Human use of Heat
---------------
Human society developed from its ability to convert prime sources
of energy into useful work. The prime source could be wind, sunlight,
flowing water, animal muscle.
These, while helpful to further human society, were catch-as-can,
erratic, unreliable, presented undesirable side effects. A horse, for
example, may take sick or spontaneously go wild. Wind mya fall calm
or blow too weakly. The river may overflow and flood its shoreline.
Early in human history we learned that fire contained energy,
mostly for light at night and heat in cold times. It was also a weapon
to fend off predators. Fire had domestic uses like cooking, curing,
drying, crafts. It was available more or less on demand and was
supplied by a wide variety of natural combustible materials.
For most of history fire was a passive energy source, used as is.
We used it to start more fires, like lighting street lamps or torching
an enemy fort. Little effort was made to improve the ability to make
and keep fire since it was an limitless resource, given the cast
amount of combustile materials to hand.
In the mid 1700s we learned that fire could boil water into steam
and the steam could do mechanical work. This was the birth of
the industrial revolution. The steam engine was vastly more reliable,
available, dependible, and orders stronger than other energy sources.
Steam engines improved rapidly in the 1700s but a nasty
realization came into society. No matter how well we made and worked
the engines, we lost a major portion of the input heat. It was
discarded as waste heat. At first we thought this wasted heat, from
the fuel we paid good money for, was poor insulation, excessive
friction, loose fittings, leaks. But even the best efforts to extract
a greater portion of useful work failed.
This irritating feature of throwing away large portions of the
ingredient heat was present in other heat-based machines. A gasoline
motor receives energy from the detonated fuel and rejects heat thru
the exhaust pipe.
Such machines are called 'heat engines'. They take in energy from
a source at high temperature, perform work from this energy, and
discard much of it into a sink at low temperature.
Temperature
--------
The common conception of 'heat' is its temperature, as read from a
thermometer placed in the heat source. The higher the temperature, the
more heat there is.
This is actually not all true. Temperature does not state the
amount of heat, a quantity of energy. It's a potential for heat, like
an elevation in mechanics or voltage in electricity. Neither of
these tell directly the amount of energy they can deliver. A large vat
and a teacup filled with boiling water have the same temperature, that
of the water in boil. the vat has far more heat energy in it than the
cup.
We have two main temperature scales. Celsius is based on two
defined degrees of heat, water-ice as 0 degrees and water-steam as 100
degrees. This is under one atmosphere of pressure. On this scale, the
one for everyday use, is also called by its older name 'centigrade'.
Nominal comfortable temperature for office of home is 20C; a really
cold day is -20C; a really hot day is +40C.
Historicly this scale was the centigrade scale, for its interval
of 100 degrees between its defining points. In 1954 it was renamed to
celsius. By luck both names begin with 'C' so older thermometers are
still usable.
The other scale has the same size degree, 100 to the internal of
freezing and boiling water. It starts at the coldest possible
temperature, -273C, rounded. This is the kelvin scale, or for older
scientists, absolute celsius. It has no negative readings, starting at
0K. Room comfort is 293K; colder day, 253K; hotter day, 313K.
Thermometers for civil use are dimensioned in celsius. Those for
scientific use may have both kelvin and celsius scales.
Kelvin is the preferred scale in physics but for very high
temperatures it doesn't matter which is stated. The Sun is about
6,000K or 6,000C, the 273 degree offset being only about 4%.
The 'o' symbol, for 'degree', is no longer part of the
specification of temperature, like '24oC', the letter 'C' or 'K' by
itself states the unit of measure. The 'o' is still in common use, for
civil use.
+-----------------------+
| TEMPERATURE RELATIONS |
| |
| kelvin = celsius + 273 |
| |
| celsius = kelvin - 273 |
+------------------------+
Nature of heat
-----------
Until the 18th century heat was considered as a product of fire or
strong sunlight. In 1744 duChatelet experimented with heat and found
it shared many properties with light. She suggested that since, in her
day, light and heat are often produced together, maybe heat and light
were somehow phases of a one emanation. under the French word 'feu'.
In the late 19th century we learned that heat is the energy within
agitated molecules. The more molecules, the more heat. The intensity
of the agitation is the temperature.
The molecules agitate, vibrate, pscillate within their host
material's structure. If the vibration is too vigorous the structure
falls apart and the material changes phase, such as ice to water or
water to steam. For lower temperatures the molecules may settle into
the lower-energy state, steam to water or water to ice.
