Equations of State Compiled by: Gan Chin Heng / Shermon Ong 07S06G / 07S06H

Equations of State

Compiled by:

Gan Chin Heng / Shermon Ong

07S06G / 07S06H

How are states represented?

Diagrammatically (Phase diagrams)

Temp

Pressure

Gas

SolidLiquid

Triple pointCritical point

How are states represented?

MathematicallyUsing equations of stateRelate state variables to describe property of

matterExamples of state variables

Pressure Volume Temperature

Equations of state

Mainly used to describe fluidsLiquidsGases

Particular emphasis today on gases

ABCs of gas equations

Avogadro’s LawBoyle’s LawCharles’ Law

ABC

Avogadro’s Law

At constant temperature and pressureVolume of gas proportionate to amount of gas i.e. V n

Independent of gas’ identity Approximate molar volumes of gas

24.0 dm3 at 298K22.4 dm3 at 273K

Boyle’s Law

At constant temperature and amounts Gas’ volume inversely

proportionate to pressure, i.e. V 1/p

The product of V & p, which is constant, increases with temperature

Charles’ Law

At constant pressure and amountsVolume proportionate to

temperature, i.e. V TT is in Kelvins

Note the extrapolated lines (to be explained later)

Combining all 3 laws…

V (1/p)(T)(n) V nT/p Rearranging, pV = (constant)nT Thus we get the ideal gas equation:

pV = nRT

Assumptions

Ideal gas particles occupy negligible volume

Ideal gas particles have negligible intermolecular

interactions

But sadly assumptions fail…Nothing is ideal in this world…

Real gas particles have considerable intermolecular

interactions

Real gas particles DO occupy finite volume

It’s downright squeezy here

Failures of ideal gas equation

Failure of Charles’ LawAt very low

temperaturesVolume do not

decrease to zeroGas liquefies insteadRemember the

extrapolated lines?

Failures of ideal gas equation

From pV = nRT, let Vm be molar volumepVm = RT

pVm / RT = 1

pVm / RT is also known as Z, the compressibility factor

Z should be 1 at all conditions for an ideal gas

Failures of ideal gas equation

Looking at Z plot of real gases…

Obvious deviation from the line Z=1

Failure of ideal gas equation to account for these deviations

So how?

A Dutch physicist named Johannes Diderik van der Waals devised a way...

Johannes Diderik van der Waals

November 23, 1837 – March 8, 1923

Dutch 1910 Nobel Prize in

Physics

So in 1873…

Scientific community

I can approximate the behaviour of

fluids with an equation

ORLY?

YARLY!

Van der Waals Equation

Modified from ideal gas equation Accounts for:

Non-zero volumes of gas particles (repulsive effect)

Attractive forces between gas particles (attractive effect)

Van der Waals Equation

Attractive effectPressure = Force per unit area of container

exerted by gas moleculesDependent on:

Frequency of collision Force of each collision

Both factors affected by attractive forcesEach factor dependent on concentration (n/V)

Van der Waals Equation

Hence pressure changed proportional to (n/V)2

Letting a be the constant relating p and (n/V)2…

Pressure term, p, in ideal gas equation becomes [p+a(n/V)2]

Van der Waals Equation

Repulsive effectGas molecules behave like small,

impenetrable spheresActual volume available for gas smaller than

volume of container, VReduction in volume proportional to amount of

gas, n

Van der Waals Equation

Let another constant, b, relate amount of gas, n, to reduction in volume

Volume term in ideal gas equation, V, becomes (V-nb)

Van der Waals Equation

Combining both derivations… We get the Van der Waals Equation

2

m2m

np + a [V-nb] = nRT

V

OR

ap + [V -b] = RT

V

Van der Waals Equation -> So what’s the big deal? Real world significances

Constants a and b depend on the gas identityRelative values of a and b can give a rough

comparison of properties of both gases

Van der Waals Equation -> So what’s the big deal? Value of constant a

Gives a rough indication of magnitude of intermolecular attraction

Usually, the stronger the attractive forces, the higher is the value of a

Some values (L2 bar mol-2): Water: 5.536 HCl: 3.716 Neon: 0.2135

Van der Waals Equation -> So what’s the big deal? Value of constant b

Gives a rough indication of size of gas molecules Usually, the bigger the gas molecules, the higher is

the value of b Some values (L mol-1):

Benzene: 0.1154 Ethane: 0.0638 Helium: 0.0237

Critical temperature and associated constants

Critical temperature?

