Energy and Human Ambitions on a Finite Planet, 2021a

13 Solar <strong>Energy</strong> 199
2. Individual photons interact with matter at the microscopic scale
<strong>and</strong> are relevant to underst<strong>and</strong>ing solar photovoltaics <strong>and</strong> photosynthesis;
3. It’s how nature really works.
13.2 The Planck Spectrum
We should first underst<strong>and</strong> where photons originate, which will help us
underst<strong>and</strong> how solar panels work <strong>and</strong> their limitations. Until recent
technological advances, photons tended to come from thermal sources.
It’s true for the white-hot sun, 6 <strong>and</strong> true for flame <strong>and</strong> inc<strong>and</strong>escent light
bulb filaments. 7 Likewise, hot coals, electrical heating elements, <strong>and</strong>
lava are all seen to glow. Physics tells us how such hot sources radiate,
as covered by the next three equations. The first (with units) is:
P AσT 4 (W). (13.3)
We already saw this equation in the context of Earth’s energy balance in
Sections 1.3 <strong>and</strong> 9.2. It is called the Stefan–Boltzmann law, describing
the total power (in W, or J/s) emitted from a surface whose area is
A (in square meters) <strong>and</strong> temperature, T in Kelvin. 8 The constant,
σ ≈ 5.67 × 10 −8 W/m 2 /K 4 is called the Stefan–Boltzmann constant, <strong>and</strong>
is easy to remember as 5-6-7-8. 9
B λ 2πhc2
λ 5 1
e hc/λk BT − 1
( ) W/m
2
, (13.4)
m
Eq. 13.4 might look formidable, but only λ <strong>and</strong> T are variable. It describes
the Planck spectrum, otherwise known as the blackbody 10 spectrum. For
some temperature, T, this function specifies how much power is emitted
at each wavelength, λ. Three fundamental physical constants from key
areas of physics make an appearance: c ≈ 3 × 10 8 m/s is the familiar
speed of light from relativity; h ≈ 6.626 × 10 −34 J · s is Planck’s constant
from quantum mechanics, <strong>and</strong> k B ≈ 1.38 × 10 −33 J · K is the Boltzmann
constant of thermodynamics. 11
λ max ≈
2.898 × 10−3
T(in K)
(m). (13.5)
6: . . . <strong>and</strong> thus stars <strong>and</strong> even the moon,
which is just reflected sunlight
7: Modern lighting like fluorescent <strong>and</strong>
LED sources rely on manipulating energy
levels of electrons within atoms <strong>and</strong> crystals.
8: Recall that temperature in Kelvin is the
temperature in Celsius plus 273 (273.15,
technically).
9: The Stefan–Boltzmann constant is actually
a witch’s brew of more fundamental
constants h (Planck’s constant), c (speed of
light), <strong>and</strong> k B (the Boltzmann constant)as
σ 2π 5 k 4 B /(15c2 h 3 ).
10: The term blackbody effectively means
a perfect emitter <strong>and</strong> absorber of thermal
radiation.
11: This last one may be more familiar to
students in its chemistry form of the gas
constant R k B N A ≈ 8.31 J/K/mol, where
N A ≈ 6.022 × 10 23 is Avogadro’s number.
Eq. 13.5 is called the Wien law <strong>and</strong> is a numerical solution identifying
the peak of the blackbody spectrum as a function of temperature. Higher
temperatures mean higher kinetic energies at a microscopic scale, so
that higher-energy (shorter-wavelength) photons can be produced. This
is why as objects get hotter, they move from red to white, <strong>and</strong> eventually
to a blue tint.
All this may seem overwhelming, but take a breath, then just look at
Figure 13.1. Everything so far in this section is captured by Figure 13.1.
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.