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I read in a textbook (Mathematics for Physical Chemistry by D. McQuarrie) an example of Euler's Theorem used in Thermodynamics. The author states "The [variables] are all extensive [quantities], and so [Euler's theorem is applicable]" (I'll provide more details below)

My question: why do variables being extensive indicate that this theorem is relevant?

The theorem starts by stating that a function is homogeneous to degree $N$ in some set of variables if those variables always form terms such that their powers sum to $N$. For example:

$f(x,y) = yx^2 + y^2x + \frac{y^4}{x} + x^3$

that would be homogeneous degree 3 for $x$ and $y$.

The theorem concludes with:

If $f(\lambda x,\lambda y) = \lambda^N f(x,y)$

Then $Nf(x,y) = x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y}$

This is the wikipedia page for the same theorem.

The example uses the above theorem to derive:

$U = S(\frac{\partial U}{\partial S})_{_{V,n}}+V(\frac{\partial U}{\partial V})_{_{S,n}}+n(\frac{\partial U}{\partial n})_{_{S,V}}$

The author's solution immediately states: "The entropy, volume, and number of moles are all extensive thermodynamic quantities, and so $U(\lambda S,\lambda V,\lambda n) = \lambda U(S,V,n)$." After that statement the solution is simple.

I know that extensive variables means that they scale with the size of the system, but the relation to this theorem is unclear to me. I suppose that if the system is "twice as big" ($\lambda$ is doubled), then energy, entropy, volume, and amount are doubled. That still seems vague to me if its even a good way to think about this.

Thank you.

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It really is that simple. Stating that a thermodynamic system observes Euler's Theorem can be considered axiomatic if the geometry of the system is Cartesian: it reflects how extensive variables of the system scale with size. In a homogeneous system with a Cartesian geometry all extensive properties scale identically and so any extensive property can be written as a homogeneous function of degree 1 of the remaining properties.

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