Photon Laws for Black Body Radiation: Starting with a relation showing the temperature volume changes for black body radiation which is undergoing an adiabatic change, it is shown how photon laws which correspond one by one with known radiant energy laws may be derived. In all instances, excepting two, namely λm′T=const and νm′T=const, the photon relations differ from the corresponding radiant energy relations chiefly by having a power of T decreased by one, a power of λ increased by one, or a power of ν decreased by one. The entropy of black body radiation is strictly proportional to the number of photons involved. Table of Black Body Radiation Constants: A table includes values in common units for the collective groups of constants entering the various photon and radiant energy equations mentioned in the paper.
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These experimentally determined values taken together form the basis for the other, computed constants. Unfortunately, the values for σ and c2 vary somewhat from those obtained with the aid of generally accepted formulae using other observed data. While in the table the number of places used in expressing the values for the bracketed constants are those to be expected, were the values given for c2 and σ accepted, it is understood that they possess actual uncertainties of the order of
to 1 percent. How to shift from the values reported here to others based on other values for σ and c2 will be apparent to the user.
Tables (2)
Table I
Symbols, quantities and units.
Symbol
Quantity
Common Unit
λ
Wave-length
μ
ν
Frequency
vibration/sec.
T
Temperature
K°
h
Planck’s constant
erg sec.
k
Boltzmann gas constant for a single molecule
erg/K°
u
Radiant energy density
erg/cm3
uλ
Radiant energy density per unit wave-length interval
erg/(cm3μ)
uν
Radiant energy density per unit frequency interval
erg/(cm3 vib./sec.)
C
Photon concentration
photon/cm3
Cλ
Photon concentration per unit wave-length interval
photon/(cm3μ)
Cν
Photon concentration per unit frequency interval
photon/(cm3 vib./sec.)
s
Entropy density
erg/(cm3 K°)
σ
Stefan-Boltzmann constant for radiant energy
erg/(cm2 sec. K° 4)
σ′
Stefan-Boltzmann constant for photons
photon/(cm2 sec. K° 3)
a
Constant of the fourth power law relating to radiant energy density
erg/(cm3 K° 4)
a′
Constant of the third power law relating to photon concentration
photon/(cm3 K° 3)
ℛ
Radiancy
erg/(cm2 sec.)
ℛλ
Radiancy per unit wave-length interval
erg/(cm2 sec. μ)
ℛν
Radiancy per unit frequency interval
erg/(cm2 sec. vib./sec.)
N
Photon emission rate per unit area
photon/(cm2 sec.)
Nλ
Photon emission rate per unit area and unit wave-length interval
photon/(cm2 sec. μ)
Nν
Photon emission rate per unit area and unit frequency interval
photon/(cm2 sec. vib./sec.)
c2
Second radiation constant
μ K°
λm
Wave-length at which, for a given T, ℛλ is a maximum
μ
λm′
Wave-length at which, for a given T, Nλ is a maximum
μ
νm
Frequency at which, for a given T, ℛν is a maximum
vib./sec.
νm′
Frequency at which, for a given T, Nν is a maximum
vib./sec.
Tm
Temperature for the maximum spectral efficiency of production of radiant energy at wave-length λ
K°
Tm′
Temperature for the maximum spectral efficiency of photons production at wave-length λ
K°
λe
Effective wave-length for the total radiation
μ
νe
Effective frequency for the total radiation
vib./sec.
Table II
Equations and values for the collective groups of constants enclosed in [ ].
These experimentally determined values taken together form the basis for the other, computed constants. Unfortunately, the values for σ and c2 vary somewhat from those obtained with the aid of generally accepted formulae using other observed data. While in the table the number of places used in expressing the values for the bracketed constants are those to be expected, were the values given for c2 and σ accepted, it is understood that they possess actual uncertainties of the order of
to 1 percent. How to shift from the values reported here to others based on other values for σ and c2 will be apparent to the user.