The Ideal Gas

Since thermodynamics is a phenomenological description of macroscopic matter, it does not take into account the underlying atomic structure. The thermodynamic relations, such as the ideal gas law, are therefore often approximations that are good when one is talking about the average properties of a large system, that is, a large number of atoms. Indeed the various thermodynamic relations can be considered as examples of emergent laws: generalities about the system that are apparent only at the macroscopic scale but are not obvious or existent at the microscopic level. Indeed even the concept of temperature may be considered a macroscopic emergent feature that is ill-defined for a system with only a few atoms. The above discussion can be illustrated for the case of the equation of state of an ideal gas that allows a simple derivation starting from an atomistic description. Consider atoms of a gas confined to a box which has a frictionless and perfectly reflecting piston at one end. Let the area of the piston be $A$ the volume of the box $V$ (see figure), and the number of atoms be $N$. As the atoms rebound from the piston, they will impart momentum to it and cause it to move. To prevent the piston from moving one must therefore apply a force to it to balance that imparted by the atoms. Let this external balancing force be $F$. We now calculate the force $F$ in terms of the parameters of the gas. Now Newtons law states that

$$F = \mbox{rate of change of momentum},$$ (4.2)

Hence in a short time interval $dt$,

$$\mbox{Force} \times dt = \mbox{change of momentum}.$$ (4.3)

Consider an atom of mass $m$ and velocity $v$, with $x$-component $v_x$ hitting the piston. It rebounds with an $x$-component $- v_x$, and so the change in momentum of the atom is $- 2mv_x$. Now for atoms with a velocity $v_x$ to hit the piston in the time interval $dt$, they must be a distance $v_x dt$ away. Assuming that the atoms are uniformly distributed, then there are $n=N/V$ atoms per unit volume and so $n A v_x dt$ atoms that will hit the piston in that time interval. Hence the net change in momentum is $- 2 n m A v_{x}^{2} dt$, and so the pressure $F/A$ imparted by the piston on the gas is $- 2 n m v_{x}^{2}$ and so by Newtons Third Law the pressure of the gas is

$$P= 2 n m v_{x}^{2} .$$ (4.4)

Since the atoms in the gas have different velocities, the above result must be averaged. Let $<v_{x}^{2}>$ be the average of the squared $x$-component of the velocities. This includes averages over both positive and negative velocities and so taking half of this (to represent atoms moving towards the piston) and substituting into the above equation we get

$$P= n m <v_{x}^{2}> .$$ (4.5)

Now

$$v^2 = v_{x}^{2} + v_{y}^2 + v_{z}^2 ,$$ (4.6)

and so

$$<v^2> = <v_{x}^2> + <v_{y}^2> + <v_{z}^2> .$$ (4.7)

But on the average the motion in the three directions is equivalent and so

$$<v_{x}^{2}> = <v_{y}^{2}> = <v_{z}^{2}> ,$$ (4.8)

and hence

$$PV = {2N \over 3} < m v^2 /2> .$$ (4.9)

For a monatomic gas $<mv^2/2>$ is the average kinetic energy of an atom and so the quantity $N <m v^2 /2>$ represents the total internal energy $U$ of the gas. So

$$P V = {2 \over 3} U$$ (4.10)

for a monatomic gas. The above is an elementary kinetic theory derivation of an ideal gas law. Note however that temperature is absent from the equation. On comparing the theoretical result (4.9) with the empirical equation (4.1), a relation is obtained between the macroscopic concept of temperature and the mean microscopic kinetic energy

$$<m v^2 /2> = { 3 \over 2} kT.$$ (4.11)

Thus at least in this simple case one has succeeded in deriving an empirical law (the ideal gas law) for a large system starting from the underlying microscopic dynamics. The derivation is complete only after associating some of the emergent macroscopic parameters with averages of microscopic quantities. In more complicated situations, especially in the case of out- of- equilibrium complex systems as defined in the introduction, one will not be able to make such simple or direct associations between emergent laws and patterns and the underlying microscopic theory. Note: A more realistic equation for gases, valid at higher densities is the Van der Waals equation of state $(V - b) (P + a/V^2) =RT$ where $a$ and $b$ are constants characteristic of the substance studied.