Thermodynamics and the state postulate: should it be a Fourth Law?

Question

The state postulate is as follows: The state of a simple compressible system is completely specified by two independent, intensive properties.

My first question is whether there is any justification for this postulate or whether it is simply an empirical law? That is, why is it not the case that 3 intensive variables are required to fix the state or even 4? If it is an empirical law with no justification and just happens to be a law our universe abides by, then why is it not one of the laws of thermodynamics as it clearly provides foundational value to the field of thermodynamics?

My second question pertains to how we can know whether 2 intensive variables are in fact independant. Say temperature $T$ and specific volume $v$ for example. If we are given a thermodynamic problem, how would we know whether these are independent or not? or perhaps specific internal energy $u$ and pressure $P$. Are there certain combinations that are simply always independent of each other or does independence between intensive variables vary from problem to problem and require a full understanding of the problem at hand? Let us take a phase change for example. From my understanding of phase changes, I know that the boiling temperature is dependent on the pressure and hence I won't be able to use $T$ and $P$ as 2 variables to completely specify the state. But how would I know know what combinations of intensive variables will be independent in this case or any other case for that matter? Does it simply require rote memorization of all independent pairs of intensive variables for all thermodynamic problems or is there a more simple way to figure out what is independent and what isn't?

I hope this question makes some sense and any help would be most appreciated!

Answer 1

My first question is whether there is any justification for this postulate or whether it is simply an empirical law?

The justification is in the unpacking of what the writer means by "simple compressible". I could write the fundamental relation for a thermodynamic system as $$dU=T\,dS+\sum_i\mu_i N_i-P\,dV+\sigma\,dA+E\,dp\cdots,$$ where $U$ is energy, $T$ is temperature, $S$ is entropy, $P$ is pressure (requiring a negative sign because pressure tends to reduce volume $V$), $\mu_i$ and $N_i$ are respectively the chemical potential and amount of species $i$, $\sigma$ is surface tension, $A$ is surface area, $E$ is the electric field, and $p$ is polarization. And I could further add conjugate pairs for magnetization, gravitational fields, shear stress, electrical charge, etc. The first term on the right ($T\,dS$) describes energy gained from heating, the second describes energy gained from adding matter or performing chemical reactions, and the rest describes energy gained from work.

But from the context ("two independent [natural variables]") I know that the writer is using "simple" to mean a closed single-species system (so that $i=1$ and $N$ is constant) and "compressible" to mean that only $P$–$V$ work will be considered. Now we have simply $$dU=T\,dS-P\,dV.$$ See also this response. The switching between extensive and intensive variables using the Legendre transform is discussed here.

Alternatively, one could write

• "The state of a closed, single-component, isentropic ($dS=0$) (or isobaric—constant volume—or isenergic—constant energy) compressible system is completely specified by one independent, intensive property." or

• "The state of a closed, single-component, compressible system that can be magnetized is completely specified by three independent, intensive properties." or

• "The state of a closed, single-component, compressible system that can be magnetized and has nonnegligibly changing surface area is completely specified by four independent, intensive properties."

It all comes back to how many terms you need to consider—or are willing to consider—on the right side of the fundamental relation.

My second question pertains to how we can know whether 2 intensive variables are in fact independ[e]nt... From my understanding of phase changes, I know that the boiling temperature is dependent on the pressure and hence I won't be able to use T and P as 2 variables to completely specify the state.

Correct. If you specify that the system is boiling, then you've applied a constraint that changes the number of independent variables. You may wish to read about the Gibbs phase rule and the Gibbs-Duhem relation as you continue to build familiarity in this area.

Answer 2

1. To your first question,

I think the state postulate is a derived consequence of the equilibrium-state postulate of thermodynamics. The latter includes the definition of thermodynamic equilibrium state and a postulate that any system will eventually reach such a state given constant environment condition. So, the state the system reaches is a result of the environment you control. You can choose to control any environmental conditions to be constant and wait for the system to reach its thermodynamic state under such conditions. And the property of the system at this state is the function of the variables of the environment condition you hold constant. There are many possibilities that one can further identify which of these variables are the "least", "independent" ones to set the state of the system. For example, a dimensional analysis can help us identify the "redundant" variables we tried to control in an experiment.

The finding that if a system is simple and compressible then no matter how you control the environment it's in its state depends on two independent intensive variables can be understood as an empirical law. It is not a foundational one because 1) the more general one should be the equilibrium-state postulate, and 2) the definition of "simple" and "incompressible" mentioned in this postulate require the conclusion derived from this postulate, becoming circulation definitions.

2. To your second question,

It is also empirical. For example, the state equation of ideal gas is summarized by several named laws in history (Boyle's, etc.), whose discoverers were all experimentalists. Without using statistical mechanics or even acknowledging atomism, state equations and constitutive equations can only be empirical. Particularly, based on expreiments one can find that the state equation of a system can be expressed as F(X1,X2, ..., XN)=0, so of N state variables only N-1 are independent.

3. To your third question (why favor intensive variables)

Because physicists tend to describe the scale-invariant law about the Nature. Laws about nature should be independent of the scale of the system described (in a certain range).