Transcript microstati1

Gases
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I Gas riempiono completamente il recipiente
perchè vi sono molti più modi di far questo, che
di lasciarlo mezzo vuoto.
Ssolid <Sliquid <<Sgas
there are many more ways for the molecules to
be arranged as a liquid than a solid.
Gases have a huge number of positions
possible.
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Entropy and multiplicity
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Motion of each molecule of a gas in a vessel
can be specified by location and velocity

multiplicity due to location and velocity
Ignore the velocity part for the time being and
look at the multiplicity due to location only
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Multiplicity due to location I
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Divide the available space up into c small cells.
Put N particles inside the space: W=cN.
For c=3, N=2: W=32=9
AB
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B
B
A
B
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B
A
A
B
B
A
AB
Multiplicity due to location II
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Increasing the available space is equivalent to
increasing the number of cells c.
The volume is proportional to the number of
cells c
Hence W  VN
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Equilibrium volume
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In general the number of microstates depends
on both the volume available and the
momentum (velocity) of the molecules
Let’s ignore the momentum part and look at
the spatial microstates only.
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Equilibrium volume II
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Say we have 3 molecules in a vessel which we
split up into 6 equal parts. A partition can be
placed anywhere between the cells. One
molecule is on the left-hand side, the other
two on the right-hand side. What is the
equilibrium volume?
Look for maximum entropy!
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Equilibrium volume III
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Number of cells on the left c1, on the right c2.
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We’ll look at c1=4, c2=2:
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BC
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B
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A
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B
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Equilibrium volume IV
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Left: W1= c1=4.
Right: W2= (c2)2=4.
s = ln 4 + ln 4 = ln 16 = 2.77
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BC
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B
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C
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BC
Multiplicity and energy
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According to quantum mechanics, atoms in a
crystal have energies 0, e, 2e, 3e, … (This is
called the Einstein model of solids)
Say we have three atoms with total energy 3e
Microstates are distinguished by the different
energies E1, E2, E3.
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The microstates
3e
2e
E1=3e e
0
Energy
3e
E1=2e 2e
e
0
3e
E1=e 2e
e
0
E1=0
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3e
2e
e
0
E1,E2,E3 E1,E2,E3 E1,E2,E3 E1,E2,E3
Generalisation
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If there are n atoms and the total energy is
qe, then the number of microstates is given by
(q  n  1)!
W( q , n ) 
q!(n  1)!
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Works for previous example (n=3,q=3):
5!
W
 10
3!2!
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Entropy, energy, temperature I
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The internal energy on the left U1 = q1e.
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Equilibrium/entropy maximum when
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ds
0
dU1
Use s = s1 + s2:
Use U2 = U – U1:
ds1 ds 2
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0
dU1 dU1
ds 2 ds 2 dU1
ds 2
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
dU 2 dU1 dU 2
dU1
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Entropy, energy, temperature II
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It follows that the most likely distribution of
energy corresponds to a situation where
ds1 ds 2
dS1 dS2
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or
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dU1 dU 2
dU1 dU 2
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We know that in this situation T1 = T2
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Clearly the two are linked!
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Entropy, energy, temperature III
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Remember: we kept V, N constant so the only
way in which energy could be exchanged was
through heat transfer
Remember:
Q  U 
S   
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T  T due to heating only
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1  S 
S 



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T  U due to heating only  U fixed external parameters
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Entropy, energy, temperature IV
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In our example the only external parameter
was the volume
In general, gravitational, electric or magnetic
fields, elastic energy etc. could all change
The definition of temperature only holds if all
of these are held fixed
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Entropy and the Second Law of Thermodynamics
Entropy (S)–Entropy is a measure of the disorder of the system
A system with relatively few equivalent ways to arrange its
components, such as a crystalline solid or a deck of cards in a
specific sequence, has relatively small disorder and low entropy.
A system with many equivalent ways to arrange its components,
such as a gas or a shuffled deck of cards, has relatively large
disorder and high entropy.
Sdisorder > Sorder
Ssys = Sfinal - Sinitial
Second Law of Thermodynamics–all processes occur spontaneously
in the direction that increases the total entropy of the universe
(system plus surroundings).
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Predicting Relative Entropy Values
Problem: Choose the member with the higher entropy in each of the
following pairs, and justify your choice [assume constant temperature,
except in(f)]:
(a) 1 mol NaCl(s) or 1 mol NaCl(aq)
(b) 1 mol SF6 or 1 mol SCl6
(c) 1 mol CO(g) or 1 mol CO2 (g)
(d) 1 mol S8 or 4 mol S2
(e) 1 mol H2O(s) or 1 mol H2O(g)
(f) Lipton’s noodle soup at 24oC or at 95oC
Plan: The less ordered a system, the greater the entropy. A higher
temperature increases entropy.
Solution:
(a) 1 mol NaCl(aq). The two samples have the same number of ions,
but in the solid they are highly ordered, and in the solution they are
randomly dispersed.
(b) 1 mol SCl6. For similar compounds, entropy increases with molar
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mass.
Solution: Continued
(c) 1 mol CO2 (g) . For equal numbers of moles of substances with the
same types of atoms in the same physical state, the more atoms per
molecule, the more types of motion available to it and, thus, the higher
its entropy.
(d) 4 mol S2. The two samples contain the same number of sulfur atoms,
but different numbers of molecules. Despite the greater complexity of
S8 , the greater number of molecules dominates in this case because
there are many more ways to arrange 4 moles of particles than one mole.
(e) 1 mol H2O(g). For a given substance entropy increases in the
sequence: solid < liquid < gas.
(f) Soup at 95oC. Entropy increases with increasing temperature.
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Entropy, S
One property common to productfavored processes is that the
final state is more
DISORDERED or RANDOM than
the original.
Spontaneity is related to an
increase in randomness.
The thermodynamic property
related to randomness is
ENTROPY, S.
Reaction of K
with water
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Directionality of Reactions
How probable is it that
molecules will react?
PROBABILITY suggests that a
product-favored reaction will
result in the dispersal of
energy or of matter or both.
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Directionality of Reactions
Probability suggests that a productfavored reaction will result in the
dispersal of energy or of matter
or both.
Matter Dispersal
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Directionality of Reactions
Probability suggests that a productfavored reaction will result in the
dispersal of energy or of matter
or both.
Energy Dispersal
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Directionality of
Reactions
Energy Dispersal
Exothermic reactions involve a release of
stored chemical potential energy to the
surroundings.
The stored potential energy starts out in a
few molecules but is finally dispersed over a
great many molecules.
The final state—with energy dispersed—is
more probable and makes a reaction
product-favored.
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Entropy, S
The entropy of a substance
increases with temperature.
Molecular motions
of heptane, C7H16
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Molecular motions of
heptane at different temps.
Entropy, S
An increase in molecular
complexity generally leads to
an increase in S.
So (J/K•mol)
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CH4
248.2
C2H6
336.1
C3H8
419.4
Entropy, S
Entropies of ionic solids depend
on coulombic attractions.
So (J/K•mol)
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MgO
26.9
NaF
51.5
Entropy
Amount of disorder
 effected by degree of motion
 complexity of shape
 presence of other species
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Randomness, Order and Evolution
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Are the following letter sequences random:
crvn, smrt, vrlo, gdje, trg?
In Serbo-Croatian, the words mean,
respectively, red, death, very, where and town
square.
Moral: the fact that something looks random
doesn't mean it is. It may convey meaning in a
way you don't understand.
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Randomness,
Order
and
Evolution
Is the following number sequence random:
592653589793238462643383279?
It not only looks random: it is random.
But lacking in meaning? No. These are the
digits of pi beginning with the fourth decimal
place.
Random does not mean “meaningless”
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Can Order Arise Naturally?
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The Second Law of Thermodynamics is often
paraphrased as:
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”Things always go from bad to worse”
”Disorder in the Universe is always increasing"
The core of the Second Law is entropy
Entropy can decrease locally if it increases
elsewhere
Intuitive notions of disorder are of no
relevance whatsoever
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