A New Thermodynamics

By Kent W. Mayhew

Implications to First Law       By Kent W. Mayhew      


 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:


                       dQsys = dEsys + Watm     (1a)


   And if the only work that is done by an expanding system is the work done onto the surrounding atmosphere i.e. lost work = PdV, then there is no path dependence hence eqn (1a) becomes:


dQsys = dEsys + (PdV)atm = dEsys + PatmdVsys     (1b)


    If the work that is done is lost work plus something else (Welse) then  1(b) becomes:


                       dQsys = dEsys + (PdV)atm + Welse     (1c)


Of course if the work done on something else is path dependent then we would need to consider infinitesimal work:

                    dQsys = dEsys + (PdV)atm + dwelse     (1d)


  If you do not understand why it is wrong to write the above solely in terms of the expanding systems parameter

then see parameters


As discussed in the work vs energy blog: Things become more obvious if we now rewrite eqn (1d) in terms of isometric heat capacity (Cv) and temperature change (dT), i.e.:


                       dQ=CvdT+ PdV+Welse      (2a)


    We could look at eqn (2a) this way. Consider System 1 receives thermal energy (dQ) 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)  +Welse   (2b)


 Or another way of writing the First law

Consider that we are extracting energy out of a system. The first law of thermodynamics is traditionally written7,8 in terms extracted energy (Qout), internal energy change (dE) and work done (W):

      Qout = dE(internal) – W               eqn 3a)

   Herein Q is the total energy into the system and W is the total work done external to the system. If we consider that the internal energy change (dE) is the summation of all of the energy changes within a system (dEtot). Then a less confusing way to rewrite eqn 3) is:  


      Qout = dE(tot) – W                     eqn 3b)

Sure eqn 3a) and 3b) are similar, but eqn 3b) gives clarity in that it states the total (whole) system’s energy changes.

Or yet another way of writing

    Now consider that energy is being put into a system, then in terms of thermal energy into  a system (Ein), the first law is written:

No matter how we choose to write it, the first law states that “0energy is conserved”.

In writing the first law W is the total work done by the system.

The first law should never be written in terms of work do onto the system because that allows for unnecessary confusion, yet often you may find it written that way! Think about it, if work is done onto a system in question, then that work may, or may not, result in an energy change to that system! And any energy change to that system is part of system’s total energy change, whether you choose to write it i.e. dE, dE(internal) or dE(tot)  

Never forget that if work is limited to the displacement of our atmosphere by the some expanding system, then that work is lost work:

    W = PdV = P(atm)dV       (4)

Never forget: The atmosphere can only do work onto a system if it’s pressure is significantly greater than the pressure within that system. And in real life this only occurs when negative work was done in creating that system in the first place.  

   If an expanding system also moves an object with mass, does work onto a car, then the total work done is always:

   W = PdV + W(car) = P(atm)dV + W(car)     (5)

If you include the work done onto the car (or anything else) in the PdV term then you are only asking for undue confusion, as can be to often witnessed in tradiational writings of thermodynamics.

In some ways we have done nothing but cut hairs as often the total energy change of a system may arguably be equal to the change of the system’s internal but this may depend upon one’s definition of internal energy. So why not keep things simple and write the first law in terms of the system’s total energy change. It avoids confusion and it tells us that the work done is done to system’s surroundings and/or devices attached to the system’s exterior.

One needs to realize that a gaseous system’s thermal energy and work are never equal. Remember is work defined by the mechanical parameter P & V. While a system’s energy is defined by its temperature See work vs energy.

    Furthermore work is most often extracted from gases. Specifically work can be extracted from a gas’s translational and rotational energy but not necessarily its vibrational energy.  See kinetic theory






   Copyright Kent W. Mayhew

   (see how differential equations are poorly applied:Differential Shuffle)



1.  “Fundamentals of Statistical and Thermal Physics”, F. Reif, McGraw-Hill, New York, 1965

2.  “Statistical Physics”, F. Reif, McGraw-Hill, New York, 1967


First Law: Implications to
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Sommerfield quote:"Thermodynamics is a funny subject. The first time you go through it, you don't understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don't understand it, but by that time you are so used to it, so it doesn't bother you any more."
This website is copyright of Kent W. Mayhew who in 2018 resides in Ottawa Ontario Canada
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