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van der Waals Equation

An example of a non-ideal gas equation

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van der Waals equation

Not all gasses act ideally. This is especially truewhen one approaches the conditions for the gas tocondense.

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One of the most widely known equations fordeviation from ideality is the van der Waalsequation. The following is a explanation.

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The ideal gas equation is modified. Starting with:

PV= nRT

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The ideal gas equation is modified.

The volume term is modified to account for thevolume of the molecules:

P(V - nb)= nRT

where bis the volume of the molecules per mole.

volume permole =b

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The ideal gas equation is modified.

The attractions between the molecule arepropoertional to the square of the number of

molecules. So the pressure correction is:(P + n2a/V2)(V - nb) = nRT

where n/Vis the molar concentration.

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This gives the van der Waals equation for a gas:

(P + n2a/V2)(V - nb) = nRT

This is used to calculate some common gasses:

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(P + n2a/V2)(V - nb) = nRT


moles in 1 L volume

acetone and ethyl alcohol at 500 K

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(P + n2a/V2)(V - nb) = nRT


moles in 1 L volume

acetone and ethyl alcohol at 500 KEach gas has its ownconstants:

acetone:

ethyl alcohol:

a = 12.02 atm L2mol-2

b= 0.0841 L mol-1

a = 13.91 atm L2mol-2

b= 0.0994 L mol-1

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(P + n2a/V2)(V - nb) = nRT

Example: Compare the van der Waals calculationof the pressure of 3.00 mol of ethyl alcohol in

2.00 L with the ideal gas calculation at 500 K.

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(P + n2a/V2)(V - nb) = nRT



From the data on the previous slide:

Substituting (leaveing the units out to save space):

[P+ (3.00)2(12.02)/(2.00)2][2.00 -(3.00)(0.0841)] = (3.00)(0.08206)(500)

a = 12.02 atm L2mol-2

b= 0.0841 L mol-1

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(P + n2a/V2)(V - nb) = nRT



From the data on the previous slide:

Solving for P:

P= 43.4 atm from the van der Waals equation

a = 12.02 atm L2mol-2

b= 0.0841 L mol-1

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(P + n2a/V2)(V - nb) = nRT




For the ideal gas calculation:

P(2.00 L) = (3.00 mol)(0.08206 L atm K-1mol-1)(500 K)

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(P + n2a/V2)(V - nb) = nRT

Example: Compare the van der Waals calculationof the pressure of 3.00 mol ethyl alcohol in



Solving for P:

P= 61.5 atmfor the ideal gas equation.

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(P + n2a/V2)(V - nb) = nRT

Example: Compare the van der Waals calculationof the pressure of 3.00 mol ethyl alcohol in


P= 43.4 atm from the van der Waals equationP= 61.5 atmfor the ideal gas equation.This difference can be seen from the graph

below.

0 0.40 0.80 1.20 1.60 2 2.40 2.80 3.200

10

20

30

40

50

60

70

80

90100

moles of ethyl alcohol

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The End

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