We humans have no means to control the vibration of individual
molecules in a body. If there are 'loose' ones in vibration, they are
sources of loose heat that usually can not be captured. This
fundamental nature of heat makes the science of heat energy so
frustrating. We can work only with the overall properties of matter,
with its agitated molecules, and try to transfer and transform its
energy into useful work while keeping the lost energy at tolerable
levels.
Thermodynamics involves capturing energy from the agitated
molecules in a source of heat and turning it into work for human
society. There arose two schools of thermodynamic, the study of heat.
One works with the macroscopic behavior of eat as a bulk flow of
energy, the scheme for industry and engineering.
The other works with the microscopic behavior of the molecules
themselfs. This is used in statistical mechanics and quantum physics.
Radiant 'heat', like from a fire or the Sun, is itself not heat
energy. It is electromagnetic radiation that imparts energy io the
receiving body in the agitation of its molecules. The body's
temperature is raised. In humans the radiation evaporates moisture
from the skin, inducing the burn sensation.
To carry heat, the hot body is physicly moved from place to place
or put in contact with other objects. The former method is convection
and the hot object is usually a fluid like water or air. The latter
method is conduction, like a soldering iron or laundry press.
Laws of Thermodynamics
--------------------
Thermodynamics is described by four laws, zero thru three. Until
the late 20th century they had weak underlying theory but were a;ways
followed by experience and experiment. Today, as odd as it seems, they
are under vigorous study banking off of blackhole physics.
= = = = =
ZEROTH LAW: A body in thermal equilibrium with its surrounds has
the temperature of the surrounds. A pie taken from an oven and put in
open air to cool delivers energy into the air until it stops doing so.
The pie then has the temperature of the open air.
This is law #0 because it was recognized after the other three
were known. By history it was numbered '0' rather than '4'.
= = = = =
FIRST LAW: Energy can be changed from one form to an other but can
not be created from noting or destroyed to nothing. This is the
conservation of energy.
Mass can be turned into energy by the E=mc2 process, as the Sun
does. Some 4 million tons of the Sun's mass are turned into energy
every second. This energy did not come from 'nothing'. It was created
from mass.
In human experience the mass turned into energy is infinitesimal,
usually beyond detection. The mass 'has energy in it' that is
released, like the chemicals in a battery or gasoline in a car.
A fascinating instance of energy turned into mass is the
blackhole. Energy, like irradiation and energetic material, fall into
the blackhole and become part of its mass.
= = = = =
SECOND LAW: A body taking energy from a high temperature reservoir
and converting it to useful work must discard some of the energy into
a reservoir of lower temperature. In common terms it is not possible
to turn ingredient heat entirely into work.
The second law has many meanings, Some relate to the concept of
'entropy', a measure of the worth or value of energy processed within
a body.
So much of human production of energy is extraction from heat,
like fuel combustion. This causes the loss of some of the input energy
is a crucial concern.
= = = = =
THIRD LAW: A body with no heat energy in it has a temperature of
absolute zero. Since heat is the motion of molecules, if this motion
is somehow arrested, the body can have no lower temperature.
The third law defines the absolute temperature scale, with 0
kelvin as the zero point.
By this law we can not actually achieve a temperature of 0K. We
have no means what so ever to completely stop molecular motion in a
body. and other reason, based on the second law, is that reaching
absolute zero means we did work, stopping the molecules, from a single
temperature, 0K, without discarding heat to a lower, nonexistent,
temperature.
Second Law of Thermodynamics
--------------------------
The realization of work from heat is stated as the second law of
thermodynamics. There are many expressions and corollaries of the law.
Here we phrase it that work can be extracted only when heat is
received by a heat engine at a reservoir of high temperature and some
of that heat is released into a low-temperature reservoir.
Not all energy from the high temperature ends up in work, typicly
mechanical movement and force. The rejected heat comes from a genuine
limitation and not just from deficiencies in the build or operation or
upkeep of the heat engine. Even in a perfect machine some input heat
is lost. This is a superset of the first law, which allows that in
general all ingredient energy could be converted into work. The second
law put out the exception for input energy in the form of heat.
The second law can be stated as
work = Qhi - Qlo
where Qlo can not be zero.
The second law is sometimes stated as 'heat always flows from the
higher to the lower temperature'. In most situations this is true. The
second law and other principles of thermodynamics apply to large
sources and sinks of heat, not small amounts. An incandescent light
bulb in front of an open window will not stop cold air from outside
from passing by the hot bulb.
