A Volumetric Interpretation of Thermodynamic Entropy

(Refer: The preceding two blog posts, long and short versions of ‘3 Measures for the environment’)

Why engineers, financiers, statesmen need to look at the environment through the lens of entropy (apart from the usual heat energy)?

Because the issue is related to the survival of our children and grandchildren which is being threatened by climate changes by 2050. See the MIT report ‘Ahead of Schedule’, which claims collapse of our civilization by 2040: https://www.youtube.com/watch?v=kVOTPAxrrP4 .

Entropy is a well-defined scientific concept but is understood vaguely, and is largely confused with ‘disorder’, which is an un-defined popular term and has no dedicated unit of measure associated with it.

Consider the following two scenarios:

A family caught in a basement in flood times, with no means to escape, cannot survive an ingress of water beyond a point:

and

Human beings caught in a small island habitat which they cannot escape, and cannot survive an ingress of entropy in to the island, beyond a point:

The similarities are striking!

Modelling of different phenomena which are similar mathematically is a standard practice in the areas of science and engineering. Diverse areas of mechanical, electrical, and other sciences, mathematics, analogue and digital computing are replete with examples of mathematical modelling.

We take advantage of the similarity between the two terms TdS and pdV , a lucky break, in the equation dU = δQ + δW = TdS + pdV, the first law of thermodynamics, expressing conservation of energy. [Instead of writing (-)pdV, fashionable for use with engines performing work on the environment, and thereby causing reduction of the internal energy of the gas, we have chosen to
use the (+)ve sign for work done on the system. Note, in writing dU = TdS + pdV, both the terms add to the internal energy dU of the gas, taken simply.

COMPARISON BETWEEN:

A family caught in a basement in flood times, with no means to escape, cannot survive an ingress of water beyond a point (assume pressure at 1 atm, vents are available):

Water level, OK: When the quantity of water accumulated is small it does not matter much
Water level, Critical: When the quantity of water accumulated rises to a point about half of the basement volume, the situation is still hopeful
Water level, Deadly: However, when quantity of water has accumulated to, say, 90% of the volume of the basement the situation has become hopeless

and,

Human beings caught in a small island habitat which they cannot escape, and cannot survive an ingress of entropy in to the island, beyond a point (atmospheric temperature at 300 K, i. e., 27° C):

Entropy level, OK: When the quantity of entropy accumulated is small it does not matter much.
Entropy level, Critical: When the quantity of entropy accumulated rises to a point when the resulting disorders are starting to be intolerable to the inhabitants of the island.
Entropy level, Deadly: However, when quantity of entropy that has accumulated to a level when flora and fauna start to die away in large numbers, then the situation has become hopeless.

There is uncanny similarities here between the two scenarios. These similarities can be graphically and picturesquely captured as follows: