Intensive and extensive properties.

Let us ask a simple question. What is matter?

You will come to realize, this is actually a hard question. Some of the hardest questions in science are often the simplest ones —and a quick search online yields the following definition: ”matter is any substance that has mass and takes up space by having volume.”[1]

We can ask ourselves another simple yet hard question. Why does matter have mass?

Well…this is not known —It is not a thing a physicist can easily define without some major assumptions about the real world. The definition of these two concepts—mass and matter—appears to be circular. Mass is a measure of the amount of matter in an object, and matter refers to any “substance” that occupies space and has mass. Weird questions arise from this circular definition. For example, can there exist matter that has no mass? what about infinite mass? Clearly, everything else we observe falls somewhere between those two. Our problem is, how can we define it in a way we can say something about it, mathematically.

In physics, a formal definition of a property is a mathematical equation that expresses the property based on other properties previously introduced through direct empirical evidence.

So, empirically and with much confidence, we can say:

Matter is a property of nature.

Properties of matter

Matter is an assumption—something we cannot really define but know we can rely on. Built into nature itself, like the concepts of mass, time, and length. All these “measurements” are the empirical evidence we seek. Each of these concepts is better explored through their properties. Through things we can reliably measure. So, let us explore the properties of matter rather than attempting an elusive definition. That is far more useful because, ultimately, physics is meant to predict things.

A better approach to find a good definition is to look outward, into nature.

We live in a three-dimensional world, and to talk about it, we use the famous 'x, y, z' coordinate system to denote the concept of 'length.' We assume length to mean a unit in any one of the axes in our coordinate system. I assume you understand what is a coordinate system. Using that definition, we can define volume as a measure of mass bounded by a region in three-dimensional space. With those definitions in mind, let us see what properties of matter we can derive from first principles.

Interesting things happen when we think about the properties of matter in terms of mass. We can intuit that some properties must be affected by the mass of the object and naturally, some must not. We call those intensive and extensive properties.

Intensive and extensive properties

The properties of matter that depend on the amount of mass present in an object are called extensive properties. Properties that do not depend on the amount of mass are called intensive properties.

[2]

I like to think of it this way, “In-tensive,” from “with-in” matter —properties that do not change with mass. Extensive, the opposite.

The terms "intensive and extensive quantities" were introduced into physics by German mathematician Georg Helm in 1898, and by American physicist and chemist Richard C. Tolman in 1917. [3]

Below is a list of some physical properties categorized between intensive and extensive properties. We will analyze some of them together and try to see what we find.

Extensive Properties:

• Mass
• Volume
• Energy
• Entropy
• Internal energy
• Enthalpy
• Gibbs free energy
• Amount of substance (moles)
• Electric charge
• Total heat capacity
• Momentum
• Kinetic energy
• Potential energy
• Surface area

Intensive Properties:

• Temperature
• Pressure
• Density
• Color
• Melting point
• Boiling point
• Refractive index
• Specific heat capacity
• Electrical conductivity
• Viscosity
• Hardness
• Magnetic susceptibility
• Heat of fusion
• Heat of vaporization
• Concentration (e.g., molarity, molality)

Let us take a quick glance at one of the intensive properties —density. From your high school physics memories, you might recall the formula…

density = mass / volume

Note that both mass and volume are extensive properties, yet density is not. As it turns out, any ratio between two extensive properties is an intensive property. Let's prove it because it is a surprising conclusion to make. Let me phrase the problem slightly differently:

If A and B are two extensive properties that change proportionately with respect to each other, then the ratio A / B is an intensive property.

Proof:

Let A and B be two extensive properties, and let k be a constant of proportionality such that A=kB We want to show that A/B is independent of the quantity of the substance, k.

Consider a system with n identical subsystems, each having A_n units of property A and B_n units of property B. For each subsystem, we have A_n=kB_n.

Now, for the entire system: A_total=A_1+A_2+…+A_n=k(B_1+B_2+…+B_n)

Divide both sides by the total amount of property B in the system (B_total=B_1+B_2+…+B_n):

A_total / B_total= k

This ratio is independent of the number of subsystems or the total amount of substance. Thus, A / B is a constant for the entire system. ◼️

The ratio A / B is an intensive property because it remains constant regardless of the quantity of the substance, given that A and B change proportionately as mass scales up or down. Speaking of scale, the thing we have not considered yet is the idea of smallness. If we shrink matter down enough, do quarks, atoms, or molecules have a say on its properties? Do they influence the macroscopic behavior of matter?

