A New Thermodynamics

Blog: Work vs Energy of a System

By Kent W. Mayhew


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  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

does not readily contribute to work, in which case the vibrational energy is part of the gas's internal energy while the rotational and translational components contribute to the mechanical parameters P, V.


See internal energy  


 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 systemís gaseous molecules will be able to contribute all of their momentum to the systemís 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-yplane 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 gasís 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 gasís ability to do work becomes: (1/4)nv/(1/6)nv = 2/3 i.e. 66.67%, of the gasís 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.


   Note based upon eqn 3) we might say that enthalpy is sort of a maesure of a system's ability to do work.

However this could be troublesome as enthalpy is reallyy used in physical chemistry for chemical reactions and in that context there could be some confusion. One must remember that the enthalpy relation has an energy term (E) and a work term (PV). i.e. H=E+PV, and yes often reactions do work which is generally the upward disp[lcement of our atmosphere in which case W=PatmdV. Accordingly how we rewrite physical chemistrey will need some thought and I would love to find someone in that field who can.  


   It must be emphasized that since not all of a system's energy can be used for work means that no mechanical device driven by gaseous system can ever be 100% efficient. 


This all also helps explain things like Helmholtz free energy but that shall be left for you to ponder and will be considered in my future book.


Copyright Kent W. Mayhew


References used herein


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

To see Reif's calculation of flux    click here   

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