State Variables

We are all familiar with the concept of temperature. The Zeroth Law of Thermodynamics formalizes our intuition and experience as follows:
“If a system A is in equilibrium with system B (that is, has no exchange of heat with it), and if system B is in equilibrium with system C, then A is in equilibrium with C”.
This law allows us to associate a quantity called temperature to each system in thermal equilibrium, so that two systems in equilibrium have the same temperature. The thermometer is a device that uses the Zeroth Law in a quantitative and practical way. In addition to the temperature, one may need more thermodynamic parameters, called state variables, to completely characterize the state of the system. For example, for a gas these are the pressure $P$ and volume $V$. Variables such as temperature and pressure that are independent of the size of the system are called intensive, while those such as the volume are called extensive. The parameters that can be used to describe a system are not all independent but related by an equation of state. Recall that real gases at moderate pressures and temperatures obey universal gas laws, the Boyle's law and the Gay-Lussac-Charles law, which can be summarised as saying that $PV = a(\theta + \theta _0)$ where $a$ and $\theta_0$ are constants and $\theta$ is the temperature in the Celcius scale where $0^{\circ}$ is the freezing point of water and $100^{\circ}$ is the boiling point. An extrapolation of the empirical curve to the limit $PV=0$, which is allowable for an ideal gas, gives the intercept $\theta_0 =273.15$ and so it is convenient to define an absolute temperature scale $T$, measured in Kelvins $K$, by

$$T = \theta + 273.15 .$$ (4.1)

to simplify the ideal gas law to

$$PV =NkT ,$$ (4.2)

where $P$ is the pressure, $V$ the volume, $k = 1.38 \times 10^{-23} J/K$ is the Boltzmann constant and $N$ the number of atoms/molecules in the gas. The coldest temperature one can attain is $0 K$, absolute zero, where all molecular motion would cease. That is, an ideal gas is the universal limiting description of real gases when their density is very low and the temperature high. In general, the equation of state of a real substance is more complicated. It is usual to plot the equation of state as a function of its parameters. One useful curve follows by keeping $V$ constant and representing the equation of state on a $P-T$ plot as shown in the figure for a generic substance. The lines mark boundaries between the different phases of the substance, where changes occur in the physical properties of the substance.