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 $$\theta = \theta(p,V).$$