An astronomy example is the heat from the Sun's photosphere. it
flows thru the corona. The photosphere temperature is 6,000K but the
corona is a couple million kelvin. There is no violation of the second
law because there is so little material in the corona compared to in
the photosphere. It's the bulb against the open window.
Carnot Limit for Work
------------------
In early days of thermophysics we believed that the rejected heat
resulted from imperfections in the machines, typicly excess friction
and deficient insulation. Carnot in 1824 demonstrated that in all
heat-based machines there is a maximum portion of the input heat that
can be turned into work. The rest of it must be thrown away into the
low-temperature reservoir, usually open water or air.
A typical heat engine is shown here in a simplified sketch.
^ Thi--3 +----------+ 4
| | |
| | |
T | Tlo--2 +----------+ 1
| + +
| + +
| + +
0K +-----------5-+----------+-6--->
V
The engine is an idealized piston pushed by heated air and
connected to a crank to run machinery. The spent air is vented into
open air. Fresh air enters the engine at ambient temperature. This is
all-new air, not the actual slug that left the motor.
T is temperature with the y-axis origin at absolute zero. V is
volume, that of the cylinder.
At step 1 the piston is fully retracted from the previous stroke
and air is let into the cylinder. The air is at maximum volume and
lowest temperature for the instant stroke.
From 1 to 2 the piston compresses the air to its minimum volume.
The piston does work on the air by inertia from the previous stroke
and rotation of the crank.
At 2 gasoline vapor is injected into the cylinder and ignited by
the spark plug.
From 2 to 3 the air temperature is increased by the combustion of
the gasoline.
From 3 to 4 the air, mixed with spent gasoline, shoves the piston
and turns the crank. The air does work on the machinery attached to
the crank. The volume increases to the full capacity of the cylinder.
At step 4 the air is released into the open air, still hotter than
the ambient temperature. In a actual engine this hot air is dissipated
and does not return to the cylinder for the next stroke. New air at
ambient temperature enters the cylinder.
From 4 to 1 for this example we pretend the open air is a cooling
stage. It bringing the hot air back to ambient temperature. The cooled
air returns to the cylinder.
Note that there is a high temperature in steps 3 and 4 and a low
temperature at steps 1 and 2. The work realized is the area bounded by
points 1 thru 4. I do know that this is not stricta mente all the
detail, but it serves to bring out the principles of getting useful
work out of heat energy.
Note well that the energy put into the engine is the area 3-4-6-5.
This came from the gasoline, which we paid to acquire and handle. The
portion of the fuel's energy content that is not turned into work,
area 1-2-5-6. We paid for this energy and must throw it away!
Carnot discovered that there is a maximum fraction of the input
heat energy that can be turned into work. The rest must be wasted.
Wmax = Qhi * (Thi - Tlo) / Thi
+---------------------------------+
| MAXIMUM POSSIBLE WORK FROM |
| FROM cARNOT HEAT ENGINE |
| |
| |
| Wmax = Qhi * (Thi - Tlo) / Thi |
+--------------------------------+
Qhi is the energy taken in at the high temperature, the are 3-4-6-
5, from the combustion of fuel.
Thi and Tlo are the high and low temperatures in kelvin, at 3-4
and 1-2.
We examine the particular kind of heat engine, the Carnot engine.
Other kinds of heat engine were devised, mostly constrained by
engineering and mechanical factors. These are usually named for their
inventors, such as otto, Stirling, Diesel, Rankine. They process heat
thru a series of pressure, volume, phase changes, temperatures.
No Work from a Single Temperature
-----------------------------
The second law shows that work can not be continuously extracted
from a single reservoir of heat energy. By setting Tlo = Thi in the
Carnot equation
Wmax = Qhi * (Thi - Tlo) / Thi
= Qhi * (Thi - Thi) / Thi
= Qhi * 0 / Thi
/ = 0 / Thi
= 0
It is the hope and dream of inventors to build a machine that does
generate work from a single reservoir of heat with no discharge of
heat to a lower reservoir. All attempts totally failed. The ship that
sails by sipping up heat from the ocean or the plane that flies by
sucking in heat from the air is impossible.
Be careful. There are ways to convert energy from a prime source
into work with high efficiency. A hydroelectric station turns some 90%
of the energy passing thru it into electric. The station takes energy
from the mechanical elevation of the water, not from the heat
contained in the water.
