Work vs Energy (of a gaseous system)

 Writing the first law, in terms of heat into the system (dQ) and changes to a system’s internal energy (dE) plus any work done by that system (W) gives:

                       dQin = dEsys + Wdone     (1a)

   And if the work is done onto the surrounding atmosphere [lost work = (PdV)atm] plus work done onto something else (Welse):

                        dQin = dEsys + (PdV)atm + Welse      (1b)

  Work is done by a system at a higher pressure onto a system at lower pressure. This can be an isothermal process if both systems are in thermal contact with the surrounding atmosphere (or one of the systems is the atmosphere) and the process occurs quasi-statically i.e. slow enough that sufficient heat can flow into or out of, the systems thus all systems remain isothermal. (Understand: There is natural relation between temperature and pressure, which is often overlooked when processes are isothermal).

 Note: For a discussion on expanding piston cylinder click here

   Systems that can actually perform work tend to be realtively high pressure gaseous systems. This differs from heat transfer which occurs between systems in thermal contact with real temperature differences. In other words work is pressure driven, while energy transfer is temperature driven. Although inherently obvious once stated, it is not so obvious when writing eqn (1a). Things become more obvious if we now rewrite eqn (1b) in terms of isometric heat capacity (Cv) and temperature change (dT), i.e.:

                       dQin = (CvdT)sys+ (PdV)atm      (2a)

   Again for emphasis, as discussed in First Law blog. We could look at eqn (2a) this way. Consider System 1 receives thermal energy (dQin) from an external source and expands hence does work onto its surroundings atmosphere. The change to a System 1’s internal energy change (whether dE or CvdT) is the change to the thermal energy within the System 1, while PdV is the work done externally onto the surroundings atmosphere, as defined by the atmosphere’s mechanical parameters.

 For blatant clarity eqn (2a) can be rewritten as follows:

                    dQ(into System 1) =CvdT (inside of System 1) + PdV (to surrounding atmosphere)      (2b)

  However the difference between work and energy goes beyond energy involving the thermal parameter temperature (T) inside of a system and work involving the mechanical parameters (P,V) done by that system. It must be emphasized that this work is done onto the system's surroundings hence CANNOT be expressed in terms of the expanding system's parameters! see parameters.

   The energy of a monatomic gas is a result of its translational plus rotational energy, both of which can perform work. Now the same does not necessarily always apply to a gas’s vibrational 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. See kinetic theory. Okay let us leave the ridiculous and get back to work vs energy.

  It should also be noted that for polyatomic gases it is most likely that it is the rotational plus translational energies of a gas that allows that gas to do work. In this way of thinking the vibrational energy of a polyatomic gas doesn’t 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).  

 The total energy of a N molecule monatomic gas is:

                    Etotal=3NkT/2   (3)

 The ability of a gas to do work is:

                    Wability = NkT   (4)

  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%      (5)

   One may ask why a gas cannot use all of its energy to perform work? 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, while a gas’s energy has no such sense of direction.

If you prefer work involves the movement of a mass whose surface area in the x-y plane experiences a force, resulting in the mass’s movement along the z-axis. The actual molecular flux that strike the x-y plane is defined by eqn B.7.14: Flux = (1/4)nv  (see Reif) or see download a copy from my book. This being the flux of molecules that can actually contribute their energy to work.  Note herin v is the 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: Flux =(1/6)nv. 

Remember it is the gas’s translational and rotational energies that can pass a net momentum onto a mass, hence invoking an acceleration of that mass i.e. actually do work. Hence the efficiency of 66.7% may apply to monatomic gases and should decrease as gas molecules get bigger)       

  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.

   Interestingly, one might say that enthalpy (H=E+PV) is sort of a measure of a system's ability to do work. However this could be troublesome as enthalpy is really 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 system’s energy term (E) and a work term (PV). And yes often reactions do work which involves the upward displacement of our atmosphere i.e. lost work: W=PatmdV. Accordingly how we rewrite physical chemistry will need some thought and I would love to find someone in that field who can actually help. see physical chemistry!  

   Remember dH=d(E+PV) only makes sense when one considers the expanding isobaric system case that being dH=dE+PdV (see Enthalpy), because dE is the change to systems energy (summation of all microscopic changes if you prefer) while PdV is the lost work done onto the surrounding atmosphere. And the atmosphere cannot do work onto any reaction when that reaction is in an environment/system that is isobaric to the atmosphere! This is no different than the atmosphere cannot transfer thermal energy to an environment/system that is isothermal to the atmosphere!    

   It must be emphasized that since not all of a system's energy cannot be used for work is another reason 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 is considered in my book.

  For those of you who have traditional theory enshrined in your hearts and minds, hence do not abide by what was discussed in this blog, I must now ask you this. In traditional kinetic theory, the energy passed onto the wall molecule’s by a gas molecule is wrongly derived using 1/6 of the gas’s total flux instead of ¼. And to you this is fine because you then attain your desired result, is this right? (to see the flux calculation from Rif click on box in right column)

   I remember reading the above in a book (probably Reif) how the approximation of one-sixth was close to one-quarter, thus things are fine with the traditional approximation (wrongly based upon 1/6), which then forms the basis of the science. All I have to say is WTF.