KINETIC THEORY OF GASES

Gases • Rather than considering the atomic

nature of matter we can classify it based on the bulk property: gaseous, liquid or solid.

• Gases are the most easily understood form of matter (we shall see why).

Air is an example of a complex mixture of gases: gases form homogeneous mixtures regardless of identities or proportions (unlike liquids and solids).

Gases expand to fill any container, and are highly compressible (unlike liquids and solids)

These characteristics arise because the molecules of gas are very far apart and don’t (mostly) interact. Different gases thus behave similarly.

Component Symbol Volume

Nitrogen N2 78.084%

99.998% Oxygen O2 20.947%

Argon Ar 0.934%

Carbon Dioxide CO2 0.033%

Neon Ne 18.2 parts per million

Helium He 5.2 parts per million

Krypton Kr 1.1 parts per million

Sulfur dioxide SO2 1.0 parts per million

Methane CH4 2.0 parts per million

Hydrogen H2 0.5 parts per million

Nitrous Oxide N2O 0.5 parts per million

Xenon Xe 0.09 parts per million

Ozone O3 0.07 parts per million

Nitrogen dioxide NO2 0.02 parts per million

Iodine I2 0.01 parts per million

Carbon monoxide CO trace

Ammonia NH3 trace

Physical Characteristics of Gases

Physical Characteristics Typical Units

Volume, V liters (L)

Pressure, P atmosphere

(1 atm = 1.015x105 N/m2)

Temperature, T Kelvin (K)

Number of atoms or

molecules, n

mole (1 mol = 6.022x1023

atoms or molecules)

Pressure • Pressure is the force that acts on a given area (P=F/A).

• Gravity on earth exerts a pressure on the atmosphere:

atmospheric pressure.

• We can evaluate this by calculating the force due to

acceleration (by gravity) of a 1m2 column of air extending

through the atmosphere (this has a mass of ~10,000kg).

25

2

5

22

/1011

101/

/000,100/8.9000,10

.

mNm

NAFP

skgmsmkgF

amF

This unit is a Newton (N)

This unit is a Pascal (Pa)

Units of Pressure S.I. unit of pressure is the N/m2, given the name Pascal (Pa).

A related unit is the bar (1x105 Pa) used because atmospheric

pressure is ~ 1x105 Pa (100 kPa, or 1bar).

Torricelli (a student of Galileo) was the first to recognise that the

atmosphere had weight, and measured pressure using a barometer

Standard atmospheric pressure was thus defined

as the pressure sufficient to support a mercury

column of 760mm (units of mmHg, or torr).

Another popular unit was thus introduced to

simplify things, the atmosphere (atm =

760mmHg).

Pressure • Atmospheric pressure and relationship between units

1 atm = 760 mmHg = 760 torr = 101.325kPa = 1.01325 bar)

Measuring Pressure: the manometer

Exercise:

On a certain day a barometer gives the atmospheric pressure as 764.7 torr. If a

metre stick is used to measure a height of 136.4mm in the open arm, and

103.8mm in the gas arm of a manometer, what is the pressure of the gas sample?

(give in torr, atm, kPa and bar).

Result

Difference in height is 32.6 mm. Gas inside has greater pressure than prevailing atmospheric pressure: 764.7 + 32.6 mmHg = 797.3 mmHg (Torr)

Convert to atm: divide by 760 = 1.049 atm

Convert to kPa: multiply by 101.325 = 106.3 kPa

Convert to bar: divide by 100 = 1.063 bar

Gas Laws

• A large number of experiments have determined that 4

variables are sufficient to define the physical condition (or

state) of a gas: the gas laws.

Boyle’s Law,

Charles’ Law,

Avogadro’s hypothesis

Boyle’s Law

• Boyle investigated the variation of the volume occupied by a

gas as the pressure exerted upon it was altered and noted that

the volume of a fixed quantity of gas, at constant temperature

is inversely proportional to the pressure

constantor 1

constant PVp

V

Boyle’s Law: P1V1 = P2V2

Boyle’s Law: P1V1 = P2V2

Charles’ Law • A century later, a French scientist, Jacques Charles discovered that the

volume of a fixed amount of gas, as constant pressure, is proportional to the absolute temperature. Cool a balloon, or a sealed plastic bottle, to verify this!

constantor constant T

VTV

It was recognised (by William

Thomson, Lord Kelvin, a Belfast

born physicist) that if the graph was

extrapolated to zero volume, an

absolute zero of -273.15 oC is

obtained.

