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)

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 written^{7,8} 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 “0*energy 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)

References

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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