CHEM 331

Physical Chemistry

Fall 2014

Real Gases

During our development of the Historical Gas Laws we had to admit that real gas behavior was

distinctly different than that of an Ideal Gas. For instance, even below 1 atm, P data for

gaseous CO2 at 0oC shows this product is not constant.

P [atm] [L/mol] P [L atm/mol]

0.2500 89.5100 22.3897

0.3333 67.0962 22.3654

0.5000 44.6794 22.3397

0.6667 33.4722 22.3148

1.0000 22.2643 22.2642

Although the properties of an Ideal Gas are derived from the properties of real gases, an Ideal

Gas is simply a model of a real gas that happens to work well at moderate temperatures and

pressures. Its behavior is extrapolated from that of real gases. We now want to consider what

happens at much higher pressures; 500-1000 atm. It is much more difficult to develop an

Equation of State for a real gas because the behavior of the gas depends not only on the state

variables (T, P, ), but on the chemical composition of the gas.

We know, for an Ideal Gas:

So, if we define a Compressibility Factor as:

22.24

22.26

22.28

22.3

22.32

22.34

22.36

22.38

22.4

0 0.2 0.4 0.6 0.8 1 1.2

PV

[L

atm

/ m

ol]

Pressure [atm]

Low Pressure PV Data for CO2

then, for an Ideal Gas, Z = 1. For a real gas, we have:

What if we are not in the limit of zero pressure, where the gas behaves as an Ideal Gas? What if

we are at a pressure of 1000 atm? The dependence of the compressibility factor on pressure is a

convenient method of presenting experimental data so as to note the deviations from Ideal

behavior.

We begin by examining P- -T data for a particular real gas. In this case, I randomly selected

data for Ammonia gas from Thermodynamic Functions of Gases, Ed. by F. Din (1956).

Calculations of Z are easy enough. For instance, at P = 1 atm and T = 320 K, we have =

26.069 L/mol. So,

Z =

= 0.9927

This data is typically plotted in the form of isotherms of Z vs. P. This is done for Ammonia for

isotherms at 320K, 340K, 360K, 380K, 400K, 420K, 440K, 460K and 480K. The general trend

in the data is typical. Note that for low pressures, the slope of the Z vs. P curve tends to skate

upward with increasing temperature.

For another example, consider P- -T for gaseous Ethane taken from the International Critical

Tables. Z vs. P isotherms are plotted for 300K and 350K.

When Z < 1, Attractive Intermolecular When Z>1, Repulsive Intermolecular

Forces Dominate Forces Dominate

0

0.2

0.4

0.6

0.8

1

1.2

0

0.2

0.4

0.6

0.8

1

1.2

-10 10 30 50 70 90 110 130 150

Z =

Co

mp

rssi

blit

y Fa

cto

r

Pressure [atm]

Ammonia Gas (320 - 480K)

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 100 200 300 400 500 600

Co

mp

ress

ibili

ty F

acto

r

Pressure [atm]

Gaseous Ethane

T = 300K

T = 350K

When the attractive forces between the gaseous molecules dominate the intermolecular

interactions, the gas will hug in on itself, giving rise to a molar volume that is smaller than it

would be if the gas were Ideal. So, Z will have values below unity. If repulsive forces dominate,

the gas will expand slightly; meaning will be larger than it would be if the gas were Ideal.

This will result in Z values greater than unity. This behavior is observed at very high pressures.

In fact, at these high pressures the gas may be more of a super-critical fluid or even a liquid; not

a gas in the traditional sense.

Additional Notes:

Z, at any given temperature or pressure, depends on the composition of the gas.

The initial slope of the Z vs. P plot depends on the temperature of the gas; increasing with

increasing temperature.

A Textbook of Physical Chemistry

Arthur W. Adamson

If the initial slope is zero, then Z ~ 1 over an extensive pressure range and the gas obeys Boyle's

Law. The temperature at which this occurs is defined as the Boyle Temperature, TB. Formally,

we define the Boyle Temperature as the temperature when:

= 0 at TB

In the above graphs, we note that gaseous N2 at 100oC is very close to its Boyle Temperature.

The Boyle temperature for several gases is presented below:

Gas M [g/mol] TB [K]

He 4 23.8

H2 2 116.4

N2 28 332

Ar 40 410

CH4 16 506

CO2 44 600

C2H4 28 624

NH3 17 995

Let us return to our gaseous Ammonia data and consider the 380 K isotherm. Below ~ 20 atm,

we see the isotherm is fairly linear.

Thus, we can write:

Z = 1 + B'(T) P

where the "constant" B'(T) is the slope of the above line and is dependent on temperature. This

constant is referred to as the Second Virial Coefficient. If we push beyond 20 atm in pressure,

the data begins to show significant curvature.

To properly represent this data, we should add a quadratic term to our equation above:

Z = 1 + B'(T) P + C'(T) P2

In fact, if we wish to represent the data over an even broader range of pressures, we should

expand Z in terms of higher powers of P:

Z = 1 + B'(T) P + C'(T) P2 + D'(T) P

3 + …..

0.9

0.92

0.94

0.96

0.98

1

1.02

0 5 10 15 20 25

Z

Pressure [atm]

Ammonia Gas at 380K

0.6

0.7

0.8

0.9

1

1.1

0 20 40 60 80

Z

Pressure [atm]

Ammonia Gas at 380K

This gives us a practical Equation of State for a real gas. All that is needed is to determine the

values of B'(T), C'(T), D'(T), etc. for the gas in question.

