Real Gases and Liquids
Phase Diagram Review
We studied the phase diagram last lecture. On phase boundaries of phase diagram the Gibbs free energy per molecule is the same for each phase. Slopes of phase boundaries determined by the entropy and volume difference:
For liquid-gas phase boundary
Ideal Gases
As we know the fundamental ideal gas equation:
When you plot the equation into a three-dimensional graph it will be smooth and the isothermal line is a hyperbola with isochore and isobar lines being straight lines.
The success of the ideal gases equation is the perfect approximation when gases are in high temperature and low pressure, thus the separations between gases are far, then we can ignore the interaction between particles and the volume of particles compared to vessel.
The flaw of the equation is that it cannot be used to predict liquidation as the isothermal line is always decreasing without the steady plateau of phase transition. Thus it cannot be used in the case of low temperature and high pressure.
The sources of failures are:
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Molecules are not without volume. Ideal gases are built on the assumption that molecules do not possess size. But in fact the real molecules possess diameter . In high pressure, molecules are compressed together and cannot ignore their volume – it decreases the free path of molecules.
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Molecules’ interactions. Ideal gases assume that the molecules are only affected by elastic collisions from each other. But in fact, there are van der Waals forces which attract far away molecules and repulse neighbour molecules. In low temperature, the molecules possess low kinetic energy, thus the attraction forces are important, leading to non-ideal behaviour.
Experimental Isothermal Lines and Critical Point
Experimental Data Violation
Take the example of in experiment. We measured the - relations at different temperatures.
We may see the critical temperature and critical point: when the temperature is below the critical temperature, its curve will contain one plateau. The behaviour proceeds in steps:
- Gas behaviour: when pressure is low and volume is large, near ideal approximation, the - curve is like a hyperbola.
- Start of liquidation: at for , the curve turns flat and pressure does not increase with decreasing volume; all energy goes into latent heat.
- Gas-liquid coexistence: the pressure on the horizontal line is the vapour pressure at this temperature.
- After liquidation, all turns to liquid: as the liquid nearly cannot be compressed, a small volume decrease makes a large pressure increase, and the curve is nearly a straight line.
Critical Constants
While the temperature increases, the horizontal plateau shortens (gas-liquid coexistence zone smaller). At critical temperature , the plateau lessens to one point: the critical point.
Three characteristics at critical point:
Supercritical fluid: When and , the substance exists in one phase and is denser than normal gas. It possesses both the expansion property of gas and the solubility of liquid – this is a supercritical fluid (SCF). In engineering, supercritical is used to extract caffeine.
The real surfaces are complex. We obtain the phase diagram when projecting onto the - plane and isothermal lines when projecting onto the - plane.
Virial Expansion
Can we find some general formula for the equation of state? We introduce the virial expansion so that the ideal gas equation is its approximation in low density:
The here are called virial coefficients and . They depend on the temperature and are determined by experimental fitting. In physics, represents the pair interactions and corresponds to three-body interactions.
The is the real molar volume instead of the ideal one. When all the virial coefficients tend to zero, the equation reduces to the ideal gas equation.
Why is negative at low temperature? means the attractive forces between molecules dominate. The gases are pulled together, therefore the real volume is smaller than ideal. When temperature increases, the thermal motion overcomes the attractive forces and turns positive as repulsion dominates.
The Boyle Temperature is the temperature when ; the gas behaves like an ideal gas.
Example: 1 mol in a 1 L container with :
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Ideal gas approximation:
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Virial equation (first-order):
The pressure with virial correction is lower than ideal by 12% because as attraction dominates.
Compressibility Factor
We can define the compressibility factor as
which tells us the ratio between real volume and ideal volume.
Physical meanings of values:
This makes sense:
- At , always possesses which indicates weak attractive force as light molecules.
- molecules possess down to 0.2, indicating strong attractive force as it is a strong polarity molecule.
- All molecules tend to at higher pressure as they are compacted together and cannot be compressed.
Van der Waals Equation
Although the virial equation is precise, it is a pure mathematical expansion with no physical meaning like “why does gas liquify?”. Van der Waals chose another way: starting from physics to do the simplest correction.
Correction 1: Molecules Repulsion Effect (Volume)
The volume of one particle can be given by:
The closest distance between two such particles is , so the excluded volume for two particles is:
The excluded volume for each mole is given by
Thus the volume becomes
Correction 2: The Attractive Forces Between Molecules
The pressure in a container comes from the collisions of molecules against the boundary. But the interior molecules attract the molecules near the surface, pulling them to the centre, reducing the effective force. The pressure reduction relates to force and collision frequency, and they are both proportional to the density of molecules . Thus we obtain the pressure reduction:
where is the constant describing the strength of attractive forces.
By composing the two corrections, we obtain the Van der Waals Equation:
But in simulations of VdW, we see the van der Waals loop when the temperature is under the critical temperature . This means : increasing leads to larger ? This is an unstable mechanism that is unphysical and needs to be corrected by the Maxwell equal-area rule.
Maxwell Equal-Area Rule
The Maxwell equal-area rule is used to correct the unphysical loop of the Van der Waals equation in the gas-liquid phase transition zone. In the real system, the zone will have a phase transition.
Maxwell equal-area construction: We use one horizontal isobaric line to replace the loop part, and make the cut area above the horizontal line equal to the replaced area below the horizontal line. Physical meaning: this ensures the amount of work is the same as in the real phase transition. As the Gibbs free energy is a state function, the of the two paths must be equal. This is actually area in geometry.
As the temperature tends to the critical temperature, the line turns shorter and collapses to one point at .
Critical Constants from Van der Waals Equation
At the critical point, the isothermal line does not increase or decrease and it is a turning point. In mathematics, its first-order and second-order derivatives are both zero:
From the VdW equation itself:
Calculate the first-order derivative, then at the critical point:
Calculate the second-order derivative:
Substituting to solve:
And substituting this result:
Limits of the VdW model: The Van der Waals equation is a mean-field theory which takes the average of the forces each molecule experiences. In the limit of low density, it returns to the ideal gas, but around the critical point its approximation violates experimental data. The practical data give different formulae compared with VdW predictions. The reason is that fluctuations near the critical point are very strong and cause the mean-field theory to break down. Understanding this point requires deeper statistical mechanics.