Re: why is some energy put into engine not converted to work?

Graham,

The second law of thermodynamics is really telling us why it is impossible 
to build a perpetual motion machine.  There is an additional level of 
complexity that is generally left out in simple proofs, presumably as a 
means of keeping the explanation simple.  It boils down to the fact that a 
"perfect" conversion of heat into work (in gas expansion, for example) is 
only possible in an adiabatic system.  By using the word "adiabatic" we've 
just cheated.  An adiabatic system is one that is artificially held at 
constant temperature by placing the system of interest (our engine) in a 
large bath.  We allow heat to exchange across the boundaries of the bath, 
and therefore can gain or lose heat according to the process that is being 
studied.

To put it in terms of thermodynamic equations, work, dw, resulting from a 
change in heat, dq, is possible when we consider changes in the internal 
energy of a system, dU, where:

	dU =dq +dw

If we hold the internal energy of a system constant, then the change in 
heat (due to gas compression or expansion) will result in an equal change 
in expansive or compressive work:

	dq = -dw

The caveat is the machine performs extra work on the system such that the 
internal energy of the system within the bath is held to zero.  The point 
is the adiabatic bath, through heat energy transfer that maintains the 
constant temperature, is doing work on the internal system such that our 
measurements can be performed.  The reality is when we include our 
adiabatic bath as part of the WHOLE system, the TOTAL change in work done 
includes dw(expansion) and dw(bath), such that dU is non-zero for the 
entire system.  That's the first law of thermodynamics (in ONE form, 
anyway).  If we ignore the work that the adiabatic bath does on the system, 
then we have a perfect engine...but you can see how that's cheating.

The laws of thermodynamics tell us that this heat-loss from our internal 
system to the adiabatic bath cancels out at absolute zero temperature, 
where a perpetual motion machine is theoretically possible.  The problem is 
by definition no work can be performed at absolute zero, as all motions and 
energy transfers are frozen out at absolute zero temperature.  (If you're 
at absolute zero, all energy of thermal motion is quenched; the system has 
uniform entropy).
If the system is not at absolute zero, then the net total entropy change is 
always non-zero; the system loses energy in the form of entropy loss to the 
surroundings.  At some point, the perpetual motion machine will run out of 
energy through heat loss to the surroundings.  Therefore, the amount of 
work we can extract from a system is always less than 100%.

It's easier for us to think about chemical systems or reactions as 
statistical ensembles.  Individual collisions among a collection of atoms 
and molecules give rise to the bulk properties (extrinsic variables) of a 
system, such as chemical potential, pressure, and temperature.  The bulk 
properties are a function of the individual components (intrinsic 
properties) of the system, such as the number of molecules, the volume they 
occupy, and the internal energy or entropy.  If we express the free energy 
of the system in it's differential form, we see how these variables relate 
to one another:

	dG = (Vdp) + (-SdT) + (udN)

Each relationship has units of energy.  The bulk properties are defined by 
a corresponding intrinsic property.  If you change the pressure, the free 
energy changes by a certain amount.  We can keep the energy fixed and allow 
the volume to change in response to pressure changes.

The key to understanding the effects of energy changes at a molecular level 
is to remember how one function affects another.  The relationship between 
entropy and temperature is an easy one to understand in the context of 
internal motions.  Above absolute zero, any system experiences molecular 
motions.  These can be in the form of bond stretching, bond bending, 
intermolecular collisions, diffusion (or tumbling), etc.  The frequency of 
ALL motions, in total, give rise to our observed temperature.  We can turn 
the relationship around and describe the same relationship as a change in 
temperature affects the frequency or energy of these motions.  Bond 
stretching is slow at low temperatures and fast at high temperatures.  
Since (Vdp), (udN) and (-SdT) are all defined as units of energy, you can 
see how each relationship affects the system as a whole.  If we keep the 
volume and free energy fixed, then a change in temperature will increase 
the pressure of a system.  We can require the pressure and free energy to 
stay fixed, and allow the volume to change.  Chemical reactions can affect 
the energy of the system, resulting in a temperature change.

This is perhaps a little more information than you wanted, but I hope it 
helps you out.  Thanks for your interesting question.

Regards,
Dr. James Kranz