of Compressibility a real gas
The compressibility factor of a gas (and, by extension, any fluid) is defined as
PVm Z= RT
For an ideal gas, the compressibility factor is always 1. For a real gas, the deviation of the compressibility factor from 1 is a convenient measure of nonideality.
This experiment allows you to measure the compressibility factor of carbon dioxide at temperatures just above and just below the critical point, and to compare your observations with a simple model.
The model
The van der Waals equation of state is a commonly used empirical equation that is more flexible than the ideal gas equation:
RT a P= - 2 Vm - b Vm
The constant a is interpreted as a measure of the attractive force between molecules, while the constant b is a measure of the excluded volume resulting from the volume of the condensed gas.
The model
The van der Waals equation can be rewritten in terms of the compressibility factor:
PVm Z= = RT
1 a b RTVm 1- Vm
Notice that, as Vm becomes large, Z approaches 1.
The gas
Carbon dioxide has been an important industrial chemical for many years. It now is receiving attention as a solvent. The fact that its critical point is only slightly above room temperature means that its solvation properties can be tailored by changing its density in a way that cannot be accomplished with ordinary liquids.
The equipment
The instrument is the Phase Monitor from Supercritical Fluid Technologies. It is designed to provide an adjustable volume (3 to 33 mL) and temperature (room temperature to ~100) at pressures up to 10, 000 PSI.
The equipment
The instrument is the Phase Monitor from Supercritical Fluid Technologies. It is designed to provide an adjustable volume (3 to 33 mL) and temperature (room temperature to ~100) at pressures up to 10, 000 PSI.
Part 1: an isotherm
The first part of the experiment will involve recording pressure as a function of volume at a constant temperature (or as near to one as we can manage). The procedure will be to maintain a constant (thermostatted) temperature while reducing the volume of the chamber. Choosing a temperature below the critical point, we expect condensation to occur at the vapor pressure of CO2 at that temperature. During condensation the pressure will not change as volume is changed (except as a result of a change in the temperature).
Part 1: an isotherm
The van der Waals equation makes a somewhat peculiar prediction about the shape of the isotherm:
Part 1: an isotherm
Clearly, the oscillatory segment should be replaced by a straight horizontal line connecting the outer and inner curves. Where should the line be drawn?
Where to draw the line?
The solution was given by Maxwell, reasoned who as follows: The Helmholtz free energy is given by
dA = - PdV - SdT
So a finite change DA is given by
DA = A2 - A1 = - 1 PdV
2
and a round trip DA is given by
DA = - 1 PvdW dV +
2
2
1
Pvap dV
which must be zero, since A is a state function.
Where to draw the line?
The solution we seek is then the pressure Pvap for which V2
(P
V1
vdW
- Pvap ) dV = 0
V1
Where volumes V1 and V2 are the inner and outer solutions of the van der Waals equation at the temperature of interest.
V2
Where to draw the line?
This is equivalent to constructing Pvap so that the areas included between the curve and the horizontal line are equal above and below the line. This allows us to calculate the vapor pressure of CO2 for as many temperatures as we like, which brings us to the next part of the experiment.
V1
V2
Part 2: the vapor pressure curve
You can easily collect pressures as a function of temperature for a range of 5C or so (or more if you want to take the time). Below the critical point these pressures are the vapor pressure whenever the liquid and vapor coexist. Above the critical point, no such interpretation is possible. The treatment in terms of the van der Waals equation is straightforward enough, but it is a bit tedious computationally. Why not use the more convenient vapor pressure equation the Clausius-Clapeyron equation?
dPvap DH vap DH vap = ; ln Pvap = 2 dT RT RT
The Clapeyron equation
The Clapeyron equation is an exact thermodynamic relationship:
dPeq DS = dT DV
This holds for any phase transition. The Clausius-Clapeyron equation is derived from it by assum...