Such machines are not heat engines. The prime source is not a
reservoir of heat. The huge discharge of usable heat imposed by the
Catnot limit applies only to true heat engines.
Clausius Emtropy
----- ------
Entropy is a concept for the part of ingredient energy that could
not be turned into work. It is tossed with the rest of the heat within
the Carnot limit. It was first described by Clausius in 1855, who
named it 'entropy'. It is measured in joule/kelvin.
From the Carnot equation
Wmax = Q hi * (Thi - Tlo) / Thi
Qhi - Qlo = Qhi * (Thi - Tlo) / Thi
(Qhi - Qlo) * Thi = Qhi * (Thi - Tlo)
Qhi * Thi - Qlo * Thi = Qhi * Thi - Qhi * Tlo
- Qlo * Thi = - Qhi * Tlo
Qlo * Thi = Qhi * Tlo
Qlo / Tlo = Qhi / Thi
Shi = Slo
Shi - Slo = 0
Thi and Tlo are the high and low temperatures; Qhi and Qlo, heat
energies at the two temperatures; Shi and Slo, the associated
entropies.
In a perfect Carnot heat engine the entropy taken in at the high
temperature equals the entropy passed out at the low temperature.
There is no change in the engine's entropy by processing heat energy
into work. While the ingredient heat is greater than the egredient
heat, it is the same 'worth' per kelvin degree of temperature.
+-----------------------+
| CLAUSIUS ENTROPY | |
| Qhi / Thi = Qlo / Tlo |
| |
| Shi = Slo |
| |
| Shi - slo = 0 |
+-----------------------+
Clausius flourished before the molecular structure of matter was
known. e studied the mass flow of heat in industrial machines. These
usually moved heat thru them by water and steam in large amounts.
Other scientists looked at entropy with slightly different
interpretations but it wasn't until the 1870s, after molecules were
recognized, that a radicly different inquiry occurred into entropy.
This is the statistical method, as first worked out by Boltzmann.
Actual Heat Engines
--------------
A Carnot heat engine realizes all the maximum work between its
high and low temperatures. A real heat engine always loses some energy
to friction, mechanical wear, radiation, conduction,, other factors.
The work realized is always LESS than the Carnot limit The energy that
would have gone into work is discarded to the lower reservoire.
Such a machine is an 'irreversible heat engine'. If it is run in
reverse, as a heat pump, the effects of friction and other losses are
NOT undone. In fact, MORE loss is generated.
The energy balance becomes
(Qhi / Thi) - (Qlo / Tlo - Qloss / Tlo
< (Qhi / T)hi - (Qlo / T)lo
< 0
del(Q / T) < 0
delS < 0
This is the Clausius Inequality,
+----------------------+
| CLAUSIUS INEQUALITY |
| FOR REAL HEAT ENGINE |
| |
| del(Q / T) < 0 |
| |
| delS < 0 |
+----------------------+
The Qloss/Tlo is the portion of ingredient heat that could have
been turned into actual work. In the stead it is lost. This Qloss/Tlo
is not available for the production of work and is dumped into the
surrounds as waste.
The inability to convert the ingredient heat into the Carnot
maximum output comes from the nature of heat. As the machine runs,
parts within it absorb some of the heat flowing thru it. Atoms in
these parts are agitated to become heat sinks.
Unless there is a way to prevent such diversion of heat away from
performing useful work, the change in entropy high-to-low is always
less than zero. The Clausius inequality is a built-in feature of the
world.
A make-do effort to reduce the heat loss is cascading heat
engines. The rejected heat from one is the input heat of the next. The
high temperature is lower than for the first engine but it can be
adequate for low-grade work.
Older electric power plants make high temperature steam for their
turbines. In the stead of recycling the exhaust steam back to the
boiler,s, the steam is sent thru surrounding streets as an other
energy service. Customers take the steam, as a cascade heat source, to
do useful work. Examples are clothes cleaning, chemical mixing, wood
curing,, food cooking, plastics molding, space heating. The exhaust
steam from t cascade engine is generally too cold for further use and
is discharged into storm drains.
Boltzmann Entropy
---------------
Boltzmann flourished in the 1870s, with the newly accepted
molecular theory of matter. He discovered that the macrostate property
of material comes from combinations of microdtate molecules.
Boltzmann found that the behavior of systems is the result of the
behavior of its microstates, like a body and its molecules. The
measure of this relationship ends up very similar to that of Clausius
entropy, altho it does not directly derive from action of heat energy.
