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Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 1

Real Gases

Molecular Interactions

Compression factor

 Virial coefficients

Condensation

Critical Constants

van der Waals Equations

Corresponding States

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 2

Molecular Interactions

Repulsive forces:

assist expansion of gas

 Attractive forces:

assist compression of gas

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 3

Compression Factor

The compression factor of a gas can be definedas:

where V m  is the molar volume, V/n.

Ideal gas:

Intermediate Pressure:

High Pressure:

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 4

Important Question

How do we go about representing this real (i.e.non-ideal) behavior in a logical manner ?

Empirical Models 

Semi-empirical Models 

Theoretical Models 

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Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 5

 Virial Coefficients

For a real gas with large molar volumes and higher temperatures,the isotherms are almost identical to those of an ideal gas.However, there are some small differences which suggest that theperfect gas law is only the first term in a power series :

These are the virial equations of state, where coefficients B and Chave to be evaluated at each temperature!

p = 0:p increases:

p higher:

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 6

 Virial Coefficients

Rewriting in terms of compression factor, Z :

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 7

 Virial Coefficients for Methane

 Virial coefficients for methane up to about 400 bar, 350oC

T (oC) B’ (x 10-3 bar-1) C’ (x 10-6 bar-2) D’ (x 10-9 bar-3)

0 -2.349 -0.877 2925 -1.727 +0.438 17

50 -1.274 +1.353 7.9

100 -0.677 +1.447 4.1

150 -0.324 +1.219 2.0

200 -0.106 +0.967 0.99

250 +0.0345 +0.749 0.56

300 +0.125 +0.583 0.31

350 +0.186 +0.461 0.16

 pV m = RT  1+ ′ B p+ ′C p2+ ′ D p

3( )

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 8

Boyle Temperature

Perfect gas: dZ/dp = 0 (since Z = 1), butin a real gas

 At low T: initial dZ/dp < 0, B is negative

 At high T: initial dZ/dp > 0, B is positive

(Correspond to the first order Bcoefficients at these temperatures)

The temperature at which the initial slopeis zero is the Boyle Temperature, T B ,where B = 0 (real gas corresponds to anideal gas)

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Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 13

van der Waals

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 14

van der Waals Equation

The van der Waals equation is written as

where a and b are van der Waals coefficients,specific to each gas.

Term a is adjusted to represent the attractive forces ofthe molecules, without giving any specific physicalorigin to these forces;

V - nb , not V , now represents the volume in whichmolecules can move.

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 15

Justifying the van der Waals Equation

The coefficients a and b are adjusted to model certain features of the gasbehavior, and are dependent upon the attractive and repulsive forcesbetween molecules in the gas.

The frequency and forces of the collisions are both reduced by attractive

forces, by a strength proportional to n/V (molar concentration); thus,pressure is reduced as the square of this concentration, and thiscontribution is written as:

The parameters a and b should not be assigned specific molecularproperties, but rather, serve as coefficients which help to accurately modelgas behavior, and can be assigned specifically to each type of gas.

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 16

Reliability of the van der Waals Equation

It is unrealistic to expect that such a simple equationwill be a perfect equation of state for all gaseoussubstances.

- van der Waals equation:

- virial equation:

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Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 17

Reliability of the van der Waals Equation

Compare these calculated isotherms with the experimentalisotherms of CO2: the most notable anomaly are the so-called vander Waal’s loops, where the isotherms drop suddenly below Tc(they suggest increase p results in increased V) - they can be fixedby a Maxwell construction (see text) - and the vdW coefficientsare found by comparison of experimental and theoretical curves.

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 18

Summary of the van der Waals Equation

(1) Perfect gas isotherms are obtained at high temperatures and largemolar volumes:

(2) Liquids and gases exist when cohesive and dispersive forces arebalanced:

(3) Critical constants are related to the van der Waals coefficients:

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 19

Summary of the van der Waals Equation

(3) Critical constants are related to the van der Waals coefficients:

- at the flat inflection, the 1st and 2nd derivatives are zero !

Solve using some algebra and find critical constants:

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 20

Principle of Corresponding States

In many areas of science it is useful to compare afundamental property of different objects on a relativescale for purposes of comparison.

For different gases, we use reduced variables:

The observation that real gases at the same V and Texert the same reduced pressure is the principle ofcorresponding states.

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Prof. Mueller Chemistry 451 - Fall 2003 Lecture 3 - 21

Principle of Corresponding States

value of T r