The non-Ideal The non-Ideal Reality of Gases Reality of Gases

CM2004 CM2004 States of Matter: States of Matter: Gases Gases

The Act of Pressing The Act of Pressing ((L.L. pressura) pressura)

• Many features related to the pressure exerted by a gas had been established in the 17th,18th and early 19th centuries

• Torricelli & Pascal measured atmospheric pressure with a barometer

• Boyle, Townley & Power demonstrated the inverse relationship between volume and pressure.

• John Dalton established in 1801 that : The total pressure exerted by a gaseous mixture is equal to the sum of the pressures that would be exerted by the gases if they alone were present and occupied the total volume.

Dalton’s LawDalton’s Law

• The gas that a diver breathes must be maintained at a partial pressure of 0.20 atm of oxygen. More oxygen could poison the diver, and less would lead to suffocation. The total pressure, however, must be equal to the external pressure to avoid collapsing the lungs. So, a special valve is used to equalize the pressure inside the divers lungs with the external pressure by adding helium gas. The valve also maintains oxygen levels by using Dalton's Law.

No Pressure No Pressure • In 1848 Lord Kelvin established the value

for the “absolute zero” in temperature. (When no pressure is exerted by the gas)

What happens when the What happens when the Pressure is on?Pressure is on?

• Daniel Bernoulli suggested (in 1734) that the pressure of the gas on the walls of its container is the sum of the many collisions made by the individual particles all moving independently

• From this idea it can be reasoned that the pressure is proportional to the developed momentum (mv) and frequency and therefore the particle density

Slowing Down: LiquefactionSlowing Down: Liquefaction• When a gas is compressed in a

confined space, its molecules are forced closer together

• Their motion is thereby reduced and so the system gets cooler

• Air initially at 273K, for example, will drop about 0.25K for each atmosphere of pressure applied

• As compression proceeds, the speed of the molecules and the distances between them continue to decrease, until eventually the substance undergoes a change of state and becomes liquid.

Linde natural gas liquefaction plant in Gablingen, Germany.

Thomas AndrewsThomas Andrews• However, it was found that some

cooling is necessary for the liquefaction to occur

• It was discovered by Thomas Andrews in 1869 that he could condense CO2 gas into a liquid by raising the pressure on the gas, as long as he kept the temperature below 304.16K (~31oC).

• It was impossible to turn CO2 into a liquid above this temperature, no matter how much pressure was applied.

• A gas must, therefore, be cooled below its critical temperature, Tc before it can be liquefied.

Thomas Andrews

Born in Belfast 1813

Critical PointsCritical Points• At 31.0oC it takes a pressure of 72.85 atm to

liquefy CO2 gas.

• Hence there are critical pressures (and critical volumes, by pV=nRT) that also exist

Some CRITICAL POINTS for Certain Gases

Isotherm PlotsIsotherm Plots• Andrews coined the term VAPOUR for the gas that

is in equilibrium with its liquid below Tc

• pV plots at single temperatures are called isotherms and help us visualize gas/liquid behaviour

T1-T5: Liquid/Vapour Equilibrium

T5: Critical Temperature

T6-T7: Boyle’s law region--- Supercritical Fluid

The Forces of van der WaalsThe Forces of van der Waals• van der Waals looked for an

explanation for Andrews' discovery of the existence of "critical temperatures" in gases.

• He saw the necessity of taking into account the real volumes of molecules and any intermolecular forces

• He thereby established an accurate relationship between the pressure, volume and temperature of gases and liquids in 1873: the van der Waals equation.

Quantifying the Imperfections:Quantifying the Imperfections:Compressibility Factor, ZCompressibility Factor, Z

• pV = nRT for an ideal gas.

• Hence (RT/p) = V/n (termed V’)

• V’ is called the MOLAR VOLUME• Z = molar volume of gas = V’/(RT/p) molar volume of ideal gas

• Z = pV’/RT

• Z= 1 for an ideal gas

• However when Z is plotted as a function of

pressure for a real gas…………

Deviations from ZDeviations from Z

pV’/RT vs p graph gives V’

information

Physical Meaning of ZPhysical Meaning of Z• Z< 1

e.g. methane, ethane ammonia

• Implication is that

their molar volume is smaller than an ideal gas. i.e molecules tend to cluster (be attracted to each other)

• Z> 1 e.g. hydrogen and all gases at high pressures

• Implication is that the molecules are repulsed by each other (or that attractive interactions are very weak)

van der Waals’ Modificationsvan der Waals’ ModificationsModifies Ideal Gas Equation by taking into account

attractive and repulsive forces

P*V* =nRT

Ideal Volume incorporates REPULSIVE FORCES.

A correction term is SUBTRACTED from the volume because molecules are not “points” but occupy space and other molecules are therefore repelled from these volumes.

Ideal Pressure incorporates ATTRACTIVE FORCES.

A correction term must be ADDED to the pressure because the attractions “soften” each wall collision and reduces the effective pressure.

V*P*

Ideal Pressures Ideal Pressures

(

The pressure exerted on the container will depend on the frequency and momentum that the gas strikes the walls with. These are BOTH dependent on the density of the gas. ………….=m/V’ ………..

Frequency At high

densities many more molecules between

wall and a given

molecule

Momentum The

attractions “soften” each wall

collision and reduces the

effective pressure.

The Observed Pressure will hence be less than the Ideal Pressure and a correction factor must be added to it. This is related to density2 (or 1/V’2)

X

=

Force =mass X accel = kg m s-2

= (kg ms-1) (s-1)

Pressure = Force per unit area

van der Waals Equation of Statevan der Waals Equation of State[P + a/(V’)2] [V’-b] = RT (for 1mol of gas)

a is a constant related to attractive forces

b is a constant related to molecular size

[P + an2/V2] [V-nb] = nRT

Virial Equations of StateVirial Equations of State• The van der Waals equation is a virial equation of

state that provides physical insight into molecular reality.

• We can write more general equations that describe all the attractive and repulsive interactions causing deviations from the“ideal”

Z = 1IDEAL

REAL Z = 1 + B/V’ + C/V’2 + D/V’3 +…….

2 molecule interaction 3 molecule

interaction

Virial is derived from the Latin word for Force

Temperature EffectsTemperature Effects• B, C, D etc are called

Virial Coefficients and depend on temperature and the gas involved

• They are determined from experiments and the overall equation for any gas is therefore empirical

• The contribution of the terms to deviation from ideality falls off because of the inverse power dependence

The higher the temperature the further apart the molecules due to their greater speeds