What do we mean by the number of components?
What do we mean by the number of phases?
How many independent thermodynamic variables are there for a one phase system containing one chemical component?
What is a function of state?
Write out the ideal gas equation?
Write out a function of state for a real gas?
Why is number of mols not a thermodynamic variable?
Equilibrium and Gibb's phase rule
As discussed in the following video, intensive variables must be homogenous (equal) across the entirety of the system when the system is at equilibrium"
https://www.youtube.com/watch?v=udqI0rq6b3U
The requirement that all intensive variables must be homogenous (equal) across the entirety of the system when the system is at equilibrium ensures that at equilibrium, not all thermodynamic quantities are independent. In particular, there is a result called Gibb's phase rule which states:
<aside> 💡 $F = C -\pi + 2$
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Here C is the number of components in the system (the number of chemically distinct species), pi is the number of phases in the system (the number of distinct regions of space in the system where the properties of the material are essentially uniform) and F is the number of thermodynamic variables that are independent. Consider the consequence this has for argon gas. Argon gas contains only one chemical component (argon atoms) hence C=1. Furthermore, the properties of the gas are uniform throughout so the number of phases in this system is also equal to 1. The thermodynamic state of argon gas (if it is at equilibrium) can thus be characterised using only 2 thermodynamic variables. If we are told the temperature, T, of the gas and the pressure, P, that it is under there should be some function, f, that we can use to calculate the volume, V, of the gas, V = f(P,T).
Now consider some argon gas placed next to some water. Some of the argon atoms will dissolve in the water so there are 2 chemically distinct species (argon atoms and water molecules) so C=2. There are also two distinct phases (the argon solution and the argon gas). The number of variables F must thus be 2. The equilibrium thermodynamic state of the system can thus be described using two variables - for instance the temperature and the concentration of argon atoms in the solution. All other quantities - e.g. the volume of the whole system, the chemical potential - must be some function of these two variables.
Ideal and Non-Ideal gasses
For gasses it is possible to write out explicit forms for the equations of state. In the first instance these relations were obtained through experimentation - i.e. looking at what happens to the volume of the gas as the pressure and temperature are changed and fitting. We thus have:
$PV = nRT$
where n is the number of mols of gas (The number of mols, n, in the system is not a thermodynamic variable as it is a fixed property of the system. When the number of atoms/mols appears as a thermodynamic variable there must be at least two phases present.) and R is a fixed constant that is the same for all gases. This is the so-called ideal gas relation. The behaviour of many real gasses is better described using the following equation:
$\left( P + \frac{a}{V^2}\right)(V-b)=nRT$
where a and b are constants that depend on the particular type of gas.
<aside> 📌 SUMMARY: The number of independent thermodynamic variables is fixed at equilibrium. Consequently, for a system at equilibrium we can write functions that relate the values of the various thermodynamic variables.
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