The Fundamental Relation of Thermodynamics

[avocadogirl]

What does it mean to say that the fundamental relation of thermodynamics is homogeneous first-order?

I struggle with the abstraction of mathematical definitions and, I'm really seeking more to understand the relation of the physical variables of thermodynamics.

Why are all of the variables interdependent by a linear relationship? Or, is it incorrect to say such a thing?

Is it because, you look at the change in every variable as a function of the changes of every other variable?

Please know, the time and assistance of all those who respond is deeply appreciated. Sincere thanks.

M.

 

[Monocles]

Homogeneous first-order is just a description of the differential equation. If you've taken differential equation you'll be familiar with what the ideas are (though you may have forgotten their names). If you haven't taken differential equations then it would behoove you to read the first chapter of a differential equations text.

 

[Count Iblis]

The physics behind this relation is explained quite well in this wiki article:

http://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation

which was mostly written by me.

 

[Ben Niehoff]

A function f(x,y) is first-order homogeneous if the following holds:

$$f(\lambda x, \lambda y) = \lambda f(x,y)$$

Linear functions are first-order homogeneous, but they are not the only possibility. Consider the function

$$f(x,y) = \sqrt{x^2 + y^2}$$

This function is also first-order homogeneous. You should be able to come up with additional examples.

The reason first-order homogeneous functions turn up in thermodynamics is that we choose to write the equations in terms of all the "extensive" thermodynamic quantities. Extensive quantities are additive, by definition. For example, when you double the volume of a system, the entropy and energy also double. This allows us to write

$$E = E(S, V)$$

where E is a first-order homogeneous function. Using this fact, the differential equation may be solved using Euler's theorem.

 

[cmos]

Edit: Ben Niehoff beat me to the punch. Everything I said he basically already said.

Homogeneous first-order is just a description of the differential equation. If you've taken differential equation you'll be familiar with what the ideas are (though you may have forgotten their names). If you haven't taken differential equations then it would behoove you to read the first chapter of a differential equations text.

Isn't it obnoxious when somebody, instead of giving you a quick answer plus possible references, they tell you to just go read a general reference? It's even worse when they are absolutely wrong.

The fundamental equation of thermodynamics is when you are able to get the internal energy (or the entropy) as a function of the other extensive parameters. Only then is ALL thermodynamics information about a system known. Mathematically:

U = U(S, V, N_i)

The fact that the fundamental equation is a homogeneous FUNCTION of first-order (different front a homogeneous differential equation of first-order) means that:

aU(S, V, N_i) = U(aS, aV, aN_i)

where a is some constant.

In general, a function is homogeneous of n-th order if:

aⁿf(x₁, x₂, ..., x_m) = f(aⁿx₁, aⁿx₂, ..., aⁿx_m)

Note that my definition of the fundamental equation differs from the definition of Count Iblis. What he calls the fundamental equation, I usually see termed as "the combined first and second law". The solution of this equation, however, yields the fundamental equation.

 

[Count Iblis]

The equation:

dU = T dS - P dV

is called the "fundamental thermodynamic relation" by F. Reif. The equation:

U = TS - P V + mu_1 N_1 + mu_2 N_2 + ..

is called the "fundamental equation" or "fundamental relation" in some books.


Thing is that dU = T dS - P dV is more fundamental than the other equation, because this is also valid (for fixed N_i and other generalized forces) when the system is not extensive. Suppose the system is completely described by S, V and N, then we have:

dU = T dS - P dV + mu dN

Then, as Ben Niehoff and cmos explain, U assumed to be homogeneous means that we assume that increasing S, V and N by a factor p will make U increase by a factor p. Euler's theorem, mentioned by Ben above implies that U(S, V, N) can be expressed as:

U(S, V, N) = T S - P V + mu N

So, when would this not be valid (apart from the trivial case when ther are other types of particles that have not been included)?
Consider e.g. a system of particles that interact with each other via long range interactions e.g. gravity. Then, combining two identical system would certainly not lead to a doubling of the internal energy.