ENTROPY AND THE SECOND LAW OF THERMODYNAMICS:
............How the universe works
The first law of thermodynamics is simply a statement of
energy conservation. That is, it states that energy can always be accounted
for, that the energy of the universe is a constant - it can be transferred
between objects and can change form, but the total doesn't change. But the
first law does not preclude things occuring that we know do not occur: A glass
of water does not spontaneously separate into ice cubes and warm water even
though the energy balance equations used in calorimetry problems would allow
it. That is, energy conservation - the first law of thermodynamics - would
allow for the possibility that a system in thermal equilibrium could separate
into two systems - one at a higher temperature than the other - and that
temperature difference could then be used to drive a heat engine to do work.
The second law of thermodynamics explains why the universe does not work that
way. It articulates the underlying principle that gives the direction of heat
flow in any thermal process. The result, of course, fits our everyday
experience. The second law states the reason why it is true.
Heat naturally flows from higher temperatures to lower
temperatures.
No natural process has as its
sole result the transfer of heat from a cooler to a warmer object.
No process can convert heat
absorbed from a reservoir at one temperature directly into work without also
rejecting heat to a cooler reservoir. That is, no heat engine is 100%
efficient.
Carnot Cycle - Maximum Thermodynamic Efficiency in a
Cyclic Process
It was observed by Sadi Carnot
- a French scientist and engineer trying to improve the efficiency of steam
engines in the mid-1800s - that there is always waste heat rejected by a heat
engine. And that waste heat limits the efficiency of the engine since energy
has to be conserved or accounted for. In trying to understand the limits of
efficiency, he stated that in any heat engine in principle there would always
be rejected heat (even in an ideal engine) - and the net work done would be the
difference between the heat absorbed and that rejected. He then set out to
determine the principles that would affect that efficiency. He stated that the
most efficient heat engine possible would be one that worked reversibly - an
ideal that could never be attained. This would mean that heat transferred into
or out of the system (the heat engine) would only occur at constant
temperatures - the high or the low temperatures between which the heat engine
operated. That is, the system would stay at the temperatures of the reservoirs
during those heat transfers - necessary for the process to be reversible since
the heat flow could not be reversed to go from the lower to the higher
temperature. And furthermore, said Carnot, the maximum conceivable efficiency
would be limited by those two temperatures. The most efficient thermodynamic
cycle operated between any two temperatures is therefore called a Carnot
cycle.
The Carnot cycle is a four
step process involving two isothermal processes (which are said to be ideal
reversible processes) at the temperatures Th and Tc and two adiabatic processes
(ie, without heat transfer) which operate between those two temperatures. In
the isothermal steps, there is no change in internal energy and the heat
exchanged is equal to the work done. In the two adiabatic processes, there is
no heat exchanged. No such system can ever be built - since it is an idealized
process (the two isothermal steps being reversible and quasistatic which means,
in effect, they occur infinitely slowly). The importance of the process is that
it gives an upper limit to the efficiency of any cyclic process between the
same two temperatures.
Entropy and the Second Law of
Thermodynamics
In trying to synthesize the
ideas of Kelvin, Joule, and Carnot - that is, that energy is conserved in
thermodynamic processes and that heat always "flows downhill" in
temperature - Rudolf Clausius invented the idea of entropy in
such a way that the change in entropy is the ratio of the heat exchanged in any
process and the absolute temperature at which that heat is exchanged. That is,
he defined the change in entropyDS of an object which either absorbs or gives
off heat Q at some temperature T as simply the ratio Q/T.
With this new concept, he was
able to put the idea that heat will always flow from the higher to the lower
temperature into a mathematical framework. If a quantity of heat Q flows
naturally from a higher temperature object to a lower temperature object -
something that we always observe, the entropy gained by the cooler object
during the transfer is greater than the entropy lost by the warmer one since
Q/Tc.>|Q|/Th. So he could state that the principle that drives all natural
thermodynamic processes is that the effect of any heat transfer is a net
increase in the combined entropy of the two objects. And that new principle
establishes the direction that natural processes proceed. All natural processes
occur in such a way that the total entropy of the universe increases. The only
heat transfer that could occur and leave the entropy of the universe unchanged
is one that occurs between two objects which are at the same temperature - but
that is not possible, since no heat would transfer. So a reversible isothermal
heat transfer that would leave the entropy of the universe constant is just an
idealization - and hence could not occur. All other processes - meaning, all real processes
- have the effect of increasing the entropy of the universe. That is the second
law of thermodynamics.
Entropy is a measure of the
disorder of a system. That disorder can be represented in terms of energy that
is not available to be used. Natural processes will always proceed in the
direction that increases the disorder of a system. When two objects are at
different temperatures, the combined systems represent a higher sense of order
than when they are in equilibrium with each other. The sense of order is
associated with the atoms of system A and the atoms of system B being separated
by average energy per atom - those of A being the higher energy atoms if system
A is at a higher temperature. When they are put in thermal contact, energy
flows from the higher average energy system to the lower average energy system
to make the energy of the combined system more uniformly distributed - ie, less
ordered. So the disorder of the system has increased - and we say the entropy
has increased. But the process of increasing the disorder has removed the
possibility that the energy that was transferred from A to B can be used for
any other purpose - for example, work cannot be extracted from the energy by
operating a heat engine between the two reservoirs of different temperatures.
So although energy was conserved in the transfer (the first law), the entropy
of the universe has increased in becoming more disordered (the second law) and
consequently the availability of energy for doing work has decreased.
The second law of
thermodynamics can be summarized in many different statements - and has been by
many thermodynamicists in the last century and a half. All of the statements
are an attempt to put a reason to the things all of us have observed - that
when two objects are in thermal contact, heat always goes from the warmer to
the cooler and never the other way. This universal result has probably as many
explanations as there are physicists trying to explain it - and is still the
subject of serious consideration by some of the best theorists. The difficulty
does not lie in what the second law says - or how it should be interpreted -
but rather in what the fundamental, underlying reason is for why nature behaves
in that way.
Any process either increases the entropy of the universe
- or leaves it unchanged. Entropy is constant only in reversible processes
which occur in equilibrium. All natural processes are irreversible.
All natural processes tend
toward increasing disorder. And although energy is conserved, its availability
is decreased.
Nature proceeds from the
simple to the complex, from the orderly to the disorderly, from low entropy to
high entropy.
The entropy of a system is
proportional to the logarithm of the probability of that particular
configuration of the system occuring. The more highly ordered the configuration
of a system, the less likely it is to occur naturally - hence the lower its entropy.
In the language of entropy, the Carnot cycle still
represents the theoretical maximum efficiency in any cyclic process. That is,
maximum efficiency would occur if the entropy of the universe did not increase,
hence there would be no loss of availability of doing work. But entropy can
only remain constant in a reversible isothermal process. So, again, any heat
transfer would have to occur isothermally. Therefore the most efficient cyclic
process possible involves only reversible isothermal steps and steps in which
no heat is transferred - ie, adiabatic. And even in this idealized reversible
process in which the entropy of the universe was left unchanged, the efficiency
of conversion of heat to work is limited by the two temperatures involved in
the isothermal steps.
Based on the ideas of Lord
Kelvin, Joule, Boltzmann, Carnot, and Clausius, the first and second laws of
thermodynamics can now be restated in two profound sentences:
The total energy of the universe is a constant.
The total entropy of the universe
always increases.
And these
two fundamental principles of nature describe how the universe works.
