By Meghnad Saha
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Additional info for A treatise on heat: Including kinetic theory of gasses [sic], thermodynamics and recent advances in statistical thermodynamics
Next, consider the function f (w) = f (ax) where a is any positive real number. We have df (w) df (w) dw = . dx dw dx By the Fundamental Theorem of Calculus (asserting that integration can be done by anti-diﬀerentiating the integrand) we have df (w)/dw = 1/w = 1/ax, whence 1 1 df (w) = a= . dx ax x We see that the derivatives of f (x) and f (w) with respect to x are the same, both equal to 1/x. This implies that the two functions must diﬀer by a constant C: f (ax) = f (x) + C. In particular we have f (a · 1) = f (1) + C = C, whence f (ax) = f (x) + f (a).
As noted earlier, in an isolated adiabatic system equilibrium is reached at the greatest value of entropy: S = Smax . 11) A good example of an isolated system is a spaceship, since in constructing the ship every eﬀort is made to ensure that its interior is insulated as far as possible from the surrounding cosmos. Of course, the spaceship, with its functioning crew of astronauts, is far from being in a state of equilibrium. Later on we shall see how the non-equilibrium state of a living organism is sustained.
Sooner or later all moving bodies come to a halt, and an equilibrium corresponding to maximum entropy is reached. Thomson concluded that the world is ultimately subject to “heat death”—while its energy remains unchanged. There is a further very general conclusion to be drawn. We saw early on that in pure mechanics there are in principle no irreversible processes, while in thermodynamics there are such processes. We have tacitly assumed that, for isolated irreversible systems, time increases in the direction of increase of entropy.
A treatise on heat: Including kinetic theory of gasses [sic], thermodynamics and recent advances in statistical thermodynamics by Meghnad Saha