Lecture 8 Flashcards
How can we represent the ideal gas law?
pV = nRT, R = Kb * Na and n = N/Na
therefore the gas is represented by:
pV = N * Kb * T
What is the compressibility factor and how does it relate to a gas’s ideality?
The compressibility factor Z is a quantity that measures the deviation from ideal behavior.
Z = pV/NKbT = pv^-/RT , where v^- = V/n
where at Z = 1 there is no deviation from ideality, at Z < 1 (low Temp) attractive forces dominate and at z > 1 (high temp) repulsive forces dominate.
It is worth noting that in reality gases aren’t ideal unless at very low densities in which the equation of state of a real gas reduces to the ideal gas equation. This is because the interaction between molecules does not affect the macroscopic properties of a gas.
What is the virial expansion of pressure? How is it used to define the compressibility factor? How can we find the second virial coefficient using that? What is the Boyle’s temp?
found in notes on page 1
What is the Van der waals theory for fluids?
Illustration found in notes on page 2.
The theory follows that at temperatures below the critical temperature, there is a discontinuous transition between the gas and liquid phase, whereas the temp approaches the critical temp the difference between the molar volumes of liquids and gases decreases until there is no difference between them at the critical point where the transition from liquid to gas is now continuous existing in a gas phase denoted as a fluid.
The critical point where the gas and liquid connect is an inflection point. Hence the conditions for critical points are:
(δp/ δv^-)(at Tc) = 0 = (δ^2p/ δv^-^2)(at Tc)
it is noted that at high T and low p, the state of the fluid is gaseous and approaches ideal gas behavior (p is inversely proportional to molar volume).
What is the derivation of the van der Waals equation of state? How is it used to relate to the compressibility factor Z?
Found in notes on pages 2-5
What is weird about the van der waals isotherms
The plots of van der waals isotherms above the critical temperature fit with the experimental data.
Below the critical temperature, the discontinuity expected between the liquid and gas phases is not present. The coexisting gas phase and liquid phase are connected by a dashed horizontal line (which corresponds to the pressure of the coexisting phases at a given temperature)
Within that regime, the van der waals isotherms exhibit a loop with decreasing pressure (even till negative pressure which is not possible) to increasing pressure with increasing volume which hints at the instability of the homogenous phase because the condition:
δp/δv^- = - δ^2 A / δ v^- ^2 > 0, indicating a maximum in the Helmholtz energy.
Note that the van der Waals equation of state is an expression for a homogenous system of uniform density, thus predicting uniform isotherms even below the critical temperature.
how can we predict the phase separation with the van der waals using the maxwell construction?
One must first look at the coexistence between the two phases to predict phase separation.
For two phases to coexist they must have mechanical, thermal, and chemical equilibrium. Therefore the coexisting phases of gas and liquid should lie on the same isotherm connected by the horizontal line at the same p in a p x v^- graph. (this shows only mechanical and thermal equilibrium)
To achieve the chemical equilibrium we use maxwell construction where the horizontal line that connects the liquid and gas phases must cut the van der Waal loop to give equal areas in both.
The way this works and shows phase separation is shown in notes on pages 5-6
What is the critical phenomena?
using the conditions of the critical temp ( (δp/ δv^-)(at Tc) = 0 = (δ^2p/ δv^-^2)(at Tc)) and using the Van Der Waals Eos a set of equations which define the set of quantities ( v^-(c), Tc, p(c)) characterizing the critical point.
v^-(c) = 3b^-
p(c) = (a^-)/27b^-^2
Tc = 8a^- / 27R b^-
From this, we can determine a and b from the critical point.
We can also get the compressibility factor at the critical point Z(c) = p(c) * v^-(c) / RTc, in which the experimental value is lower than the theoretical value of 3/8.
it is worth noting that from experimental data it is seen that v^-(c) doesn’t exactly equal 3 b^-
What is the Law of corresponding states?
Found in notes on page 6
