By Kent W. Mayhew

*Expanding Piston-Cylinder *by Kent W. Mayhew

**Expanding Piston Cylinder**

Imagine that we have a hermetically sealed expanding
piston-cylinder.Consider fig 1. As the vertically oriented piston-cylinder expands by a distance/height *dh*, its volume increases
by a dV, where:

*dV* = *Adh*, (1)

with “*A*” being the piston’s surface area.

Imagine
that the hermetically sealed piston-cylinder is filled with an ideal gas, and the expansion is isothermal, then based upon the ideal gas law (*PV=NkT*) we know that for isothermal processes:

* PdV*=-*VdP * (2)

Interestingly
based upon empirical findings (experiments) the work done is accepted as being:

*W*= *PdV * (3)

Strangely
traditional thermodynamics fails to provide clarity onto what this work is done, which has led to most thinking that the
work is done within the expanding system. This lack of logic can be witnessed in many texts and forums e.g. the utube
video by the Khan Academy ob thermodynamics of expanding systems (work-from-expansion). A reason that this is problematic: If
one considers the expanding system then based iupon eqn (2) one must now ask: Why choose an isobaric volume increase (PdV) over
an isometric pressure decrease (VdP)? Certainly both would equally fit with any empirical findings for the magnitude of
work done.

The reality is that the work goes through the expanding walls/wall into the surrounding atmosphere and this is lost work done onto the atmosphere. So for the case of the piston-cylinder the expanding wall is the moving piston!

Note: Interestingly, in many texts you will find that this work as defined by eqn (3) is claimed to go into the walls. Although this is
a mathematical plausible result, it is not a logical result. Based upon work going into walls Enrico Fermi once illogically stated
that the work of an expanding universe "goes into the hands of god".

I do find it interesting that some now acknowledge that this work is done onto the atmosphere but they all fail to understand that it is irreversible work i.e. lost work. Remember in order for the atmosphere to do work onto a system then the atmosphere must be at a higher pressure than that system. And at no time is this the case!

** Heating**

In the above, we never discussed what drove the expansion. Let us say that the expansion
was driven by heating as defined by *dQ*. Applying the first law we can now write:

*dQ=dE*+*PdV* (4)

Basically eqn (4) states that as we heat the piston-cylinder then its temperature increases thus increasing its
internal energy (dE) and this results in a pressure increase which drives the piston outward from within the cylinder, thus
doing work (lost work) onto the surrounding atmosphere.

If the ideal gas’s isometric heat capacity was *C**v* then we could rewrite
eqn (4) as:

*dQ=C**v**dT *+*PdV* (5)

And in terms of isobaric heat capacity (*C**p*) of the ideal gas we can rewrite eqn (5) as:

* dQ=C**p**dT * (6)

To better understand eqn (5) vs (6) see specific heats . In simplest terms isobaric heat capacity includes the work done onto the atmosphere by the expanding system. I.e.

Cp=Cv+PdV (7)

Note in Fig 2 we use dx instead of dh. It does not matter which direction the volume expands, because the Earth's atmosphere when displaced can only move in one direction that being upwards.

Negative work

If you want to understand expansion by creating a vacuum i.e. pulling on a hermetically sealed piston-cylinder then click here Negative work

**Removal of weight**

As is done in Khan Academy (work-from-expansion).video , let us know consider that several pebbles act a weight on top of the piston-cylinder, as is shown in Fig 3 .
Next a pebble is removed so the system isothermally expands. Again eqn (1) through (3) still applies. However, now the work
comes from within the piston-cylinder as shown in Fig 4.

If the piston-cylinder is fully insulated then as it expands it cools down.
Now the work comes from the ideal gas’s internal energy. Therefore in terms of eqn (4) we know that dQ=0, therefore:

*dE*= -(*PdV)atm* (8)

If the piston-cylinder is not insulated, then if it expands rapidly it will cool down until enough thermal energy passes through its walls from the atmosphere to heat it back up. Conversely if it expands quasi-statically then there will be enough time for thermal energy to pass through the piston-cylinder walls the process isothermal. And it is this quasi-static process that the Khan academy is considering although they get the logic somewhat wrong. Okay its not their logic per say rather it what they have been wrongly taught concerning the sciences.

It must be stated that eqn (8) is in terms of work done onrto the surrounding atmosphere!

What if we were to contemplate the work done in terms of the expanding system itself? Since the expanding system's pressure decreases as its volume increases then we could apply the ideal gas law and write:

P=NkT/V (9)

Since it is the gas that does the work rather than heat from an external source, then one must understand that as the gas expands (does work) its pressure decreases while its volume increases, hence the work that gas can do is actually a logarithmic function.Thus substituting in for pressure [eqn (9)] and integrating one would obtain the following logarithmic function:

W=(NkT)In(Pi/Pf) = (NkT)In(Vf/Vi) (10)

where the subscripts f and i repectively signify final and intial states

We now see where logarithmic functionalty comes in. It should be stated that (10) is only valid for isothermal process which limits us to quasi-static expansion i.e. a process that is slow enough that a sufficient amount of heat can pass through the piston-cylinder walls thus keeping the process isothermal. Note: Such a concept is traditionally discussed as a process in thermal equilbrium.

If the expansion was rapid then eqn (10) the gas's temperature would decrease as it does work and eqn (10) becomes a rough approximation at best. Of course after rapid expansion one could wait for enough thermal energy from the atmosphere to pass through the piston-cylinder's walls, and eventually thermal equilibroium would be re-attained.

Next one might ask what happen if the piston-cylinder is insulated? Since the gas inside the piston-cylinder does work then it cools in which case the isothermal relation as defined by (10) is a very rough approximation at best. and the cooler the gas inside of the expanding piston-cylinder is then the less valid eqn (10) becomes.

** Entropy Change**

Now consider entropy change. In subjects like physical chemistry we often write isothermal entropy change in terms of an eaxpanding system's volume change as:

dS=kIn(Vf/Vi) (11)

Comparing eqn (10) and (11) one can see the illogical correlation of isothermal entropy with work done. It must be emphasized that both (10) and (11) are in written in terms of the expanding system but when the system expands the work done is onto the atmosphere so should be written in terms of [(VdP)atm].

One can now see how confusion has allowed isothermal entropy change to feaster although in its its pureset thermodynamic contexrt it maybe nothing short of the over-complication of our reality.

** Work Done**

I repeat that the work done by the expanding is done onto the atmosphere and is defined by:

W=(VdP)atm (12)

It maybe of interest that the work done as defined by eqn (12) equals the work done as defined by eqn (10) for infintesimal processes. It makes one wonder if again we got the applicability of infinitesimal processes as backward or at least we certainly placed the cart brefore the horse in this one.

And if there is other work being done by the expanding system, i.e. the movement of man and/or machine, then this other work is to be added to the work done onto the atmosphere!

Blog: Expansion of a Piston-Cylinder