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Why is the temperature of a sample of matter intensive?

I don't think it would be extensive, but I also don't see why it would be intensive, either.

It's a very basic and easy question, but I can't seem to rationalize why temperature is independent on the sample size. Nor is it dependent on the sample size. In fact, to me it seems that temp would be unrelated to the type of material, amount of material, and I don't even know if I would necessarily through temp into a "property of matter" so to speak.

This all started with this basic quiz question:

Which of the following are correctly referred to as “intensive properties”?
(i) density of a substance 
(ii) mass of a substance
(iii) temperature of a substance
(iv) boiling point of a substance
(v) volume of a substance 

  
(i) and (iv) - my answer

(ii) and (v)
  
(i) (iii) and (iv) - correct answer

(ii), (iii), and (v)
  
(i), (ii) and (iv)

I understand how Density and Boiling point are intensive, and I understand how Volume and Mass are both extensive, but what I am confused about is how exactly Temperature is intensive.

Because if the definition of an intensive property is simply that said property does not rely on the amount of the material, then I have a hard time understanding how temp is independent of the amount of material.

The way I always think of it is that if I had, for example, 30 grams of iron in a closed system, and then added 30 more grams of iron to it, the mass would increase, the volume would increase, the weight would increase, the number of mols would increase, etc (all extensive properties). Whereas the density, melting point, etc would remain constant, as they are just a property of the material itself, and can’t be altered unless the substance itself is changed (say it oxidized to iron oxide, then those properties would change).

;TL/DR

How would temperature be intensive, as the temperature can be changed by a plethora of factors, and has little to nothing to do with the quantity of mass of the substance itself (aside from the the specific heat, for example). Temperature is just the average kinetic energy of the particles in the material, so if 30 grams of iron had x amount of energy put into it to, which set the iron's temp to be 20 degrees C, and then 30 more grams of iron were added to the system spontaneously, the kinetic energy of the first 30 grams of iron would transfer to the second 30 grams, resulting an overall lower temperature, correct?

Alternatively, if the 30 grams of Iron were 20 C, and then 30 grams of iron at 50 C were added, the temperature would change and adjust to be 40 C on average, correct? Thus it would technically depend on the sample size, as when the sample increased, the temp increased also.

Not to mention that the temperature of any material isn’t an intrinsic property of the material itself, but is based on the system/circumstances surrounding the material, whereas intensive properties rely on nothing but the properties of the matter itself (such as the bonds between atoms, structure of any lattice formed when solid, polarity, and even how many subatomic particles make up the atom), while temperature is just a temporary state of the energy applied to the atoms/particles, causing them to move and creating a temperature.

Thanks in advance.

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Your major confusion may be thinking intensive property must be property of material.

There are 3 types of quantities, depending how they are related to system scalability:

  • Extensive properties are additive wrt(with respect to) the system scaling. If there is twice as big system, there is twice as big value of an extensive property. Examples are mass, volume, various forms of system energy.

  • Intensive properties are invariant wrt the system scaling. They may be intrinsic properties of system components, or may be state variables of the system conditions. Examples are density, concentration, refraction index, chemical potential, specific and molar heat capacity, temperature. Two times $\pu{1 L}$ of $\pu{20 ^{\circ}C}$ water gives $\pu{2 L}$ of $\pu{20 ^{\circ}C}$ water.

  • Properties, that are wrt system scaling neither additive neither invariant. Examples are system surface, heat conductance or ohmic conductance, as these depend on the system scaling geometry.

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The question was poorly designed with incorrect choices. Density is also intensive property. Density of 1 kg of water is same 0.001 gram of water so your answer is also correct.

I feel the source of your confusion is between heat and temperature. This concept has bewildered big physicists and philosophers in the 18- and 19-century, so no need to feel bad about this confusion.

Let us say you have 1 kg of water at and 5 gram of water in a container. You insert a thermometer, keep heating and you measure the temperature. When all water has converted into steam, you measure the temperature again. The themrometers registers 400 K in both containers. Which container has higher heat content, yet the temperature is the same? Temperature is the manifestation of the amount of heat, it is not heat itself.

Temperature indeed is a peculiar intensive quantity which is experimentally measurable. In thermodynamics, it is defined rather circulary. For example, Wikipedia say temperature is an intensive property because it is ratio of partial derivative of internal energy and partial derivative of entropy.

At a deeper level, what causes objects to have mass to begin with and what is charge itself? What causes molecules of continually move and never rest? There are certain quantities and observations which are not fully understood.

Your second example,

30 grams of Iron were 20 C, and then 30 grams of iron at 50 C were added, the temperature would change and adjust to be 40 C on average, correct? Thus it would technically depend on the sample size, as when the sample increased, the temp increased also.

This is again an example transfer of heat, this is a special case. If the masses of iron were different you have to do proper calculations, as discussed in calorimetry examples. Example

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