volume of the gas collected was

0.650 L, and the vapor pressure of 0.650 L, and the vapor pressure of

water at 22º C is 21 torr.water at 22º C is 21 torr.

Calculate the partial pressure of OCalculate the partial pressure of O22

in the gas collected and the mass of in the gas collected and the mass of

KClOKClO33 in the sample that was in the sample that was

decomposed.decomposed.

SolutionSolution

Partial pressure of OPartial pressure of O22 = 733 = 733 torrtorr

2.12 g KClO2.12 g KClO33

KINETIC MOLECULAR KINETIC MOLECULAR THEORY OF GASESTHEORY OF GASES

Assumptions of the Assumptions of the MODEL:MODEL:

1. All particles are in constant, 1. All particles are in constant, random, motion.random, motion.2. All collisions between particles are 2. All collisions between particles are perfectly elastic.perfectly elastic.3. The volume of the particles in a 3. The volume of the particles in a gas is negligible.gas is negligible.4. The average kinetic energy of the 4. The average kinetic energy of the molecules is its Kelvin temperature.molecules is its Kelvin temperature.

This neglects any intermolecular This neglects any intermolecular

forces as well. forces as well.

Gases expand to fill their container, Gases expand to fill their container,

solids/liquids do not.solids/liquids do not.

Gases are compressible, Gases are compressible, solids/liquids solids/liquids

are not appreciably compressible.are not appreciably compressible.

This helps explain:This helps explain:

Boyle’s Law Boyle’s Law P & V P & V

If the volume is decreased, that If the volume is decreased, that means that the gas particles will means that the gas particles will hit the wall more often, thus hit the wall more often, thus increasing pressure.increasing pressure.

Boyle’s Law Boyle’s Law P & V P & V

VnRTP

1)(

Constant

Charles’ Law: V & TCharles’ Law: V & T

When a gas is heated, the speeds When a gas is heated, the speeds

of its particles increase, thus of its particles increase, thus

hitting the walls more often and hitting the walls more often and

with more force. with more force.

The only way to keep the P The only way to keep the P

constant is to increase the constant is to increase the

volume of the container.volume of the container.

TP

nRV

Constant

Charles’ Law: V & TCharles’ Law: V & T

Gay-Lussac’s Law: P & Gay-Lussac’s Law: P & TT

When the temperature of a gas When the temperature of a gas

increases, the speeds of its particles increases, the speeds of its particles

increase. The particles are hitting increase. The particles are hitting

the wall with greater force and the wall with greater force and

greater frequency. greater frequency.

Since the volume remains the Since the volume remains the same, same,

this would result in increased this would result in increased gas gas

pressure.pressure.

Gay-Lussac’s Law: P & Gay-Lussac’s Law: P & TT

TV

nRP

Constant

Avogadro’s Law: V & nAvogadro’s Law: V & n

An increase in the number of particles An increase in the number of particles

at the same temperature would cause at the same temperature would cause

the pressure to increase, if the volume the pressure to increase, if the volume

were held constant. were held constant.

The only way to keep constant The only way to keep constant P is P is

to vary the V.to vary the V.

Avogadro’s Law: V & nAvogadro’s Law: V & n

nP

RTV

Constant

Dalton’s Law Dalton’s Law

The P exerted by a mixture of gases The P exerted by a mixture of gases

is the SUM of the partial pressures is the SUM of the partial pressures

since gas particles are independent since gas particles are independent

of each other and the volumes of of each other and the volumes of

the individual particles the individual particles DO NOT DO NOT

matter.matter.

Distribution of Distribution of Molecular SpeedsMolecular Speeds

Plot # of gas molecules having Plot # of gas molecules having

various speeds vs. the speed and various speeds vs. the speed and

you get a curve. you get a curve.

Changing the temperature affects Changing the temperature affects

the shape of the curve, NOT the the shape of the curve, NOT the

area beneath it. area beneath it.

Change the # of molecules and all Change the # of molecules and all

bets are off!bets are off!

Maxwell’s equationMaxwell’s equation

FW

RTuu rms

32

Use 8.314510 J/K• mol for this equation—the “energy” R since it’s really kinetic energy that is at work here!

Exercise 19Exercise 19 Root Mean Square Root Mean Square VelocityVelocityCalculate the root mean square Calculate the root mean square

velocity for the atoms in a velocity for the atoms in a sample sample

of helium gas at 25º C.of helium gas at 25º C.

SolutionSolution

1.36 x 101.36 x 1033 m/s m/s

If we could If we could

monitor the monitor the path path

of a single of a single

molecule it molecule it

would be very would be very

erratic.erratic.

Mean Free PathMean Free Path

the average distance a particle the average distance a particle

travels between collisions. travels between collisions.

It’s on the order of a tenth of a It’s on the order of a tenth of a

micrometer. WAY SMALL! micrometer. WAY SMALL!

Examine the effect of temperature Examine the effect of temperature

on the numbers of molecules with a on the numbers of molecules with a

given velocity as it relates to given velocity as it relates to

temperature. temperature.

HEAT ‘EM UP, SPEED ‘EM UP!!HEAT ‘EM UP, SPEED ‘EM UP!!

Drop a vertical line Drop a vertical line

from the peak of from the peak of

each of the three each of the three

bell shaped curves—bell shaped curves—

that point on the x-that point on the x-

axis represents the axis represents the

AVERAGE velocity of AVERAGE velocity of

the sample at that the sample at that

temperature. temperature.

Note how the bells are “squashed” Note how the bells are “squashed”

as the temperature increases. You as the temperature increases. You

may see graphs like this on the AP may see graphs like this on the AP

exam where you have to identify exam where you have to identify

the highest temperature based on the highest temperature based on

the shape of the graph!the shape of the graph!