Since heat energy is stored in the vibrations and agitation of
molecules, the statistics of this motion sare a measure of the heat.
This ties Boltzmann and other related entropies to thermodynamics.
We play with two dice whose pips display a count from two to
twelve. The dice are in the left and right cells of a shuffling box.
after shuffle we examine the cells and read the pip values.
The overall property of the two dice is the number displayed, say,
five. Boltzmann called this a 'state'. The microstate property is the
many ways five can be shown on the dice, as 1-4, 2-3, 3-2, 4-1. Out of
all the ways two dice can be thrown, four of them present dice by
their pips. Boltzmann called the individual ways to make a state a
'complexion'. The state '5'e has four complexions.
A table of states and complexions for two dice is given here. the
complexion is the reading of the left and right die.
-------------------------------------------------------
state | complexions
------+------------
2 | 1-1
3 | 1-2 2-1
4 | 1-3 2-2 3-1
5 | 1-4 2-3 3-2 4-1
6 | 1-5 2-4 3-3 4-2 5-1
7 | 1-6 2-5 3-4 4-3 5-2 6-1
8 | 2-6 3-5 4-4 5-3 6-2
9 | 3-6 4-5 5-4- 6-3
10 | 4-6 5-5 6-4
11 | 5-6 6-5
12 | 6-6
-----------
The possible states of the dice have different complexions, from 1 for
states 2 and 12 to 6 for state 7. Boltzmann reasoned that for a random
throw of the dice the state 7 would turn up most frequently because
there are more ways to make that state. states 12 and 12 come up with
lowest frequency because there is only one way to make thee states.
The most frequent states represent the most disordered, random,
disorganized complexions while the rarer states have the neatest
organized complexions.. In this sense Boltzmann described entropy as a
measure of disorder. He noted that in nature activity left to itself
tends toward disorder and higher entropy.
He at first defined entropy as the number of complexions that make
that instant state of the material. The entropy of the two ice in
state 5 is 4.
As the number of states and complexions increases the value of
entropy rises to immense values. Boltzmann in the stead defined the
entropy as the natural logarithm, ln, to keep the numerical value to
manageable levels. He introduced a proportion constant k to fix up the
units, as determined from experiments and comparison with Clausius's
work.
+-------------------+
| BOLTZMANN ENTROPY |
| FOR A GIVEN STATE |
| |
| S = k * ln(N) |
| |
| k = 1.381e-23 J/K |
+-------------------+
The numerical value of Clausius and Boltzmann entropy for a given
situation are different and can not easily be translated between them.
Arrow of Time
-----------
In our world time flows in one direction with no possibility of
modifying it. We can only passively use time as it comes to us, else
lose it forever. The other dimensiions of our world are freely
modified, these being X, Y, and Z directions of length.
We speculate that inside a blackhole we can freely move around in
time but are trapped in the one-directional flow of space. This is
toward the singularity.
Staying in our world, the study of entropy suggests that it
defines the direction of time flow. Entropy continuously increases,
from less in past time to more in future time.
We neffer-effer see a natural process that decreases entropy on a
large scale. On the small scale we can and do decreases entropy. When
we build a wall of bricks we organize and arrange the bricks against
their natural tendency to fall into a random heap. we purposely do
work on the bricks opposing the natural forces toward disorder.
Entropy, by Boltzmann, is a measure of disorder, randomness,
disorganization. These qualities grow on their own in nature. Examples
are rotting, SPalling, crumbling, erosion, corrosion, cracking.
Natural disasters are disorganizing events, like earthquakes,
tornados, floods, conflagrations. All of these turn order into
disorder. There is no 'natural proaster' that somehow spontaneously
arranges the world into an orderly state.
Philosophers and scientists try to concoct situations where
entropy on its own decreases in nature with utterly no success. The
best effort is on a minuscule scale, while he rest of the universe
goes its merry way toward ever-growing entropy, flying with the arrow
of time.
Sun as a Heat Engine
------------------
The Sun is a heat engine running between the central temperature
of about 15,000,000K and the photosphere of about 6,000K. Because the
Sun is a stable quiet star the power radiated off of the photosphere
equals the power generated in the core, Qhi = Qlo.
The Sun as a machine produces no work! Except for the minimal
amount of matter heaved up in solar wind, flares, convection cells, &c
all of the input heat is sent to the lower reservoir as waste heat.
Certain other stars, like pulsating stars and cataclysmic stars ,
do actual work. They shove against gravity substantial amounts of
mass.
For the Sun
delS = del(Q / T)
= (Q / T)hi - (Q / T)lo
= ((3.82e26 J/s) / (15e6 K)) - ((3.82e26 J/s) / (6e3 K))
= (2.547e19 J/s.K) - (6.367e22 J/s.K)
= -6.364e22 J/s.K
The universe receives this energy, as a positive input, and
increases its own entropy. The input of entropy to the universe from
the Sun is +6.364e22 J/s.K.
If the Sun somehow converted the heat energy from its core into
the Carnot maximum work, that work would be
W = Qhi * (Thi - Tlo) / Thi
= (3.82e26 J/s) * ((15e6 K) - (6e3K) / (15e6 K))
= (3.82e26 J/s) * (0.9996)
= 3.818e26 J/s
This stays inside the Sun, not radiated away. No, I have no idea what
can be the 'useful work' performed by this energy.
Qlo = Qhi - W
= (3.82e26 J/s) - (3.18e26 J/d)
= 1.52e23 J/s
This is about the amount from a cool red dwarf star! Or it is the
amount from the real Sun as received from about 50 AU, deep in the
Kuiper Belt.
Entropy from Starlight
--------------------
Taking the Sun as a typical star we can estimate the contribution
from starlight into the entropy of the universe. The Sun's rate of
entropy injection into space is, from above, 6.364e22J/s.K.
Normalizing this to the matter density of the universe and the
lifetime of the Sun we have
Sstar = SSun * lifeSun * densityMass * 10% / massSun
= (6.364e22J/s.K) * (3.156e17s) * (4e-28kg/m^3) * (0.10) /
(1.989e30kg)
= 4.039e-19J/K.m^3
The 10% factor recognizes that only about 10% of the mass of the
universe actually produces starlight. The rest is inert, outside
stellar cores and in disorganized clouds of gas and dust.
Sstar / Scmb = (4.039e-19J/K.m^3) / (2.063e-14J/K.m^3)
= 1.958e-5
-> 0.1958%%
The universe started out with an entropy of quite its present
amoount and added the remaining 0.2% over its 13-odd billion years of
its life.
Energy and Entropy in CMB
-----------------------
The cosmic microwave background is a pure blackbody radiation with
temperature .2.73K. Its entropy is easy to compute because we avail of
the standard blackbody formulae.
The energy density of blackbody radiation is
density = 4 * sigma * T^4 / c
I went directly to the blackbody formula for energy density in the
stead of first getting the irradiation
density = 4 * sigma * T^4 / c
sigma is the Stefan-Boltzmann constant, 5.670e-8J/s.m2.K4.T is the
CMB temperature, 2.73K. c is lightspeed, 2.998e8m/s
density = 4 * (5.670e-8 J/s.m2.K4) * (2.73 K)^4 / (2.998e8 m/s)
= 4.256e-14 J/m3
This value is for the entire spectral range of blackbody radiation, not
for a particular band. 'Microwave' in the name comes from the
discovery and initial measurements done in the microwave band. The CMB
peak radiation is at about 1 miilimeter wavelength.
Entropy implies that heat is exchanged from the heat engine and
its surrounds. The universe in the whole has to surrounds. No heat is
exchanged into or out of the universe. Any entropy dumped from objects
within the universe like stars, supernovae, merging neutron stars,
earthly machines stays 'inside; and continuously increase the entropy
of the universe.
We also have no simple notion of size or volume for the universe
to sum up the contributions of entropy. We work with a representative
volume where we can be free of local energy floes thru it. For the
moment we ignore any local floes and let this volume be penetrated
only by radiation from the cosmic microwave background. That's why we
worked out the energy density, joule/neter3, and not energy for the
whole universe in joule.
Continuing with blackbody rulees, the entropy of the CMB is
entropy = = (4 / 3) * density / T
= (4 / 3) * (4.256e-14 J/m3) / (2.73 K)
= 2.018e-14 J/m3.K
These equations are a bit different from the ones for other
kinds of radiation, notably the factors '4' and '4/3'. I skipped their
derivations for this article.
