1.4.1 Universal Constants: Dimensionless Conversion Factors
and Dimensional Universal Constants
At this point, it is useful to point out that the formulation of physical theories involves a
number of universal constants. A partial list includes g c (the constant in Newton’s
second law of motion relating force to mass and acceleration), G (gravitational constant), R (universal molar gas constant), and k (Boltzmann constant), N A (Avogadro
number), h (Planck constant), and J (the constant relating heat to mechanical energy).
Note that the Boltzmann constant is k ¼ 1:38054 Â 10
À23
joule
KÀmolecule and the Avogadro
number is N A ¼ 6:02252 Â 10
26
molecules/kmol. The universal gas constant R is
related to the Boltzmann constant k: R ¼ N A k ¼ 8314:47 j/kmol K.
Progress in physics has been accompanied by increasing precision in the
determination of these constants as well as the evolving understanding of the
theoretical nature of universal constants, which is open to two different interpretations (see paragraphs below). The choice made on each universal constant is, in
some cases, viewed to be a matter of whether a particular choice is capable of
yielding useful applications, such as dimensional analysis in engineering. However,
instead of suggesting possible arbitrariness in the choice, a choice that yields useful
practical applications reflects a correct decipher of Nature.
Some of the universal constants such as g c and J underwent a transformation
from erstwhile dimensional universal constants to become dimensionless conversion factors: for instance, g c became Newton’s conversion factor relating force to
mass; J became Joule’s constant relating thermal energy to mechanical energy.
When this happens, a new understanding of the constants emerged. For instance,
with an arbitrary set of units, the Newton second law is written
F ¼
1
g c
m
d
2 x
dt 2
With the choice of newton, kg, meter, and second as the units of F, m, x, and t,
respectively,
g c ¼ 1
kg À meter
newton À sec 2
and the Newton second law assumes the form
F newton
ð
Þ¼m kg
ð Þ
d
2 x meter
ð
Þ
dt sec
ð Þ
2
By becoming unity, g c disappears from the physical formula. Similarly, J disappears from the physical formula with the adoption of the same mechanical energy
unit of “joule” for the unit of heat and thermal energy (see Sect. 3.3 in Chap. 3).
10
1 Introduction: Temperature and Some Comment on Work
and Dimensional Universal Constants
At this point, it is useful to point out that the formulation of physical theories involves a
number of universal constants. A partial list includes g c (the constant in Newton’s
second law of motion relating force to mass and acceleration), G (gravitational constant), R (universal molar gas constant), and k (Boltzmann constant), N A (Avogadro
number), h (Planck constant), and J (the constant relating heat to mechanical energy).
Note that the Boltzmann constant is k ¼ 1:38054 Â 10
À23
joule
KÀmolecule and the Avogadro
number is N A ¼ 6:02252 Â 10
26
molecules/kmol. The universal gas constant R is
related to the Boltzmann constant k: R ¼ N A k ¼ 8314:47 j/kmol K.
Progress in physics has been accompanied by increasing precision in the
determination of these constants as well as the evolving understanding of the
theoretical nature of universal constants, which is open to two different interpretations (see paragraphs below). The choice made on each universal constant is, in
some cases, viewed to be a matter of whether a particular choice is capable of
yielding useful applications, such as dimensional analysis in engineering. However,
instead of suggesting possible arbitrariness in the choice, a choice that yields useful
practical applications reflects a correct decipher of Nature.
Some of the universal constants such as g c and J underwent a transformation
from erstwhile dimensional universal constants to become dimensionless conversion factors: for instance, g c became Newton’s conversion factor relating force to
mass; J became Joule’s constant relating thermal energy to mechanical energy.
When this happens, a new understanding of the constants emerged. For instance,
with an arbitrary set of units, the Newton second law is written
F ¼
1
g c
m
d
2 x
dt 2
With the choice of newton, kg, meter, and second as the units of F, m, x, and t,
respectively,
g c ¼ 1
kg À meter
newton À sec 2
and the Newton second law assumes the form
F newton
ð
Þ¼m kg
ð Þ
d
2 x meter
ð
Þ
dt sec
ð Þ
2
By becoming unity, g c disappears from the physical formula. Similarly, J disappears from the physical formula with the adoption of the same mechanical energy
unit of “joule” for the unit of heat and thermal energy (see Sect. 3.3 in Chap. 3).
10
1 Introduction: Temperature and Some Comment on Work