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 considered the most easily understood form of matter because the molecules of gas are very far apart and don’t (mostly) interact. Different gases thus behave similarly (usually).

Gases• A single gas, a simple mixture (2

gases) or a complex mixture all can be analyzed in a similar fashion

• Air is an example of a complex mixture of gases

• Gases form homogeneous mixtures

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

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 millionHelium He 5.2 parts per millionKrypton Kr 1.1 parts per millionSulfur 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 millionXenon Xe 0.09 parts per millionOzone O3 0.07 parts per million

Nitrogen dioxide NO2 0.02 parts per million

Iodine I2 0.01 parts per millionCarbon monoxide CO traceAmmonia NH3 trace

Pressure• Pressure is the force that acts on a given area (P=F/A). • Due to gravity acting on atmospheric gases, these gases execrt

a force on earth (an everything else): 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).

F m.a

F 10,000kg 9.8m /s2 100,000kgm /s2

P F / A 1105 N1m2 1105 N /m2

This unit is a Newton (N)

This unit is a Pascal (Pa)

Units of PressureS.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 later 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).

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 LawRobert Boyle: (1627-1691) the first modern chemist,

known as the father of chemistry.

His 1661 book The Sceptical Chymist marks the introduction of the scientific method, a definition of elements and compounds and a refutation of alchemy and magic potions.

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

• Commonly used format of P1V1 = P2V2

V constant 1p

or PV k (constant)

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

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

constantor constant TVTV

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.

Putting it all together

V 1P

, V T

V TP

P1V1

T1

P2V2

T2

Boyle, Charles

Combine

Combined Gas Law (no change in amount of gas)

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

Gay-Lussac 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) 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

Putting it all together

nRTPVP

nTRV

PnTV

nVTVP

V

, ,1

Boyle, Charles, Avogadro

Combine

Call proportionality constant R

(gas constant)

Ideal Gas Equation

A note on units and dimensional analysisSI unit for R is J/mol·K or m3·Pa/mol·K (R=8.3145 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. (R = 8.3145 kPa·L/mol·K)

If you wish to use atm and L (as in USA and Textbook) R=0.08206 L.atm/mol/K.

Always use absolute temperature scale (K)Exercises:

What is the volume of 1 mole of an ideal gas under standard temperature and pressure (STP)?

How many moles (g) of CO2 is liberated into a 250mL flask when a pressure of 1.3atm is found upon heating calcium carbonate to 31oC?

If a metal cylinder holds 50L of oxygen at 18.5atm and 21oC, what volume will the gas occupy at 1atm and same T?

Gases in chemical reactionsIn an car airbag, NaN3 decomposes explosively into its elements. If an air bag has a volume of 36L and is to be filled with nitrogen gas at a pressure of 1.15atm and 26oC, how many grams of NaN3 must be decomposed?

More ExercisesIf the pressure of a gas in an aerosol can is 1.5atm at 21oC, what would the pressure be if can is heated to 450oC?

What is the density of carbon tetrachloride vapour at 714torr and 125oC?

Gases in chemical reactions

PdRTMolarMass

Another useful variation of the ideal gas law involves molar mass and density. The equation is derived in your textbook.

d = density

Q: Uranium hexafluoride is a solid at room temperature, but it boils at 56°C. Determine the density of uranium hexafluoride at 60.0°C and 745 torr.

Gas mixtures• Dalton’s Law of partial pressuresThe 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

Exercise: A gaseous mixture is made from 6.00g oxygen and 9.00g methane placed in a 15L vessel at 0oC. What is the partial pressure of each gas and the total pressure in the vessel?

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

dimensionless number between 0 and 1.

P1

PT

n1RT /VnT RT /V

n1

nT

P1 n1

nT

PT

P1 X1PT

Mole fraction of N2 in air is 0.78, therefore if the total baormetric pressure is 760 torr, the partial pressure of N2 is (0.78)(760) = 590 torr.

Practice• The partial pressure of CH4(g) is 0.175atm and that of O2(g) is 0.250 atm in a mixture of the

two gases.– What is the mole fraction of each gas in the mixture?– If the mixture occupies a volume of 10.5 L at 65°C, how many grams of methane are in the mixture?

Collecting Gas Over Water

• When a gas is collected over water we get a picture of gases – the intended gas plus water vapor.

Q: Helium is collected over water at 25°C and 1.00 atm total pressure. What total volume of gas must be collected to obtain 0.586 g of helium. (At 25°C, the vapor pressure of water is 19.8 torr)

Kinetic –Molecular TheoryTheory 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.

Temp (K) indicates average kinetic energy

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

MRTu 3

M is the molar mass in kg

This gives rise to interesting consequences: effusion

(KE )avg 32

RT

Effusion• Thomas Graham (1846)

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

1

2

2

1

MM

rr

Derived from comparison of rms speeds

Graham’s Law of Effusion

Diffusion is the mixing of gases. Similar behavior, but very difficult to describe theoretically (i.e. with an equation)

REAL GASES

Deviations from ideal gas lawWHY?

1. Molecules have volume

2. Molecules have attractive forces (intermolecular)

1. V-nb

2. -a(n/V)2

Van der Waals Equation of State2

Vn

anbV

nRTP

Van der Waal’s constantsvan der Waals Coefficients

Gas a (L2·atm/ mol2) b(L/mol)

Helium 3.46 x 10-2 2.371 x 10-2

Neon 2.12 x 10-1 1.710 x 10-2

Hydrogen 2.45 x 10-1 2.661 x 10-2

Carbon dioxide 3.59 x 100 4.269 x 10-2

Water vapor 5.47 x 10-1 3.052 x 10-2

a correlates with boiling point (see later)

b can be used to estimate molecular radii