Another useful way to look at the difference between heat and work is in molecular terms, merging the ideas of position probability and velocity or energy probability. If a confined gas is allowed to expand until its volume doubles, considerations of position probability tell us that, so far as its spatial arrangement is concerned, it has experienced an entropy increase, having spread out into an intrinsically more probable arrangement. In doing so, however, it has done work on its surroundings and has lost internal energy. This means that, with respect to its molecular velocity and energy, it has approached a state of greater order and less entropy. Its increase of spatial disorder has in fact been precisely canceled by its decrease of energy disorder, and it experiences no net change of entropy. Had we instead wanted to keep its temperature constant during the expansion, it would have been necessary to add heat (equal in magnitude to the work done). Then after the expansion, the unchanged internal energy would provide no contribution to entropy change, so that a net entropy increase would be associated with the expansion— arising from the probability of position. This would match exactly the entropy increase ∆H/T predicted by the Clausius formula, for this change required a positive addition of heat.
Although the macroscopic entropy definition of Clausius and the submicroscopic entropy definition of Boltzmann are, in many physical situations, equivalent, Boltzmann’s definition remains the more profound and the more general. It makes possible a single grand principle, the spontaneous trend of systems from arrangements of lower to higher probability, that describes not only gases and solids and chemical reactions and heat engines, but also dust and disarray, erosion and decay, the deterioration of fact in the spread of rumor, the fate of mismanaged corporations, and perhaps the fate of the universe.