13.2. DYNAMIC EQUILIBRIUM 193
4. Energy is transported outward in the star by either convection or radiation (depending on
some easily-tested physical criteria).
5. e star is fusing hydrogen into helium in its core.
One can apply these assumptions to make models of specific stars with different masses, spanning
the range of known masses of stars. e models will then tell you—given the chosen mass—the
visible temperature, radius, and luminosity of the star for each mass. We can then compare these
results to observations of real stars.
We can, for example, calculate such models for many different masses and plot their tem-
peratures and luminosities on the H-R diagram. When we do so, the model stars trace out the
observed main sequence! More specifically, they lie along the ZAMS. And so we now have a
physical meaning for the main sequence—these are stars that are in hydrostatic equilibrium and
are fusing hydrogen into helium in their cores.
Hydrostatic equilibrium in a main sequence star is an example of a stable equilibrium. If
one were to try to squeeze the star from the outside somehow, it would make the center of the
star hotter. is would make the fusion reactions run faster, which would generate more energy,
and that would make the star want to expand. And so our (hypothetical) squeezing is balanced
by the very physical consequences of our squeezing.
13.2 DYNAMIC EQUILIBRIUM
ere are many processes in the universe whereby different opposing tendencies are balanced,
and so it is a kind of equilibrium, even though the opposing processes involve rapid changes in
detail. For although the microscopic processes do not balance each other in every detail, they
may still average out statistically to a kind of macroscopic balance.
is more subtle form of balance is called a dynamic equilibrium. Imagine a popular night
club with a long line of individuals waiting to be allowed in, ending at a large bouncer standing at
the entrance. e bouncer lets people in only as others exit. And so the names of the individual in
the club are constantly changing, but the number of people in the club remains the same. e reason
for the constant number of people inside is because there is a dynamic equilibrium between the
process of people entering and the process of people leaving.
Another way to look at the same example is this: if one were to keep record of how many
people, on average, entered and left per minute, one would find the following: the rate at which
people enter the club is equal to the rate at which people leave the club.
Although the terms “stable” and “unstable” are not usually applied to the case of a dy-
namic equilibrium, there is a distinction that has similar importance. A given system may have
feedback mechanisms—physical properties that increase or decrease the rate of something happen-
ing, depending upon the state of the system. And these feedback mechanisms can either drive
the system away from equilibrium (positive feedback), or they can help to restore equilibrium
(negative feedback) when the system is affected by external forces.
194 13. EQUILIBRIUM
Sometimes, a positive feedback drives the system further and further away from equi-
librium until the system changes so much that a new negative feedback occurs that brings the
system to a new equilibrium—one with very different properties. As an illustration, let us again
consider our night club example. At the beginning of the evening, when the doors of the empty
club first open, the bouncer intentionally drives the system away from equilibrium, allowing
the rate of people entering to greatly exceed the rate at which people leave. When the club
reaches its legal capacity, the bouncer consciously applies negative feedback—adjusting the rate
at which people may enter in response to the rate at which people are leaving—in order to establish
equilibrium.
What if, on some summer night, the air conditioning cannot keep up with a full house,
and it gets too hot in the club? e bouncer decreases the rate at which people may enter to
below that at which people are leaving. e night club is then “out of equilibrium” and the
number of people decreases with time. When there are few enough people in the club that the
air conditioning can keep up, the bouncer establishes a new equilibrium by again adjusting the
rate at which people enter to match the rate at which people leave.
13.2.1 THERMODYNAMIC EQUILIBRIUM
A hot gas consists of individual particles—some combination of atoms, ions, molecules, and
photons (particles of light)—all bouncing off of each other randomly. Each collision involves
a transfer of energy from particle to particle, and the particular circumstances are essentially
random. e gas is in thermodynamic equilibrium when the average rates of transfer of energy
between all of the particles, photons included, are equal.
ermodynamic equilibrium is most easily established if the gas is contained with an “ideal
box” that allows no inflow or outflow of energy (see, for example, Carroll and Ostlie [2017, p.
239]). In this state, the physical properties of the gas can be defined by a single temperature, di-
rectly proportional to the average energy of the individual particles in the gas. e photons have
a range of energies as well, and this results in a spectrum of different wavelengths or frequencies.
e spectrum of light for such a gas in thermodynamic equilibrium is the ideal thermal blackbody
spectrum, described in Section 12.2.3.
Deep inside a star, the gas is, strictly speaking, not in thermodynamic equilibrium. ere
is a net flow of energy from the center outward, and so the temperature of the gas also changes
from layer to layer in the star. But the individual particles in the gas don’t “know” about this; the
changes in temperature and energy flow are on a much larger scale than the collisions between
individual particles. And so we still have a local thermodynamic equilibrium, and this means we can
describe the gas at any given point in the star as having a particular, well-defined temperature.
13.2.2 ATOMIC LEVEL POPULATIONS
e absorption or emission of spectral lines by a gas is a good example of a dynamic equilibrium.
Because of the strange rules of quantum physics, the electrons in an atom can exist only with
13.2. DYNAMIC EQUILIBRIUM 195
a specific, predetermined list of energies. It is as if these energies are a set of steps, and the
electrons cannot exist with an energy in between the steps. We call these specific steps energy
levels. e specific energy levels are a property of the atom as a whole—they do not exist for an
electron off by itself, not bound by the electrical attraction it feels for the atomic nucleus.
e particular numerical values of these possible energies are different for every atom.
Furthermore, they are different for every ion of every atom. e energy levels for hydrogen are
completely different for those of helium, for example. But also, the energies levels for neutral
helium (called He I) are completely different from those of helium that is missing one electron
(called He II).
