Transcript Physics 201 - University of Virginia

Entropy
Lecturer:
Professor Stephen T. Thornton
Reading Quiz:
Entropy is
A) another form of energy.
B) the name of a famous heat engine.
C) related to the 4th Law of
Thermodynamics.
D) the primary reason your bedroom is
(or may be) so messy.
Answer: D
Entropy is related to disorder and the
probability of events in the universe.
There is not much you can do about
the fact that your bedroom naturally
becomes messy, unless you expend a
good deal of work. Unfortunately
that leads to a heat death of the
universe.
Last Lecture
Second Law of Thermodynamics
Carnot cycle and Carnot engine
Heat engines
Refrigerators
Air conditioners
Heat pumps
Today
Entropy
Statistical interpretation - entropy
3rd Law of Thermodynamics
Entropy
There are several ways to look at entropy,
but eventually they are all equal.
Entropy is related to disorder in a system.
A messy bedroom has more entropy than a
clean one.
The natural order of the universe is to
increase entropy. Your bedroom never
naturally becomes clean; it always naturally
becomes messy.
Entropy is also related to probability.
There is a higher probability that a block of
ice will melt at room temperature than it
will get colder. Thermodynamics does not
prevent either action. The probability of
the latter is incredibly small.
Conceptual Quiz:
Humpty Dumpty falls off and breaks. Can
he get back together again?
Humpty Dumpty sat on a wall,
Humpty Dumpty had a great fall;
All the King's horses and all the
King's men,
Couldn't put Humpty together again.
A) Yes, very easily.
B) Yes, but with
extremely low
probability.
C) No, there is no
possibility.
D) Are you kidding us?
Answer: B
From what we just learned, this is
only a question of probabilities. And
no, I am not kidding!
Entropy is a very fundamental property and
is a state variable. It is determined by the
heat flow Q divided by the temperature T.
Q
S 
T
dQ
dS 
T
for constant T
for nonconstant T
Consider the heat flow change to be
reversible. T is in kelvin.
Heat added to system, S > 0.
Heat removed from system, S < 0.
When the temperature changes, things are a
little more complicated. Consider the change
of entropy from state a to state b.
b
b
a
a
S  Sb  Sa   dS  
dQ
T
This is for a reversible process. ΔS
depends only on states a and b, because
entropy is a state variable (like P, V, T, but
not like Q which is not a state variable).
S total
For a reversible heat engine, the
total entropy of the engine cycle is
Qh Qc
 
0
reversible
Th Tc
Real engines have friction and can't quite
be reversible.
S total
Qh Qc
 
0
Th Tc
real engine
All irreversible processes cause an increase
in entropy. (Q is positive here.)
Any reversible cycle can be written as
a succession of Carnot cycles;
therefore, what is true for a Carnot
cycle is true of all reversible cycles.
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Entropy and the
Second Law of Thermodynamics
The total entropy always increases when
heat flows from a warmer object to a
colder one in an isolated two-body
system. The heat transferred is the
same, and the cooler object is at a lower
average temperature than the warmer
one, so the entropy gained by the cooler
one is always more than the entropy lost
by the warmer one.
Q Q
Stotal  
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Th

