Page 90 - Coincidences in the Bible and in Biblical Hebrew

CHAPTER 3 “RANDOMNESS” AND “COLD”
CHAPTER 3 “RANDOMNESS” AND “COLD” 69 69

simplify the scenario, assume also that both objects are big enough so that at a

small interval of time, dt, when the amount of heat flowing from the hot object
to the cold object is dQ, no change in temperature occurs in either objects. Then,
from Boltzmann’s entropy formula, it may be shown that for an object with tem-
perature T, the amount of entropy gained, dH, is equal to

dH = dQ/T

Note, that although we refer to this expression only with a simplified scenario
that does not require the use of calculus, this expression for the entropy change,
resulting from heat transfer between two objects, is much more general. In fact,
one can say that this ratio measure is the “driving force” for every spontaneous
chemical reaction in the universe. Gibbs’s known equation, which describes how

energy, formerly bound in reactants, is spread out in the products, can actually be
put in terms of this equation.
Let us calculate the total change in entropy for the two objects in thermal con-
tact. Conveniently denoting dQ as negative for outflow of heat and positive for

inflow, one obtains from the above equation
dH 1 + dH 2 = dQ [1/T 2 – 1/T 1] = dQ(T 1 – T 2) / (T 1T 2) ≥ 0,

where nonnegativity for this expression is implied from the (arbitrary) assumption
that T 1 ≥ T 2.
This result implies that the total change in entropy is nonnegative, in accor-
dance with the Second Law of Thermodynamics. The same would have been

obtained if we calculated the total amount of entropy change throughout the
whole process, assuming that the objects’ temperatures are changing as a result of
the heat flow (only then we would have to use calculus, which we avoid here in

order to keep the level of the required mathematics to the possible minimum).
An interesting property of the above result, which can be proved also for the
whole process of a heat transfer, is that the lower (relative to T 1) is the tempera-
ture T 2, the higher the increase in total entropy of the two objects as a result of
the heat transfer. Mathematically, this can be seen from the above right-hand side
of the formula: decreasing T 2 results both in an increase in the numerator and a
decrease in the denominator. In other words, the colder T 2, the higher the increase
in entropy.
A similar pattern can be traced for any transfer that is activated as a result of a
gradient between two objects that are in a state of disequilibrium—for example,