According as many cosmology theories, the entropy of the cosmic
background radiation was constant since it was created, some 350,000
years after the Bigbang.
entropyBB = entropyNOw
(Q / T)BB = (Q / T)Now
Qbb = Qnow * Tbb / Tow
= (4.256e-14 J/m3) * (3,000 K) / (2.73 K)
= 4.677e-11 J/m3
The temperature of the universe when the CMB emerged was around
3,000K, that of a red dwarf star. This temperature is when the plasma
was cool enough to let electrons and ions unite into atoms and become
transparent to radiation. It's like the photosphere os a star and the
era of CMB release is sometimes called the photospheric era.
This isn't an impressive result, the energy density is a thousand
times greater near the Bigbang than now. But we must recall that while
the radiation from CMB was flowing into space, space was undergoing
Hubble expansion. the cubic meter of today was far 'smaller' back
then. The Hubble redshift, the ratio of the universe scale factor now
to that when the radiation was emitted, is quite 1000. We today are a
thousand times 'larger' than at the photospheric time.
A meter now was only a millimeter, by our meterstick, so a cubic
meter now was a cubic millimeter, on billionth as large.
The energy density was packed into a cubic millimeter, ot to fill
today's cubic meter we need one billion time more energy. The
equivalent energy density, measure by today's meterstick, in more like
densityBB/Now = densityBB * (m3Now / m3BB)
= 4.677e-11 J/m3 * (1000 / 1)^3
= 4.677e-2 J/m3
And the irradiation for an observer in the CMB right after the
universe turned transparent is
irradiation = density * c
= (4.677E-2 J/m3) * (2.998e8 m/s)
= 1.402e3 J/s.m3
This approximates the energy output from the photosphere of a red
dwarf star.
CMB versus starlight
------------------
Eddington in 1925 worked up the amount of energy in the universe
by summing the light from stars. He observed in the optical waveband
within the Milky Way. Astronomy then was constrained to only the
visual waveband and obscuring by interstellar medium. Interstellar
medium was poorly studied and the cosmic microwave background was a
far-future discovery. He came up with a 'cosmic temperature' of some
3,000K.
Firm estimates of the energy density due to the stars are hard to
find. By starlight we mean detectable radiation, except the CMB
itself, over the entire spectrum.
Because the CMB is pure blackbodybody radiation it spans all
wavelengths and contaminates the measurements. Until the cMB was
discovered this contamination was not properly recognized.
An estimate can be made by taking the Sun as a typical star and
normalizing its radiation for the mass density of the universe.
energy of Sun = (power of Sun) * (life of Sun) / (mass of Sun)
= (3.82e26 J/s) * (3.156e17 s) / (1.989e30 kg)
= 6.063e13 J/kg
Note how minuscule this is compared to the ultimate E = mc2
converting mass into energy. One kilogram equals 9e16 joule. The Sun
in its lifetime converts only (6.063e13J)/(9e16J) = 6.74e-4 of its
mass -- barely 7% of 1%! Recall that in a star only about 10% of the
mass is ever involved in generating radiation. The rest is outside of
the core, too 'cold' to undergo nuclear reactions.
Of this 10%, only 0.7% turns into energy by E = mc2. Our 7% of 1%
is very close to the value obtained by stellar evolution theory.
We next reduce this unit energy production to the mass density of
the universe. This density includes matter outside of stars, like
extended gas and dust. A good estimate for the matter engaged in
energy production is 10% of the total observed mass. This does not
consider the 'missing mass' or 'dark matter', which may be as much as
30 times the observable mass.
starlight density = (mass density) * (10%) * (energy/mass)
= (4e-28 kg/m^3) * (0.10) * (6.063e13 J/kg)
= 2.425e-15 J/m^3
(starlight / CMB) = (starlight density) / (CMB energy density)
= (2.425e-15 J/m^3) / (4.235e-14 J/m^3)
= 0.0573
-> 6%
We should allow for other energy processes, like supernovation and
quasars, blackhole activity. We still get only a couple more
percentage points. Essentially all of the energy ever produced in the
universe comes from the CMB. And it was generated only 350,000 years
after the Bigbang! The universe did its thing all in one shot and
little much else happened for the ensuing 14 billion years.
What is more astounding, few modern treatments emphasize this
feature. For all of astronomy history, prior to discovery of the CMB
in 1965, we built our knowledge of the universe thru maybe 8 percent
of the total radiation in it!
To compare this with the overall mass density of the universe, we
have
(density starlight) = (energy density) / c^2
= 2.43e-15 J/m^3) / (9e16 m^2 / s^2)
= 2.70e-32 kg/m^3
The overall mass density of the universe is about 4e-28kg/m^3 for
a ratio of starlight mass to overall mass of 1.75e-5! This minuscule
fraction of mass turned into starlight lets us treat the total mass in
the universe as constant over time. What mass was formed in the
Bigbang exists today.
