chapter 9 - gases...THE KINETIC-MOLECULAR THEORY OF GASES • A gas consists of a large collection of individual particles that are very small (no volume). • Gas particles are in

CHANGING CONDITIONS The Ideal Gas Law can also explain the

manner in which a gas changes when

conditions change (e.g. increase P,

decrease T...):

= R P1V1

n1T1

P2V2

n2T2

=

Subscript 1: Initial conditions

Subscript 2: Final conditions

QUESTION What is the effect of increasing the

pressure by a factor of four on an ideal

gas, if the number of moles and

temperature are held constant?

The volume increases by a factor of four

The volume decreases by a factor of four

The volume remains the same

The volume doubles

Answer

A -

B -

C -

D -

• Initially discovered through observations of

gas behavior - the changing of gas volumes.

• Macroscopic behavior of gases explained by

Kinetic-Molecular Theory.

UNDERSTANDING THE

IDEAL GAS LAW

BOYLE’S LAW The volume of an ideal gas is inversely

proportional to the external pressure - at a fixed

T and n:

P1V1 P2V2

n1T1 n2T2

= P1V1 = P2V2

V2 = (P1/P2)V1

MOLECULAR VIEW OF

BOYLE’S LAW

Pext increased

T, n constant

More frequent collisions (force) over a smaller

surface area.

CHARLES’ LAW The volume of an ideal gas is directly

proportional to the temperature of the gas - at a

fixed P and n.

P1V1 P2V2

n1T1 n2T2

=

V2 = (T2/T1)V1

V1 = V2

T1 T2

MOLECULAR VIEW OF

CHARLES’ LAW

CHARLES’ LAW V2 = (T2/T1)V1

CHARLES’ LAW AND

ABSOLUTE ZERO

• Celsius scale - arbitrary zero point

• Kelvin scale (K) - absolute

temperature scale; 0 K is the lowest

possible temperature

• -273.15°C = 0 K

• At 0 K (for ideal gases) - molecular

motion stops; zero volume for gases

QUESTION What is the effect of increasing the

temperature by a factor of two and

increasing the pressure by a factor of two on

an ideal gas, if the number of moles of the

gas is held constant?

The volume increases by a factor of four

The volume increases by a factor of two

The volume remains the same

The volume decreases by a factor of two

The volume decreases by a factor of four

Answer

A -

B -

C -

D -

E -

AVOGADRO’S LAW

The volume of an ideal gas is directly proportional to

the amount of gas - at fixed P and T.

P1V1 P2V2

n1T1 n2T2

=

V2 = (n2/n1)V1

V1 = V2

n1 n2

At a fixed P and T, equal volumes of an ideal

gas contain an equal number of particles.

MOLECULAR VIEW OF

AVOGADRO’S LAW

STOICHIOMETRIC

RELATIONSHIPS BETWEEN

GASEOUS REAGENTS

P, V, T

of gas A

moles of

gas A

moles of

gas B

P, V, T

of gas B

Ideal gas

law

Ideal gas

law

Molar ratio

from

balanced

equation

QUESTION Magnesium metal (0.100 mol) and 1.00 L of 0.500 M

hydrochloric acid are combined and react to completion.

How many liters of hydrogen gas, measured at 273.15 K

and 1.00 atm are produced?

Mg(s) + 2HCl(aq) → MgCl2(aq) + H2(g)

2.24 L of H2

4.48 L of H2

5.60 L of H2

11.2 L of H2

22.4 L of H2

Answer:

A

B

C

D

E

GAS DENSITY AND MOLAR MASS

n = mass

MW

n = PV

RT

MW = (mass)RT

PV

MW = dRT

P

d = mass

V

• The ideal gas law (PV=nRT)

can be used to determine the

density (d) and/or molecular

weight (MW) of the gas.

• The MW of a gas will always

remain the same, as it is based

on the mass of the atoms of

the gas.

• The density of a gas will

change with pressure, and

temperature.

PROBLEM

A 0.50 L bottle contains an unknown gas under a pressure of

3.0 atm at T = 300.K. The mass of the container is 267.37 g

when evacuated and 269.08 g when filled with the gas.

Which of the following could be the identity of the gas?

nitrogen

oxygen

fluorine

argon

carbon dioxide

DENSITY OF AN IDEAL GAS The density of an ideal gas depends on its

chemical identity (molar mass).

X moles of ideal gas H2(g) will occupy the

same volume as X moles of CH4(g).