For theoretical reasons, the real gas Equation of State is typically given as an expansion of Z in

terms of

. This equation of state is referred to as the Viral Equation of State, and the

coefficients are referred to as Virial Coefficients; with B(T) being the second virial coefficient,

C(T) the third, etc.

Z = 1 + B(T)

+ C(T)

+ D(T)

+ …..

This form is more useful than [Z vs. P] because molar volume [ ] is a measure of the average

distance between molecules and an expansion in terms of is thus an expansion in terms of

intermolecular distance. The virial coefficients can then in turn be estimated by means of various

theories for intermolecular forces of attraction and repulsion.

A Textbook of Physical Chemistry

Arthur W. Adamson

So, an expansion of Z in terms of P is more experimentally useful. But, an expansion of in terms

of

is more theoretically valid. Not to worry, one expansion can be expressed in terms of the

other. So, measured values of B'(T), C'(T), D'(T) etc. can be used to determine B(T), C(T), D(T),

etc. Of course, the reverse is also true.

Let's show this. First, use the Virial expansion in

to determine P:

P =

Now equate the two expansions:

1 + B(T)

+ C(T)

+ D(T)

+ ….. = 1 + B'(T) P + C'(T) P2 + D'(T) P

3 + …..

Insert our expression for P into the expansion on the right:

1 + B(T)

+ C(T)

+ D(T)

+ …..

= 1 + B'(T) [

]

+ C'(T) [

2

+ …..

On the right, collect terms in powers of

:

1 + B(T)

+ C(T)

+ D(T)

+ …..

= 1 + B'(T) RT

+ [B'(T) RT B(T) + C'(T) (RT)

2]

+ …..

Coefficients of

on either side of this equation must be equal. So,

B(T) = RT B'(T)

C(T) = B'(T) RT B(T) + C'(T) (RT)2 = (RT)

2 [(B'(T))

2 + C'(T)]

etc.

As mentioned, this process can be inverted and we can solve for B'(T), C'(T), etc. in terms of

B(T), C(T), etc. This gives:

B'(T) =

C'(T) =

etc.

Finally, real gases also differ from an Ideal gas in that if cooled or compressed, they will

typically condense into a liquid. This behavior can be illustrated by examining P- isotherms for

the gas.

According to Boyle's Law, for a given temperature, pressure varies inversely with volume.

0

20

40

60

80

100

0 5000 10000

Pre

ssu

re [

atm

]

Molar Volume [cc/mole]

Ethane at 350K

P

P (Ideal)

If the gas is sufficiently cool, at some point, as it is compressed, it will condense. During the

compression, as the gas is condensed into a liquid, the pressure will remain constant. Once the

condensation to a liquid has occurred fully, further compression of the liquid will cause the

pressure to rise rapidly because of the incompressibility of the liquid.

At higher pressures and temperatures, the gas is, again, behaving more as a super-critical fluid

and so will not "condense" into a liquid. It is already extremely dense and behaving somewhat

as a liquid already.

The density of a supercritical fluid is more like that of a liquid than a gas, but is significantly less

than for the liquid at ordinary conditions. For example, … the density of H2O at the critical point

[is] 0.32 g/cm3, compared with 1.00 g/cm

3 at room T and P. In ordinary room-temperature liquids,

there is little space between the molecules, so diffusion of solute molecules through the liquid is

slow. In supercritical fluids, which have a lot of space between molecules, diffusion of solutes is

much faster that in ordinary liquids and the viscosity is much lower than in ordinary liquids.

Moreover, the properties of supercritical fluids in the region near the critical point vary very

rapidly with P and T, so these properties can be "tuned" to desired values by varying P and T.

Supercritical CO2 is used commercially as a

solvent to decaffeinate coffee and to extract

fragrances from raw materials for use in

perfumes. Supercritical and near-critical water

are good solvents for organic compounds and

are being studied as environmentally friendly

solvents for organic reactions.

Physical Chemistry

Ira N. Levine

Above the Critical Isotherm (Tc), only the super-critical fluid can exist. The Critical Pressure

(Pc) and Critical Volume ( ) occur at the inflection point of the critical isotherm.

Values for the critical point of many gases are reported below:

A Textbook of Physical Chemistry

Arthur W. Adamson

It should be noted that experimental determinations of are difficult to make and the results are

not typically accurate.

Near the Critical Point, the fluid will exhibit Critical Opalescence.

A capillary tube is evacuated and then partly filled with liquid, the remaining space containing no

foreign gases but only vapor of the substance in question; the tube is then sealed. On heating,

opposing changes take place. The liquid phase increases its vapor pressure, and the vapor density

increases as vaporization occurs; this acts to diminish the volume of the liquid phase. On the other

hand, the liquid itself expands on heating. If just the proper degree of filling of the capillary was

achieved, these two effects will approximately balance, and the liquid-vapor meniscus will remain

virtually fixed in position as the capillary is heated. A temperature will then be reached at which

the meniscus begins to become diffuse and then no longer visible as a dividing surface. At this

temperature the system often shows opalescence; the vapor and liquid densities are so nearly the

same and their energy difference is so small that fluctuations can produce transient large liquidlike

aggregates in the vapor and vice versa in the liquid.

A Textbook of Physical Chemistry

Arthur W. Adamson

Carbon Dioxide at the Critical Point

http://en.wikipedia.org/wiki/File:Critical_carbon_dioxide.jpg