Well, what we have been, so far, calling an “object” or “system” or “substance” is a ridiculously large collection of individual elemental particles. Luckily for us, the interaction between small particles, like molecules, is so frequent that the physical properties of matter do not depend on individual interactions. [4] I find this to be a beautifully elegant and counterintuitive way nature keeps itself simple. Though, different atoms act differently because they have different mass.

Yet another thing we have not considered yet, and we might have wrongly assumed is, could there be anything in between the two? do properties that may or may not depend on mass exist?

Yes! The answer is yes.

Once matter gets complex enough, we have to start thinking of a new kind of emergent properties. Mechanical properties, which could be intensive or extensive. For example:

• Malleability
• Ductility
• Elasticity
• Rigidity

What we are getting at here is that scale does play a role. The classification of mechanical properties, such as intensive or extensive, may depend on the specific context in which matter is being analyzed. Let us take a closer look at some of these mechanical properties.

Malleability and Ductility:

1. Example of Malleability:
   • Consider a sheet of gold
   • Scenario 1 (Intensive): If the malleability of gold is examined at the atomic or molecular level, this property can be considered intensive, as the ability of gold atoms to deform does not depend on the amount of gold present.
   • Scenario 2 (Extensive): However, if a considerable amount of gold sheets are taken and their malleability is evaluated collectively, the ease with which they can be deformed could depend on the total amount of gold present.
2. Example of Ductility:
   • Consider a copper wire.
   • Scenario 1 (Intensive): At a microscopic level, the ductility of copper can be considered intensive, as it is related to the ability of copper atoms to form elongated structures.
   • Scenario 2 (Extensive): If a significant set of copper wires is taken and their ductility is evaluated collectively, the ease of stretching the wires could depend on the total amount of copper present.

Elasticity and Stiffness:

1. Example of Elasticity:
   • Consider a steel spring
   • Scenario 1 (Intensive): The elasticity of steel at a microscopic level can be considered intensive, as it is related to the ability of the bonds between steel atoms to recover after deformation.
   • Scenario 2 (Extensive): When considering a set of steel springs , tough, and evaluating their elasticity collectively, the resistance to elastic deformation could depend on the total amount of steel present.
2. Example of Stiffness:
   • Consider a wooden beam
   • Scenario 1 (Intensive): The stiffness of wood at a microscopic level can be considered intensive, related to the structure of the wood's cellular components.
   • Scenario 2 (Extensive): When considering multiple wooden beams and evaluating their stiffness collectively, the resistance to rigid deformation could depend on the total amount of wood present.

These examples illustrate how the interpretation of mechanical properties as intensive or extensive can vary depending on the study approach and scale level.

In summary, the exploration of the nature of matter has led us to appreciate its inherent complexity and the challenges in providing definitive definitions for concepts like mass. By focusing on properties rather than elusive definitions, we have uncovered the distinction between intensive and extensive properties—essential classifications that help us understand how matter behaves in our three-dimensional world.

References

1: R. Penrose (1991). "The mass of the classical vacuum". In S. Saunders; H.R. Brown (eds.). The Philosophy of Vacuum. Oxford University Press. pp. 21–26. ISBN 978-0-19-824449-3.

2: Tolman, Richard C. (1917). "The Measurable Quantities of Physics". Phys. Rev. 9 (3): 237–253.

3: Vendatu (2023) “Difference between Intensive and Extensive Properties.” www.vedantu.com/jee-main/chemistry-difference-between-intensive-and-extensive-properties.

4: ChatGPT (2024) “Properties Categorized: Intensive vs. Extensive”.

5: Yan, Claire Yu (2022). “1.3 Extensive and Intensive Properties.” Introduction to Engineering Thermodynamics, BC Campus, pressbooks.bccampus.ca/thermo1/chapter/extensive-and-intensive-properties/.

6: Seurat, Georges "Seascape at Port-en-Bessin, Normandy.” 1888, National Gallery of Art, Washington, D.C.