+------------------------------------+
| ENERGY CONTRIBUTION FROM STARLIGHT |
| |
| starlight = 2.43e-15 J/m^3 |
| = 2.70e-32 kg/m^3 |
+------------------------------------+
Number Density as Entropy
-----------------------
Some cosmologists say that the entropy of the universe is the same
as the ratio of the photons to baryons. The value cited is around a
billion to one. The entropy of the CMB is 2.063e-14J/K.m3 as found
from the Clausius method. By the boltzmann statement of entropy we
have
S = k * ln(N)
= 2.063e-14 J/K.m^3
ln(N) = (2.063e-14 J/K.m^3) / k
= (2.063E-14 J/K.m^3) / (1.381e-23 J/K)
= 1.494e9 1/m^3
This is an entropy measure as a pure number, rather than joule/kelvin.
The densities of photons and baryons in the universe, taken from
litterature, are demsityP = 4.157e8 photon/m3 and densityB = 0.239
baryon/m3. Their ratio is
demsotuP / densityB = (4.157e8 photon/m^) / (0.239 baryon/m^3)
= 1.733e9 photon/baryon
Given the uncertainty in assessing the density of baryons, this is in
good agreement with the entropy value based on Clausius.
+--------------------------------+
| ENTROPY AND PHOTON/BARYON |
| |
| densityP / densityB = 1.733e9 |
| |
| entropy = 1.494e9 |
| |
| entropy = densityP / densityB |
+--------------------------------+
Blackholes
--------
When first seriously considered as real physical bodies, we didn't
believe there would be an application of thermodynamics for them.
Below the event horizon there are no molecules to vibration as heat
energy and there was no obvious way to assign a temperature to a
blackhole. The interior of a blackhole was supposed to be a thoro
vacuum because all mater in it went straight into the central
singularity.
In 1974 Hawking and also Bekenstein worked out that a blackhole
must have thermodynamic properties in order that its behavior emgage
properly with the physics above the event horizon.
Explanation of the new thermodynamics -- still in the 2010s under
development -- requires a whole other discussion than available in
this here piece. I offer only a couple features as seed s for external
menticulture.
Hawking and Bekenstein showed that a blackhole has a temperature
and an interanl energy or entropy. We can reason that a blackhole must
have entropy because it is the fate of energy and mass entering the
blackhole. These do not disappear but add to the energy of the
blackhole.
For a Schwarzschild blackhole, one with only mass as its defining
parameter, there being no angular momentum or electric charge, this
internal energy is converted into more mass for the blackhole. This is
the only significant instance of nature turning energy into mass by
inanimate processes. The temperature of a blackhole is
Tbh = h * c^3 / (16 * pi^2 * gamma * M)
where H is Planck constant; c, speed of light; gamma, Newton constant;
M, blackhole mass. For blackholes formed from stars, at least 3 solar
masses, the temperature is incredibly low, about 1e-7 kelvin.
Altho such a blackhole should radiate thermal blackbody radiation,
it is utterly smothered by the irradiation absorbed by the blackhole
from the ambient 2.7K background radiation and has no astronomy
concern. For a blackhole to have a Hawking-Bekenstein temperaature
equal to the cosmic background, it has to be of mass order the Moon.
As yet we know of none among astronomy objects of interest.
The radiated energy comes from conversion of the blackhole's mass,
being a loss of its entropy. The blackhole entropy is
Sbh = area * (2 * pi * k * c^3 / (h * gamma))
where the k is Boltzmann's constant. This entropy is that built into
the original blackhole when it was created, plus that added by intake
of external mass and energy like from an accretion disc. Of this
entropy some goes into the Bekenstein-Hawking radiation, altho for
astronomy purposes this is negligible. In fact, for blackholes formed
from stars it would take orders of cosmic lifespans to radiate it
away.
I must warn that in some texts the Planck constant is replaced by
the Dirac constant, 'h-bar'. In typeset text the 'h' has a slash thru
its mast. h-bar is just h/2pi, which i use in the above formulae.
From these two basic thermodynamic properties of blackholes, there
is now a vigorous investigation of traditional thermodynamics. This
discipline is still, at least for industry and engineering, bases on
experiment and empirical work. perhaps as the blackhole thermodynamics
evolves, it can apply its theory to traditional thermodynamics