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The state postulate says that the state of a simple compressible system is completely specified by two independent, intensive properties. Two properties are independent if one property can be varied while the other one is held constant.

Most textbooks claim that temperature and specific volume are always independent, and therefore sufficient to fix the state of a system. But when the temperature changes wouldn't the volume(and the specific volume) of the system tend to change as well?

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  • Put an oxygen tank in the freezer; does it shrink?
  • Let all air out of a balloon (air volume expands); is the air hotter?

Temperature and volume are independent. But you might be confusing them with pressure. Because the pressure in the oxygen tank and on the balloon air will definitely decrease.

The ideal-gas-law tells the relationship (for idealized gas):

$$pV=nRT$$

$p$ being pressure, $V$ volume, $n$ amount of sumbstance (number of molecules in moles), $R$ the gas constant and $T$ temperature.

This is pressumably what the textbooks mean with their claim. $n$ is a constant number (you don't add or remove particles from the gas) and $R$ is a constant as well, so knowing just two of the remaining parameters ($T$ and $V$ or $T$ and $p$ or $p$ and $V$) let's you know the rest - in other words, knowing two means that you know all about that specific state.

Pressure $p$, volume $V$ and temperature $T$ are three independent parameters, but are in math said to be in a linear combination; knowing two will fix the last one, so any two constitute an independent linear combination. Therefore two are "sufficient to fix the state of a system".

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It depends on the system you are considering.

If you think of your system like a balloon, then yes -- when you increase the temperature, the volume will increase. In this system, the pressure does not change and so it is called a constant pressure system. Temperature and pressure are independent. So are pressure and volume.

On the other hand, think of a closed box made of huge slabs of steel welded together. If you increase the temperature in the box, there is no way the steel will move. The volume does not change, but the pressure does. This is a constant volume system. In this system, temperature and volume are independent. So are volume and pressure.

In any system, you just need two of the variables to define the third. Any two will work.

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