11 But sadly assumptions fail…Nothing is ideal in this world… It’s downright squeezy here
12 Failures of ideal gas equation Failure of Charles’ LawAt very low temperaturesVolume do not decrease to zeroGas liquefies insteadRemember the extrapolated lines?
13 Failures of ideal gas equation From pV = nRT, let Vm be molar volumepVm = RTpVm / RT = 1pVm / RT is also known as Z, the compressibility factorZ should be 1 at all conditions for an ideal gas
14 Failures of ideal gas equation Looking at Z plot of real gases…Obvious deviation from the line Z=1Failure of ideal gas equation to account for these deviations
15 So how?A Dutch physicist named Johannes Diderik van der Waals devised a way...
16 Johannes Diderik van der Waals November 23, 1837 – March 8, 1923Dutch1910 Nobel Prize in Physics
17 I can approximate the behaviour of fluids with an equation So in 1873…I can approximate the behaviour of fluids with an equationScientificcommunityORLY?YARLY!
18 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)
19 Van der Waals Equation Attractive effect Pressure = Force per unit area of container exerted by gas moleculesDependent on:Frequency of collisionForce of each collisionBoth factors affected by attractive forcesEach factor dependent on concentration (n/V)
20 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]
21 Van der Waals Equation Repulsive effect Gas molecules behave like small, impenetrable spheresActual volume available for gas smaller than volume of container, VReduction in volume proportional to amount of gas, n
22 Van der Waals EquationLet another constant, b, relate amount of gas, n, to reduction in volumeVolume term in ideal gas equation, V, becomes (V-nb)
23 Van der Waals Equation Combining both derivations… We get the Van der Waals Equation
24 Van der Waals Equation -> So what’s the big deal? Real world significancesConstants a and b depend on the gas identityRelative values of a and b can give a rough comparison of properties of both gases
25 Van der Waals Equation -> So what’s the big deal? Value of constant aGives a rough indication of magnitude of intermolecular attractionUsually, the stronger the attractive forces, the higher is the value of aSome values (L2 bar mol-2):Water: 5.536HCl: 3.716Neon:
26 Van der Waals Equation -> So what’s the big deal? Value of constant bGives a rough indication of size of gas moleculesUsually, the bigger the gas molecules, the higher is the value of bSome values (L mol-1):Benzene:Ethane:Helium:
28 Critical temperature? Given a p-V plot of a real gas… At higher temperatures T3 and T4, isotherm resembles that of an ideal gas
29 Critical temperature?At T1 and V1, when gas volume decreased, pressure increasesFrom V2 to V3, no change in pressure even though volume decreasesCondensation taking place and pressure = vapor pressure at T1Pressure rises steeply after V3 because liquid compression is difficult
30 Critical temperature?At higher temperature T2, plateau region becomes shorterAt a temperature Tc, this ‘plateau’ becomes a pointTc is the critical temperatureVolume at that point, Vc = critical volumePressure at that point, Pc = critical pressure
31 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 = 0We shall look at a gas obeying the Van der Waals equation
32 VDW equation and critical constants Using VDW equation, we can derive the following
33 VDW equation and critical constants At Tc, Vc and Pc, it’s a point of inflexion on p-Vm graph
37 Compressibility Factor Recall Z plot?Z = pVm / RT; also called the compressibility factorZ should be 1 at all conditions for an ideal gas
38 Compressibility Factor For real gases, Z not equals to 1Z = Vm / Vm,idImplications:At high p, Vm > Vm,id, Z > 1Repulsive forces dominant
39 Compressibility Factor At intermediate p, Z < 1Attractive forces dominantMore significant for gases with significant IMF
40 Boyle Temperature Z also varies with temperature At a particular temperatureZ = 1 over a wide range of pressuresThat means gas behaves ideallyObeys Boyle’s Law (recall V 1/p)This temperature is called Boyle Temperature
41 Boyle Temperature Mathematical implication Initial gradient of Z-p plot = 0 at TdZ/dp = 0For a gas obeying VDW equationTB = a / RbLow Boyle Temperature favoured by weaker IMF and bigger gas molecules
43 Virial Equations Recall compressibility factor Z? Z = pVm/RTZ = 1 for ideal gasesWhat about real gases?Obviously Z ≠ 1So how do virial equations address this problem?
44 Virial Equations Form B,B’,C,C’,D & D’ are virial coefficients pVm/RT = 1 + B/Vm + C/Vm2 + 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
45 Virial Equations Most flexible form of state equation Terms can be added when necessaryAccuracy can be increase by adding infinite termsFor same gas at same temperatureCoefficients B and B’ are proportionate but not equal to each other
47 Summary States can be represented using diagrams or equations Ideal Gas Equation combines Avagadro's, Boyle's and Charles' LawsAssumptions of Ideal Gas Equation fail for real gases, causing deviationsVan der Waals Gas Equation accounts for attractive and repulsive effects ignored by Ideal Gas Equation
48 Summary Constants a and b represent the properties of a real gas A gas with higher a value usually has stronger IMFA gas with higher b value is usually biggerA gas cannot be condensed into liquid at temperatures higher than its critical temperature
49 SummaryCritical temperature is represented as a point of inflexion on a p-V graphCompressibility factor measures the deviation of a real gas' behaviour from that of an ideal gasBoyle Temperature is the temperature where Z=1 over a wide range of pressuresBoyle Temperature can be found from Z-p graph where dZ/dp=0
50 SummaryVirial equations are highly flexible equations of state where extra terms can be addedVirial equations' coefficients are temperature dependent and can be derived experimentally or theoretically
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