By Kent W. Mayhew

Work vs Energy of a Gaseous System

The only systems that can actually perform appreciable work are gaseous ones. Accordingly
we shall limit our discussion to such system.

The energy of a monatomic gas is a result of its translational plus rotational energy. Note this differs from traditional thermodynamics wherein monatomic gases are illogically considered not to have rotational energy. So strange, it is like saying a baseball has no rotational energy and then trying to explain the curve ball. All so that their theory based upon the mathematical conjecture of degrees of freedom matches empirical data. Okay let us leave the ridiculous and get back to work vs energy.

It should also be note that for polyatomic gases it is this author's belief that it is mainly the rotational and translational energies of a gas that allows that gas to do work. In this way of thinking the vibrational energy of a polyatomic gas

The
total energy of a N molecule monatomic gas is:

Etotal=3NkT/2 eqn 1)

The
ability of a gas to do work is:

Wability = NkT eqn 2)

The ratio of Wability/Etotal defines
the maximum efficiency of a gas, wherein the efficiency is really how much work it can do. The Maximum efficiency is:

MaxEfficiency = NkT/(3NkT/2) = 2/3 = 66.67% eqn 3)

One
may ask why can a gas not use all of its energy to perform work. It may seem odd that a system of gas cannot convert all of the energy
into work. But its not! The 66.67% upper limit to the efficiency exists because not all of the system’s gaseous molecules will be
able to contribute all of their momentum to the system’s expansion.

One must realize the following:

1) Work involves the movement of
mass in a unique clearly defined direction i.e. along the positive *z*-axis or if you prefer the movement of a mass whose surface area
in the* x-y* plane feels the force, resulting in the movement being along the *z*-axis. The actual molecular flux that strike the *x*-*y*plane is defined by the following eqn 4): Flux = (1/4)nv (see Reif). This being the flux of molecules that can actually contribute
their energy to work. Note v is mean velocity and n = N/V where N is total number of molecules and V is the volume.

2) An enclosed
gas’s translational plus rotational energy is due to the energy obtained from interactions with the surrounding walls, and this energy
is the summation of the energies from the three orthogonal walls. Accordingly, it as if the energy flux was a summation of energy
from all six surrounding walls such that the flux of energy from each wall was proportional . Value used in kinetic theory eqn 5):Flux
=(1/6)nv. See Kinetic Theory

Accordingly the energy of system does not equal the ability
of that system to do work. The upper limit of a gas’s ability to do work becomes: (1/4)nv/(1/6)nv = 2/3 i.e. 66.67%, of the gas’s
translational plus rotational energy. And remember that this is for monatomic gases. The majority of gases are not be ideal monatomic
gases hence also have vibrational energy, hence have even lower efficiencies.

We can look at this another way. When we heat a
gaseous, all the molecules within that system will experience an increase in kinetic (translational plus rotational) and vibrational
energies. Now the 2/3 only concerns itself with the kinetic energy of that gas, hence is an upper limit. Moreover, all the gaseous
molecules cannot impart all of their increased momentum onto the systems walls during the expansion process.

1) Reif pg 271 : Fundamentals of Statistical and Thermal Physics”, F. Reif, McGraw-Hill, New York, 1965

To see Reif's calculation of
flux click here