# Clarification regarding Homogenous systems in Thermodynamics

So i have just started with Classical Thermodynamics..and i am reading Thermodynamics by Bruno LinderSo I could not understand what he means in this portion-

If intensive properties are uniform throughout the system or they are continuous (as air in a gravitational field) then it is called a homogenous system..

So i dont understand when he says continuous.

What does it mean for an intensive variable to be continuous?

If it means that it is a continuous function of something..in that case what is that something ?

Any sort of help will be great Thanks in advance

• A continuous function of position within the system... Mar 27, 2016 at 15:12
• @lemon can you tell how can we define an intensive prop. Say pressure at a point (or a position) ...within the system Mar 27, 2016 at 15:15

Imagine dividing your system into a grid of cells, each with a volume $\delta V$, and pick a single cell centred on $x$. Now multiply that cell an infinite number of times in all directions, like so:
The standard thermodynamic definition of whatever intensive variable you're interested in (e.g. pressure, temperature, density, concentration, etc) can be applied approximately to the cell centred on $x$ within the context of this infinite ensemble.
If the property varies smoothly (continuously) over space, then you can take the limit $\delta V\to 0$ and get a well-defined scalar field. Note that a continuous property is locally uniform and may therefore be treated as homogeneous.
• @HiteshPathak Sort of. We do assign a variable to each individual point, but it's interpreted to mean its value within an infinitesimal volume $dV$ centred on that point. A bit like the wave function in quantum mechanics, say. Mar 29, 2016 at 7:34