Tc
0
The fact that after every interaction the
entropy of the system plus the
environment increases is another way of
putting the second law of
thermodynamics:
The entropy of an isolated system never
decreases. It either stays constant
(reversible processes) or increases
(irreversible processes).
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The total entropy of the universe increases
whenever an irreversible process occurs.
The total entropy of the universe is
unchanged whenever a reversible process
occurs.
This is another way to state the 2nd Law of
Thermodynamics.
There is some really bad news here.
Because the universe actually works
through irreversible processes, the
entropy is gradually increasing. There
will eventually be a gradual “heat death”
of the universe. The universe will be full
of energy which cannot be used to
perform work! We are doomed!
Order, Disorder, and Entropy
As we have stated, entropy is related to
disorder.
As the entropy of a system increases, its
disorder increases as well.
GOOD NEWS: When you go home, and your
mother fusses about how messy your bedroom
is, tell her it is because entropy is increasing,
and it is the natural order of the universe. There
is little you or your mother can do about it
(without doing a lot of work!). She will be
impressed by how much physics you have
learned!
Expanding Ideal Gas. An ideal
gas expands isothermally (T = 410
K) from a volume of 2.50 L and a
pressure of 7.5 atm to a pressure
of 1.0 atm. What is the entropy
change for this process?
Entropy is a measure of the disorder of a
system. This gives us yet another
statement of the second law:
Natural processes tend to move
toward a state of greater disorder.
Example: If you put milk and sugar in your coffee
and stir it, you wind up with coffee that is
uniformly milky and sweet. No amount of stirring
will get the milk and sugar to come back out of
solution.
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Another example: When a tornado hits a building,
there is major damage. You never see a tornado
approach a pile of rubble and leave a building
behind when it passes.
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Unavailability of Energy; Heat Death
Another consequence of the second law:
In any natural process, some energy becomes
unavailable to do useful work.
If we look at the universe as a whole, it seems
inevitable that, as more and more energy is
converted to unavailable forms, the ability to do
work anywhere will gradually vanish. This is
called the heat death of the universe.
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Statistical Interpretation of
Entropy and the Second Law
Microstate: a particular configuration of atoms
Macrostate: a particular set of macroscopic variables (in
thermodynamics P, V, T, S)
This example uses coin tosses:
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Statistical Interpretation of
Entropy and the Second Law
The most probable distribution of speeds in a
gas is Maxwellian:
Highly
unlikely
The most probable state is
the one with the greatest
disorder, or the greatest
entropy. With k being
Boltzmann’s constant and
W the number of
microstates, Boltzmann
showed
S = k ln W
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Statistical Interpretation of
Entropy and the Second Law
In this form, the second law of
thermodynamics does not forbid
processes in which the total entropy
decreases; it just makes them
exceedingly unlikely.
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Conceptual Quiz:
According to the second law of
thermodynamics, for any process that may
occur within an isolated system, which one of
the following choices applies?
A) Entropy increases.
B) Entropy remains constant.
C) Entropy decreases.
D) Both A and B are possible.
E) Both A and C are likely.
Answer:
D) Both A and B are possible.
Entropy can increase for an irreversible
process or remain constant for a
reversible process, but that is all that is
likely.
Thermodynamic Temperature
Since the ratio of heats exchanged between
the hot and cold reservoirs in a Carnot engine
is equal to the ratio of temperatures
(QL/QH = TL/TH), we can define a temperature
scale using the triple point of water:
T = (273.16K)(Q/Qtp).
Here, Q and Qtp are the heats exchanged by a
Carnot engine with reservoirs at temperatures
T and Ttp.
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Also, since the maximum efficiency of a
heat engine is
TL
e = 1TH
there is no way to achieve a temperature of
absolute zero. This is the third law of
thermodynamics:
It is not possible to reach absolute zero in
any finite number of processes.
Third Law of Thermodynamics
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Thermal Pollution, Global Warming,
and Energy Resources
Over 90% of the energy used in the U.S. is
generated using heat engines to drive turbines
and generators—even nuclear power plants
use the energy generated from fission to heat
water for a steam engine. The thermal output
QL of all these heat engines contributes to
warming of the atmosphere and water. This is
an inevitable consequence of the second law
of thermodynamics.
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Dice Probabilities. Calculate
the relative probabilities, when
you throw two dice, of obtaining
(a) a 7, (b) an 11, (c) a 4.
Coin Probabilities & Entropy. (a) Suppose
you have four coins, all with tails up. You
now rearrange them so two heads and two
tails are up. What was the change in entropy
of the coins? (b) Suppose your system is the
100 coins in a given table; what is the change
in entropy of the coins if they are mixed
randomly initially, 50 heads and 50 tails, and
you arrange them so all 100 are heads? (c)
Compare these entropy changes to ordinary
thermodynamic entropy changes, such as
given in textbook examples.
Refrigeration Units. Refrigeration units can be
rated in “tons.” A 1-ton air conditioning system can
remove sufficient energy to freeze 1 British ton
(2000 pounds = 909 kg) of 0°C water into 0°C ice in
one 24-h day. If, on a 35°C day, the interior of a
house is maintained at 22°C by the continuous
operation of a 5-ton air conditioning system, how
much does this cooling cost the homeowner per
hour? Assume the work done by the refrigeration
unit is powered by electricity that costs $0.10 per
kWh and that the unit’s coefficient of performance is
15% that of an ideal refrigerator. 1 kWh = 3.60 x
106 J.
Conceptual Quiz
A) positive
In the closed thermodynamic
cycle shown in the P-V diagram,
B) zero
C) negative
the work done by the gas is:
P
V
Conceptual Quiz
In the closed thermodynamic
A) positive
cycle shown in the P-V diagram,
B) zero
the work done by the gas is:
C) negative
The gas expands at a higher pressure
and compresses at a lower pressure.
In general, clockwise = positive work;
P
counterclockwise = negative work.
V
Conceptual Quiz:
Which of the following is a statement of the third
law of thermodynamics?
A) If two objects are in equilibrium with a third, then
they are in thermal equilibrium with one another.
B) All reversible engines operating between the
same two temperatures have the same efficiency.
C) The entropy of the universe cannot decrease.
D) The entropy of the universe cannot increase.
E) It is impossible to lower the temperature of an
object to absolute zero in a finite number of steps.
Answer:
E) The other answers do not refer to the
third law of thermodynamics.
Conceptual Quiz
Which of the following statements is not true?
A) Entropy of an isolated system tends to
increase.
B) Natural systems move to a state of greater
order.
C) In any natural process, some energy
becomes unavailable to do useful work.
D) It is not possible to reach absolute zero in
any number of finite number of processes.
Conceptual Quiz
B) Natural systems move to a state of greater
disorder, not order. Remember that your
bedroom always become messy, not clean.
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