I was reading Frank Lambert's article Entropy is Simple if We Avoid the Briar Patches, which develops the idea of entropy as energy dispersal (and discourages entropy as disorder). On p13 is this paragraph:
Entropy increase without energy increase
Many everyday examples of entropy increase involve a simple energy increase in a particular
'system' (a part of the totality of 'system plus surroundings').. This energy increase is usually
evident from a rise in temperature (caused by more rapidly moving molecules) in the system after
some occurrence than before, e.g., when a pan or water in the pan is warmed or when a room is
warmed, their entropy increases. Additional energy has been dispersed in them from some
outside source, the 'surroundings'. The outside source is often combustion, the chemical reaction
of petroleum products — natural gas or fuel oil — with oxygen to yield the lesser energetic carbon
dioxide and water plus heat. The energy dispersed from the chemical bonds of gas or oil and
oxygen is dispersed tothe slower moving molecules of the pan, the water, or the air of the room. If an energy increase occurs inside a system, there must be an entropy increase in it. More energy
has been dispersed within the system and this is what entropy measures.
Is that statement in bold true in general? I know that
$$ \left ( \frac{\partial E}{\partial S} \right )_{V, N} = T > 0,$$
so that internal energy and entropy are positively correlated for changes at constant volume and number of molecules. But for an open system that can change volume and can also exchange other kinds of work with the surroundings, it seems like it wouldn't be true generally.
I realize this may be a somewhat informal article, and therefore the statement might not be intended too literally. I can't tell. So, is it actually possible for an energy increase to be accompanied by an entropy decrease in some thermodynamic processes?
Best Answer
For the closed system you are describing, in you reversibly compress and ideal gas isothermally, its internal energy stays constant and its entropy decreases. If you then just add a little heat at constant volume, the internal energy increases, while, provided not too much heat is added, its entropy increases just a little (but not to offset the decrease from the compression). So, the net effect is an increase in the internal energy of the gas and a decrease in its entropy.