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