Yet X moles of H2(g) has less mass than

CH4(g).

density = (molar mass)P

RT

Gases are

compressible: density

will vary greatly with

changes in T and P

CALCULATION QUESTION A 30.0 L balloon is filled with gas at 292.5 K

and has a pressure of 1.2 atm. How many

moles of gas are in the balloon?

Does it matter if the gas in the balloon is air,

helium, nitrogen or methane...?

STANDARD MOLAR VOLUME

One mole of any

idea gas will

occupy 22.4 L at

standard

temperature and

pressure.

The difference

will be in the

mass of the gas

(and the density

of the gas).

Standard Temperature and Pressure

(STP) = 0°C (273.15 K) and 1.00 atm

QUESTION List the following gases in order of increasing

density. Assume temperature and pressure are

constant.

Cl2<Kr<SO2

Kr<SO2<Cl2

SO2<Cl2<Kr

SO2<Kr<Cl2

Cl2<SO2<Kr

Answer:

A

B

C

D

E

PROBLEM What will be the density of CO2 at 4.00 atm

pressure and T = 300. K?

DALTON’S LAW OF

PARTIAL PRESSURES • When there is a mixture of gases, the total pressure of the

mixture is due to the sum of the individual gas pressures.

• These are termed partial pressures - the amount of

pressure produced by each individual type of gas.

Pair = Pnitrogen + Poxygen + Pcarbon dioxide + Pargon + ...

Ptotal = P1 + P2 + P3 + P4 + ...

Py = Xy × Ptotal

Xy = mole fraction of gas y = (moles of y)÷(total moles)

DALTON’S LAW PROBLEM What is the total pressure (in atm) when

1.00 mol of Ar, 0.400 mol of He, and 1.60

mol of N2 gases are injected into a 9.12 L

flask at 0.00°C?

What is the pressure of each gas in the mixture?

PROBLEM A 5.63 g sample of methane (CH4) is combusted with a

stoichiometric amount of oxygen. The gas produced from

the combustion is captured in a 6.48 L canister and is at a

temperature of 126°C. What is the pressure inside the

cylinder?

CH4(g) + 2O2(g) → CO2(g) + 2H2O(g)

PROBLEM Small metal cylinders, filled with 16.0 g of compressed

CO2(g) are often used by bicyclists to inflate flat tires. If

such a cylinder has a volume of 25.0 mL, and can fill a

1.25 L bicycle inner tube to a pressure of 7.05 atm at

22.4°C, what is the pressure of CO2(g) inside a sealed

cylinder at the same temperature?

DIFFUSION

The random thermal motion

of gases causes gas

particles to spread out.

Gas will diffuse from areas

of high concentration to

areas of low concentration.

Given enough time the gas

particles will be distributed

evenly (homogeneous

mixture).

DIFFUSION

• If gas molecules move so fast (~500 m/s) why is diffusion so

slow?

• Each molecule collides once every ~1 ns.

• The mean free path of a gas molecule (the average distance it

moves before it hits something) is ~70 nm (or 103 molecular

diameters).

EFFUSION • Effusion - The process by

which a gas molecule escapes

from a container through a tiny

hole(s).

• Graham’s Law of Effusion -

The rate of effusion of a gas is

inversely proportional to the

square root of its molar mass.

• Root-mean-squared speed

(urms)

rateA

rateB MA

MB

=

Rate: volume or number

of moles of gas per unit

time.

EFFUSION The escape of

molecules through a

tiny hole into a

vacuum is fastest for

smaller mass gases.

rateA

rateB MA

MB

=

QUESTION If it takes 20.0 minute for 0.350 moles of H2S(g) to diffuse

through a porous wall, how long would it take for 0.175

moles of krypton gas (Kr(g)) to diffuse through the same

barrier?

rateA

rateB MA

MB

=

Answer:

A - 10.0 min

B - 15.7 min

C - 20.0 min

D - 31.4 min

E - 42.1 min

QUESTION The rate of effusion for nitrogen gas has been

measured in an apparatus, and found to be 79 mL/s.

If measurement is repeated with sulfur dioxide, at the

same temperature and pressure, what will be the

effusion rate for sulfur dioxide?

THE KINETIC-MOLECULAR

THEORY OF GASES

• A gas consists of a large collection of individual particles that are

very small (no volume).

