Is Intensive property/Intensive Property an Extensive Property?

Question

We know that Extensive property/Extensive Property is Intensive is most of the cases, but is Intensive/Intensive an Extensive property ? if so, is there any examples

Answer 1

See the other answer on intensive property

Let's think about functions.

Take two functions $f(x)$ and $g(x)$, which are intensive. Hence $f(x+x) = f(x)$ and $g(x+x) = g(x)$
And let's take fraction of them $$\frac{f(x)}{g(x)}$$ What you are asking is if I am changing to $x+x$, how will my $\frac{f(x+x)}{g(x+x)}$ change? $$\frac{f(x+x)}{g(x+x)} = \frac{f(x)}{g(x)}$$ They still don't change.

Answer 2

No, intensive divided by intensive is always intensive. Here is the math.

A function $f(x)$ is homogeneous with degree $\alpha$ if $$f(\lambda x) = \lambda^\alpha f(x)$$ for all $\lambda>0$. Intensive properties are homogeneous with degree $\alpha =0$, extensive properties are homogeneous with degree $\alpha=1$.

• Ratio of extensive properties: Define $h(x) = f(x)/g(x)$ where $f$ and $g$ are extensive: $$ h(\lambda x) = \frac{f(\lambda x)}{g(\lambda x)} = \frac{\lambda f(x)}{\lambda g(x)} = h(x) = \text{intensive} $$

• Ratio of intensive properties: Define $h(x) = f(x)/g(x)$ where $f$ and $g$ are intensive: $$ h(\lambda x) = \frac{f(\lambda x)}{g(\lambda x)} = \frac{f(x)}{g(x)} = h(x) = \text{intensive} $$ In both cases the ratio is intensive. Actually, we have proven a more general result: the ratio of homogeneous properties with the same degree of homogeneity is always intensive.