Euler's Theorem and Extensive properties

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

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.

• – MaxW
Jan 14, 2020 at 23:12
• Thanks, this part was especially relevant: "In fact, any extensive thermodynamic quantity must be a homogeneous function of degree 1 in its extensive arguments." Jan 16, 2020 at 0:36