# Do we always have a choice of state variables, and how is temperature in general a state function?

I am reading Steane's fantastic Thermodynamics: A Complete Undergraduate Course and am a bit stumped as to the assertion being made that, for example, temperature is a state function.

Suppose I model my cup of water as a simple pV system. Fixing p = 1 atm, then it is a fact that "if a kilogram of water has a volume 1000.1 $$cm^3$$ at 1 atmosphere, its temperature could be either 2 or 7 degrees Celsius". How then is it fair to think of temperature as a state function? I have chosen to specify two DOF for my simple system (which has two independent DOF), and yet temperature is not uniquely specified. This seems to fly in the face of the argument Steane makes that temperature is uniquely specified by state. He seems to add that there is a choice of state variables (perhaps VT) which uniquely specifies the system, but I'm not sure I understand why that rescues T as a state function.

The direct quote from Steane about temperature as a state function is (here R is a reference system used to empirically define temperature by virtue of the state in which it is in equilibrium with the system being probed):

We have shown that every equilibrium state of every system has a temperature, defined by the value θ, that identifies which standard state of R it is in thermal equilibrium with. This temperature is a single-valued function of the state variables. For a pV system it can be written $$\theta = \theta(p,V).$$

• "its temperature could be either 2 or 7 degrees Celsius". How did they get these values? Dec 27, 2021 at 19:08
• @BobD From experiment, water < 4 C expands as it cools.
– Ben
Dec 27, 2021 at 19:09
• @Ben I'm questioning how they came up with two different temperatures for the same specific volume, not the fact that it expands < 4 C Dec 27, 2021 at 19:11
• @BobD The relationship between temperature and volume at constant pressure is a curve with a minimum at 4 degrees. If you only know the volume, there are two possible temperatures. If you know the temperature, there is only one possible volume.
– Ben
Dec 27, 2021 at 19:12
• @Ben First of all, you need to use the specific volume since it is the specification of any two intensive properties. Secondly, if you use the Engineering Toolbox on line calculator for 2 C and 7 C you get two different specific volumes. Dec 27, 2021 at 19:15

Water is an unusual substance.

Most materials expand as they get warmer and contract as they get cooler. This means that at higher temperatures either the volume increases, or the pressure increases, so that knowing any two of the temperature, pressure, and volume, will allow you to calculate the other.

Water is unusual because below 4 degrees Celsius it expands. This means that you can no longer use pressure and volume to derive temperature; instead temperature becomes compulsory.

With the temperature and pressure you can get the volume, and with the temperature and volume you can get the pressure, but with the pressure and volume, you may not be able to get a unique value for the temperature.

This is the sense in which temperature is a state variable. It is the one you can't do without.

Now you may well ask, "why does the textbook say For a pV system it can be written $$θ=θ(p,V)$$."

I expect the answer is that "water is not a well-behaved $$pV$$ system under 4 degrees" and the textbook was drawing your attention to the fact that not every system is a $$pV$$ system.

In physics, in engineering, and in life in general, things are well-behaved and well-understood within limits. Outside those limits, the regular equations we learn stop working.

• "but with the pressure and volume, you may not be able to get a unique value for the temperature." I agree with that. But with the same pressure and specific volume you will have a unique temperature. Dec 27, 2021 at 20:06
• @BobD I am sure the FE exam advice is adequate for engineering because you don't often make things out of water. In physics we have to accept that nothing we know is the absolute truth, there are only more accurate theories.
– Ben
Dec 27, 2021 at 20:11
• Forget the FE exam reference. Did you check out the website I gave you? Do you dispute their data? Dec 27, 2021 at 20:15
• @BobD Engineering toolbox gives specific volume of 1.0001 at 0.8 C and same at 6.5 C. This disagrees with the textbook quote but makes the same point, it does not uniquely determine the temperature from the volume and pressure.
– Ben
Dec 27, 2021 at 20:17
• Point made, I hadn't noticed that. I stand by the FE exam definition specifying "intensive" properties, but clearly water is an exception. In any event, a related post is here physics.stackexchange.com/questions/133917/… Dec 27, 2021 at 20:38