Charles’ Law: V1/T1 = V2/T2

Avogadro’s Law • Relationship between quantity of gas and volume established by

Gay-Lussac (balloon science!) and Avogadro in the 19th Century.

Result was Avogadro’s hypothesis: equal volumes of gases at the same temperature and pressure contain equal numbers of molecules

Experiments show that 22.4L of gas at 0oC and 1atm

(STP), or 24.8L of gas at 298.15 K and 1 bar (SATP),

contains 6.022 x 1023 molecules (Avogadro’s number, NA)

Avogadro’s law: volume of gas at constant temperature and

pressure is proportional to the number of moles of gas (n)

constant nV Remember:

1 mole = Avogadro’s

number of objects

Avogadro’s Law: V1/n1=V2/n2

Putting it all together

nRTPV

P

nTRV

P

nTV

nVTVP

V

, ,1

Boyle, Charles, Avogadro

Combine

Call proportionality constant R

(gas constant)

Ideal Gas Equation

At constant volume,

pressure and absolute

temperature are

directly related.

P = k T

P1 / T1 = P2 / T2

Gay-Lussac Law

The total pressure in a container

is the sum of the pressure each

gas would exert if it were alone

in the container.

The total pressure is the sum of

the partial pressures.

PTotal = P1 + P2 + P3 + P4 + P5 ...

(For each gas P = nRT/V)

Dalton’s Law

Dalton’s Law

Water evaporates!

When that water evaporates, the vapor has a

pressure.

Gases are often collected over water so the vapor

pressure of water must be subtracted from the

total pressure.

Vapor Pressure

Units and dimensional analysis

SI unit for R is J/mol.K or m3.Pa/mol.K (R=8.315 of these units)

Need to use the units of Pa for pressure and m3(=1000L) for volume in any calculation.

Alternatively you can use units of kPa and L.

If you wish to use atm and L

R=0.0826 L.atm/mol/K.

Always use absolute temperature scale (K)

Gas mixtures • Dalton’s Law of partial pressures

The total pressure of a mixture of gases equals the

sum of the pressures that each would exert if it

were present alone

PT=P1+P2+P3+….Pn

Mole Fractions

• The ratio n1/nT is called the mole fraction (denoted x1), a

dimensionless number between 0 and 1.

T

T

TTT

Pn

nP

n

n

VRTn

VRTn

P

P

11

111

/

/

Mole fraction of N2 in air is 0.78, therefore if the total

barometric pressure is 760 torr, the partial pressure of N2 is

(0.78)(760) = 590 torr.

Kinetic –Molecular Theory

Theory describing why gas laws are obeyed (explains both pressure and temperature of gases on a molecular level).

• Complete form of theory, developed over 100 years or so, published by Clausius in 1857.

Gases consist of large numbers of molecules that are in continuous, random motion

Volume of all molecules of the gas is negligible, as are attractive/repulsive interactions

Interactions are brief, through elastic collisions (average kinetic energy does not change)

Average kinetic energy of molecules is proportional to T, and all gases have the same average kinetic energy at any given T.

Because each molecule of gas will have an individual kinetic energy, and thus

individual speed, the speed of molecules in the gas phase is usually characterised

by the root-mean-squared (rms) speed, u,(not the same though similar to the

average speed). Average kinetic energy є = ½mu2

Application to Gas Laws

• Increasing V at constant T:

Constant T means that u is unchanged. But if V is increased the likelihood of collision with the walls decreases, thus the pressure decreases (Boyle’s Law)

• Increasing T at constant V:

Increasing T increases u, increasing collisional frequency with the walls, thus the pressure increases (Ideal Gas Equation).

Molecular speeds and mass • The average kinetic energy of gases has a specific value at

a given temperature. The rms speed of gas composed of light particles, He, is higher than that for heavier particles, Ne, at the same temperature.

• Can derive an expression for the rms speed (from kinetic theory)

M

RTu

3 M is the molar mass

This gives rise to interesting consequences:

effusion

Effusion • Thomas Graham (1846)

discovered that effusion is inversely proportional to the square root of molar mass.

1

2

2

1

M

M

r

r

Derived from comparison

of rms speeds

REAL GASES

Deviations from ideal gas law

WHY? 1. Molecules have volume

2. Molecules have attractive forces

(intermolecular)

1. V-nb

2. -a(n/V)2

Van der Waals Equation of State

2

V

na

nbV

nRTP

Differences Between Ideal and Real Gases

Obey PV=nRT Always Only at very low

P and high T

Molecular volume Zero Small but

nonzero

Molecular attractions Zero Small

Molecular repulsions Zero Small

Ideal Gas Real Gas