For a given atom or ion, there is a lowest energy level for the electrons, called the ground
state. All of the other higher energy levels are called excited states. Left to itself, an atom most
wants to be in the ground state. If it somehow finds itself in an excited state, it usually takes only
a fraction of a second before it finds its way back to the ground state.
But there is also a highest possible energy level, called the ionization energy. For if the
electron is given too much energy, it will escape from the atom altogether, and a new ion results,
with a completely new set of energy levels. e energy levels get closer and closer together as they
approach this ionization energy. And so although there is a both a lowest and highest energy,
and there are only specific values allowed, it is still true that there are infinitely many levels!
Figure 13.1 shows an energy level diagram for neutral hydrogen. Only the first four energy
levels are shown, and they are numbered, symbolized with the letter n; n D 1 is the ground state,
and n D 8 (infinity) is the ionization energy. All of the other levels—n D 5; 6; 7; 8, etc.—are in
between n D 4 and n D 8, getting closer and closer to each other as n D 8 is approached.
e energies are labeled in electron volts (eV), a minuscule unit of energy that is more
appropriate for the goings-on within individual atoms. Notice that the energies are negative,
with the highest energy level equal to zero electron volts. ere is nothing odd about this; energy
is physically meaningful only in terms of changes.
e arrows between levels in Figure 13.1 represent transitions—changes from one energy
level to another. In this example, all of the transitions shown are downward—from higher energy
to lower energy. But the opposite happens as well (called upward transitions).
Many of the energy levels have more than one possibility, in this case labeled, s, p, d, and
f. We will not consider the reasons for this here, but it implies that there are, for example, four
distinct ways that a hydrogen atom can be in the n D 4 energy state (E D ´0:85 eV), but only
one way for it to be in the ground state (n D 1, E D ´13:6 eV). is greatly alters the statistics,
when an atom changes from one energy to another, and for atoms more complex than hydrogen
these levels may even have different energies.
Level Transitions
So how do electrons make upward or downward transitions? ere are several ways, but in every
case the following must be true: the electron cannot make an upward transition to a higher
196 13. EQUILIBRIUM
0,0 eV
−0,85 eV
−1,51 eV
−3,4 eV
−13,6 eV
1
2
3
4
∞
n
E
s
p
d
f
Figure 13.1: An energy level diagram for neutral hydrogen. (Graphic by Wersje rastrowa wykonal
uzytkownik polskiego projektu wikipedii: Artura Jana Fijalkowskiego (WarX), Zwektoryzowal:
Krzysztof Zajaczkowski, GFDL.)
energy unless some process transfers to it the exact amount of energy gained. Similarly, the
electron cannot make a downward transition to a lower energy unless it can transfer the energy
lost to some other process. And so what are the processes that can transfer energy to or from an
atom, and so allow for upward or downward transitions? e most common of such processes
are listed below.
• e atom can collide with free electrons or other atoms in the gas. In the process, collisions
can transfer some of the other particle’s energy to the atom to provide the energy needed
for an upward transition. is is called collisional excitation. Similarly, an electron can make
a downward transition with the excess energy transferred to the other atom, in a process
called collisional de-excitation.
• An atom can absorb a particle of light—a photon. e photon disappears, but its energy
does not, and it is used to excite the atom to a higher energy level, in a process called
photoexcitation. is is an all-or-nothing thing; the atom cannot absorb only part of the
photon’s energy. And so it only happens if the incoming photon has exactly the right
amount of energy to bring the electron to one of the possible higher energy levels.
13.2. DYNAMIC EQUILIBRIUM 197
Similarly, the opposite can happen. An electron in an excited state can spontaneously de-
crease its energy, going to a lower energy level, in a process called spontaneous emission. In
so doing, the excess energy is transformed into a photon of that same energy, emitted by
the atom, that travels away at the speed of light. us, an excited atom can emit light.
An individual photon has energy E D hf D hc=, where h is the tiny Planck constant
and f , , and c are, respectively, the frequency, wavelength and speed of the light (see
Section 12.5). To say that photons of only particular energies can be absorbed or emitted
by the atoms is also to say that photons of only particular wavelengths or frequencies may
be emitted or absorbed.
• An already-ionized atom can re-capture a free electron in a process called recombination.
e new electron more often than not recombines to an excited state, rather than the
ground state.
• An atom can be ionized, with one or more electrons removed entirely, which then become
free electrons in the gas. is can happen by the same processes that produced excitation—
collisions with free electrons or other atoms and absorption of photons of light. In both
cases, the energy transferred must be greater than the energy required to reach the maxi-
mum n D 8 level. Ionization by absorbed photons is called photoionization and if from a
collision it is called collisional ionization.
• For all of these processes, transitions need not occur one step at a time. An electron can
jump directly from n D 1 to n D 5, for example.
ere are, however so-called selection rules, set by the physics of quantum mechanics, that
make certain transitions extremely unlikely, or forbidden. Quantum physics is required to
calculate these transition probabilities.
Atomic Level Populations and Collisions
In a given situation, some combination of all of these processes might occur. For example, in a
gas atoms are constantly bumping into each other, and at a given temperature, these collisions
have an average energy. ese collisions produce both excitation of atoms and de-excitation of
atoms that are already excited. Furthermore, excited atoms undergo spontaneous de-excitation,
and emit photons of light.
ese events, for a given atom, are random. And so the electron in some particular hydro-
gen atom, for example, will change rapidly up and down because of collisions and spontaneous
emission. Similar things are also happening for all of the other hydrogen atoms in the gas, all of
which are experiencing the same overall conditions, but very different individual circumstances
at any one moment.
But the overall result of all this random jumping up and down the stairs of energy levels
is not random. For example, because the temperature of the gas dictates the average energy
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.