Graham’s Law of Graham’s Law of Diffusion and EffusionDiffusion and Effusion

Effusion is closely related to Effusion is closely related to diffusion.diffusion.

DiffusionDiffusion

is the term used to describe the is the term used to describe the

mixing of gases. mixing of gases.

The The raterate of diffusion is the of diffusion is the raterate of of

the mixing. the mixing.

EffusionEffusion

is the term used to describe the is the term used to describe the

passage of a gas through a tiny passage of a gas through a tiny

orifice into an evacuated chamber orifice into an evacuated chamber as shown above.as shown above.

The rate of effusion measures The rate of effusion measures the the

speed at which the gas is speed at which the gas is

transferred into the chamber.transferred into the chamber.

The rates of effusion of two gases The rates of effusion of two gases

are inversely proportional to the are inversely proportional to the

square roots of their molar masses square roots of their molar masses

at the same temperature and at the same temperature and

pressure.pressure.

FW

FW = 2 gas ofeffusion of Rate

1 gas ofeffusion of Rate

1

2

REMEMBER,REMEMBER,

raterate is a change in a quantity is a change in a quantity

over time, over time,

NOT just the time!NOT just the time!

Exercise 20Exercise 20 Effusion Rates Effusion Rates

Calculate the ratio of the effusion Calculate the ratio of the effusion

rates of hydrogen gas (Hrates of hydrogen gas (H22) and ) and

uranium hexafluoride (UFuranium hexafluoride (UF66), a gas ), a gas

used in the enrichment process to used in the enrichment process to

produce fuel for nuclear reactors.produce fuel for nuclear reactors.

SolutionSolution

13.213.2

ExerciseExercise

A pure sample of methane is found A pure sample of methane is found

to effuse through a porous barrier in to effuse through a porous barrier in

1.50 minutes. Under the same 1.50 minutes. Under the same

conditions, an equal number of conditions, an equal number of

molecules of an unknown gas effuses molecules of an unknown gas effuses

through the barrier in 4.73 minutes. through the barrier in 4.73 minutes.

What is the molar mass of the What is the molar mass of the

unknown gas?unknown gas?

DiffusionDiffusion -- This is a classic! -- This is a classic!

Distance traveled by NH3 Distance traveled by NH3 = =

Distance traveled by HCl Distance traveled by HCl

urmsurms for NHfor NH33 = =

urms for HClurms for HCl

5.117

5.36

3

NH

HCl

FW

FW

The observed ratio is LESS than a The observed ratio is LESS than a

1.5 distance ratio. Why?1.5 distance ratio. Why?

This diffusion is slow considering This diffusion is slow considering

the molecular velocities are 450 and the molecular velocities are 450 and

660 meters per second. Which one 660 meters per second. Which one

is which?is which?

This tube contains air and all those This tube contains air and all those collisions slow the process down in collisions slow the process down in the real world. the real world.

Speaking of real world….Speaking of real world….

REAL, thus NONIDEAL GASESREAL, thus NONIDEAL GASES

Most gases behave ideally until Most gases behave ideally until you you

reach high pressure and low reach high pressure and low

temperature. temperature.

(By the way, either of these can (By the way, either of these can

cause a gas to liquefy, go figure!)cause a gas to liquefy, go figure!)

van der Waals Equationvan der Waals Equation

corrects for negligible volume of corrects for negligible volume of

molecules and accounts for inelastic molecules and accounts for inelastic

collisions leading to intermolecular collisions leading to intermolecular

forces (his real claim to fame)forces (his real claim to fame)

a and b are van der Waals constants. a and b are van der Waals constants.

No need to work problems, it’s the No need to work problems, it’s the

concepts that are important!concepts that are important!

nRT = bn] - [V ])V

na( + [P 2

Notice, pressure is increased Notice, pressure is increased

(intermolecular forces lower real (intermolecular forces lower real

pressure, you’re correcting for pressure, you’re correcting for this) this)

and volume is decreased (corrects and volume is decreased (corrects

the container to a smaller “free” the container to a smaller “free”

volume).volume).

The following graphs are The following graphs are classics and make great classics and make great multiple choice questions on multiple choice questions on the AP exam.the AP exam.

When PV/nRT =1.0, the gas is When PV/nRT =1.0, the gas is ideal.ideal.

All of these are All of these are

at 200K. Note at 200K. Note

the P’s where the P’s where

the curves crossthe curves cross

the dashed the dashed line [ideality].line [ideality].

This graph is just for nitrogen This graph is just for nitrogen gas. gas.

Note that although nonideal Note that although nonideal

behavior is behavior is

evident at each evident at each

temperature, temperature,

the deviations the deviations

are smaller at are smaller at

the higher Ts.the higher Ts.

Don’t underestimate the power Don’t underestimate the power of of

understanding these graphs. We understanding these graphs. We

love to ask questions comparing love to ask questions comparing the the

behavior of ideal and real gases. behavior of ideal and real gases.

It’s not likely you’ll be asked an It’s not likely you’ll be asked an

entire free-response gas problem on entire free-response gas problem on

the real exam in May. the real exam in May.

Gas Laws are tested extensively in Gas Laws are tested extensively in

the multiple choice since it’s easy to the multiple choice since it’s easy to

write questions involving them! write questions involving them!

You will most likely see PV=nRT as You will most likely see PV=nRT as

one part of a problem in the free-one part of a problem in the free-

response, just not a whole problem!response, just not a whole problem!

GO FORTH AND RACK UP THOSE GO FORTH AND RACK UP THOSE

MULTIPLE CHOICE POINTS!!MULTIPLE CHOICE POINTS!!