Given a p-V plot of a real gas…

At higher temperatures T3 and T4, isotherm resembles that of an ideal gas

Critical temperature?

At T1 and V1, when gas volume decreased, pressure increases

From V2 to V3, no change in pressure even though volume decreases

Condensation taking place and pressure = vapor pressure at T1

Pressure rises steeply after V3 because liquid compression is difficult

Critical temperature?

At higher temperature T2, plateau region becomes shorter

At a temperature Tc, this ‘plateau’ becomes a point

Tc is the critical temperature Volume at that point, Vc =

critical volume Pressure at that point, Pc =

critical pressure

Critical temperature

At T > Tc, gas can’t be compressed into liquid

At Tc, isotherm in a p-V graph will have a point of inflection1st and 2nd derivative of isotherm = 0

We shall look at a gas obeying the Van der Waals equation

VDW equation and critical constants Using VDW equation,

we can derive the following

m2m

2m m

ap + [V -b] = RT

V

RT ap = -

V -b V

VDW equation and critical constants At Tc, Vc and Pc, it’s a

point of inflexion on p-Vm graph

2

2

0

0

m T

m T

dp

dV

d p

dV

VDW equation and critical constants

2 3

2

2 3 4

m,c c c2

c m,cc

c

2

( )

2 6

( )

Rearranging...

a 8aV = 3b; p = ; T =

27b 27Rbp V 3

Z = = RT 8

m m mT

m m mT

dp RT a

dV V b V

d p RT a

dV V b V

VDW equation and critical constants Qualitative trends

As seen from formula, bigger molecules decrease critical temperature

Stronger IMF increase critical temperature Usually outweighs size factor as bigger molecules have

greater id-id interaction Real values:

Water: 647K Oxygen: 154.6K Neon: 44.4K Helium: 5.19K

Compressibility Factor

Compressibility Factor Recall Z plot? Z = pVm / RT; also called

the compressibility factor

Z should be 1 at all conditions for an ideal gas

Compressibility Factor For real gases, Z not

equals to 1 Z = Vm / Vm,id

Implications:At high p, Vm > Vm,id, Z

> 1Repulsive forces

dominant

Compressibility FactorAt intermediate p, Z <

1Attractive forces

dominantMore significant for

gases with significant IMF

Boyle Temperature

Z also varies with temperature At a particular temperature

Z = 1 over a wide range of pressures That means gas behaves ideally Obeys Boyle’s Law (recall V 1/p) This temperature is called Boyle Temperature

Boyle Temperature

Mathematical implication Initial gradient of Z-p plot = 0 at T dZ/dp = 0

For a gas obeying VDW equation TB = a / Rb Low Boyle Temperature favoured by weaker IMF

and bigger gas molecules

Virial Equations

Virial Equations

Recall compressibility factor Z?Z = pVm/RT

Z = 1 for ideal gases What about real gases?

Obviously Z ≠ 1 So how do virial equations address this

problem?

Virial Equations

FormpVm/RT = 1 + B/Vm + C/Vm

2 + D/Vm3 + …

pVm/RT = 1 + B’p + C’p2 + D’p3 + …

B,B’,C,C’,D & D’ are virial coefficientsTemperature dependentCan be derived theoretically or experimentally

Virial Equations

Most flexible form of state equationTerms can be added when necessaryAccuracy can be increase by adding infinite

terms For same gas at same temperature

Coefficients B and B’ are proportionate but not equal to each other

Summary

Summary

States can be represented using diagrams or equations

Ideal Gas Equation combines Avagadro's, Boyle's and Charles' Laws

Assumptions of Ideal Gas Equation fail for real gases, causing deviations

Van der Waals Gas Equation accounts for attractive and repulsive effects ignored by Ideal Gas Equation

Summary

Constants a and b represent the properties of a real gas

A gas with higher a value usually has stronger IMF

A gas with higher b value is usually bigger

A gas cannot be condensed into liquid at temperatures higher than its critical temperature

Summary

Critical temperature is represented as a point of inflexion on a p-V graph

Compressibility factor measures the deviation of a real gas' behaviour from that of an ideal gas

Boyle Temperature is the temperature where Z=1 over a wide range of pressures

Boyle Temperature can be found from Z-p graph where dZ/dp=0

Summary

Virial equations are highly flexible equations of state where extra terms can be added

Virial equations' coefficients are temperature dependent and can be derived experimentally or theoretically