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# Equations of State (2013)

last edited by 6 years, 1 month ago

Working ContentThermodynamics and Statistical Physics

We have learned about many quantities in thermodynamics and statistical physics.

For example, you can characterize a gas by a few variables:

N  - the number of atoms or molecules in the gas.

T -  the temperature of the gas.

p - the pressure the gas exerts on the boundary.

V - the volume of the gas.

U - the internal energy of the gas.

v - the characteristic speed of atoms/molecules in the gas.

Our intuition may tell us that we cannot set all these variables independent of each other.  In fact, in earlier sections of this chapter on Thermodynamics and Statistical Physics we have covered a number of equations that relate these variables to each other. Indeed this is a hallmark of thermodynamics  - a system of a large number of atoms/molecules can be characterized by multiple, interconnected variables.  Since the variables are dependent on each other, we do not have to measure them all at the same time to fully characterize a gas.

Examples of equations that relate the variables:

• When thinking about pressure generated by a gas, we recall that pressure is generated by the collision of atoms/molecules with the wall so pressure depends on the number of molecules in a given volume and the speed of these atoms/molecules.  See the wiki page on the microscopic viewpoint on the ideal gas law.
• We also recall that the speed of atoms/molecules in a gas is related to temperature 3/2 kbT=1/2mv2
• We all should remember the ideal gas law that also relates these quantities:  pV=NkbT

How many variables do we have to measure in a gas to characterize the system?

Let's look at an example.  If we measure T, p in a gas and fix N, we can deduce the other two from equations

In an ideal gas we have three degrees of freedom per atom/molecule so, U=3/2 NkbT and from the ideal gas law we get  V=NkbT/p

In other words U and V can be calculated if we know N, T and p

The equations that yield us the missing variable are called the "Equations of State" in thermodynamics,  since they allow you to characterize the state of the system.  (Later, we will call this the macro-state to distinguish it from the micro-state, which involves much more detailed information).  As you can see from the example above, the equations of state require us to have a model for our system - in this case the model is the "ideal gas", which is a good model for simple gases.

Why is it useful to understand that a limited number of variables suffices to characterize the system?

As we add more quantities and variables  it is important to keep in mind that they are related to the more familiar ones in a way determined by the internal properties of the system under study.
Let's say I sum up and multiply the variables that characterize the gas and make up a new variable.   For example, let's invent a variable H, call that enthalpy, and say H=U+PV.

That new variable does not characterize the system any more or any less than any of the other variables.  The new variable must in some way be a function of the old variables.  However, if I know H, I do not need to know one of the other variables.

If three variables are enough (for an ideal gas) to characterize the state of the system, why would anyone add more variables in thermodynamics?

The additional variables come in very handy when we want to characterize a PROCESS where the system changes.  Examples are exchange of heat between two gases, mixing of two gases, or compression of a gas, or some combination of these processes.

We could describe adding energy to a system at constant pressure and temperature using the basic list of variables that you see at the top of this page.  However, when energy is added, the volume V and internal energy U both must change in such a way as to keep pressure and temperature constant.  This is where enthalpy is a useful variable to consider.

Tom Antonsen and Wolfgang Losert 2/15/2014