• Gas particles are in constant, random, straight-line, motion

(except for collisions)

• Collisions between particles are elastic - their total kinetic

energy (Ek) is constant.

• Between collisions, the gas particles do not influence each other

in any way (act independently).

HOW FAST DO THE

PARTICLES OF A GAS MOVE? Very fast!

500 m/s = 1800

km/h Speed increases with

temperature.

= (3/2)(R/NA)T Ek

Temperature is a

measure of molecular

motion.

NA=Avogadro's number

AVOGADRO’S LAW REVISITED

Why do equal numbers of particles of

an ideal gas occupy equal volumes at

constant temperature and pressure?

He (4 g/mol) Ar (40 g/mol)

= (3/2)(R/NA)T = ½mu2 Ek m = mass, u =

speed

MASS VERSUS SPEED

Gas particles with lower mass have higher speeds

Root-mean-squared speed (urms)

QUESTION Pressure is a measure of force per unit area.

What does this imply regarding the velocity of

gas particles?

•The velocity of gas particles is independent of

the mass of the particle.

•The velocity of gas particles is directly

proportional to the mass of the particles.

•The velocity of gas particles is inversely

proportional to the mass of the particles.

•All of the above statements are true.

•None of the above statements are true.

Answer

A

B

C

D

E

CALCULATION QUESTION What is the root-mean-square velocity for a

molecule of nitrogen gas at 30°C?

How fast does a molecule of SF6 travel at

the same temperature?

QUESTION A sample of an ideal gas is heated in a steel container

from 25°C to 100°C. Which quantity will remain

unchanged?

The average kinetic energy of the gas

particles.

The collision frequency.

The density.

The pressure of the gas.

Answer:

A

B

C

D

POSTULATES OF KINETIC-

MOLECULAR THEORY Postulate 1: Particle volume

Because the volume of an ideal 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 of

the particles is constant.

REAL GASES

REAL VS IDEAL GASES

• Real gases do not act exactly as we predict ideal gases would

behave.

• Intermolecular Attractions - are much weaker than bonding, so

only seen under extreme conditions. Intermolecular attractions

reduce the force of the impact with the walls of the container.

• Molecular Volume - as the free volume (empty space) decreases,

the volume of gas molecules becomes significant.

QUESTION At very high pressures (~1000 atm) the

measured pressure exerted by a real gas is

greater than that predicted by an ideal gas.

Why is that?

•Because it is difficult to measure high pressures accurately.

•Because real gases will condense to form liquids at that pressure.

•Because gas phase collisions prevent the molecules from colliding with

the container walls.

•Because of the attractive intermolecular forces between the gas

molecules.

•Because the volume occupied by the gas molecules becomes

significant.

Answer:

A

B

C

D

E

INTERPARTICLE

ATTRACTIONS

• Interparticle attractions are very weak forces - much

weaker than the bonds in a molecule.

• At low pressures gas particles are far from each other so

the Interparticle attractions have little influence.

• At high external pressures the particles are closer

together and the Interparticle attractions becomes

significant. This also happens at very low temperatures.

INTERPARTICLE

ATTRACTION • Impact of interparticle attractions is a reduction in the velocity of the

particles.

• A lower particle velocity reduces the force of the collision with the walls.

• Overall this means the gas exerts less pressure on the container walls.

MOLECULAR VOLUME

• In the ideal model of a gas we presume that the volume of

the gas molecule is negligible in comparison to the free

space around each particle.

• As the pressure increases, the amount of free space for

particles to move is reduced, to the point where the gas

particles become a meaningful amount of that “free space”.

REAL VS IDEAL GASES

• Real gases do not act exactly as we predict ideal gases would

behave.

• Collisions are not elastic - Intermolecular attraction, though

very weak, and seen at low pressures, becomes a factor.

Intermolecular attractions reduce the force of the impact with

the walls of the container; slow down the gas particles.

• Molecular Volume - As the free volume (free space)

decreases, the volume of gas molecules becomes significant.

The gas molecules have less space to move without having a

collision, so the pressure is higher than predicted.

PVDW=45.9 atm

ADJUSTING THE IDEAL GAS LAW • To better describe real gasses we need to:

• Adjust P up, to account for inter particle

attractions.

• Adjust V down, to account to particle

volume.

• The values a and b were determined

experimentally by Johannes van der

Waals.

4.89 mol CO2 in 1.98 L at 299K:

Preal=44.8 atm

PIGL=60.6 atm