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Material Type: Notes; Professor: Thaler; Class: Univ Physics: Thermal Physics; Subject: Physics; University: University of Illinois - Urbana-Champaign; Term: Fall 2008;
Typology: Study notes
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Physics 213: Lecture 11, Pg 1
Physics 213: Lecture 11, Pg 2
Lecture 11 Lecture 11
Equilibrium Between Particles Equilibrium Between Particles
Agenda for today Agenda for today
Free Energy and Chemical Potential
Simple defects in solids
Ideal gases, revisited
Reference for this Lecture:
Elements
Ch 11
Reference for Lecture 12:
Elements
Ch 12
Physics 213: Lecture 11, Pg 4
The path ahead... The path ahead...
We have thus far studied systems in which volumes can beexchanged
this led to mechanical equilibrium and equal
pressures in the two volumes.
We then studied systems in which energy can be exchanged
this
led to thermal equilibrium and equal temperatures of the systems.
Now we consider systems in which
particles
can be exchanged, e.g.,
Particles can move from place to place
Particles can combine (chemistry, vacancy-interstitialrecombination, electron-hole recombination, nuclear reactions…)into new types.
This will lead to “chemical equilibrium”, in which the “chemicalpotential” is equalized, the free energy of the system is minimized,and the
total
entropy is maximized (which after all is the most
probably state of affairs...)
And lots and lots of applications...
We start with a concrete example...
Physics 213: Lecture 11, Pg 5
Atoms can pop out of position and sit in interstitial sites.
These defects (and others) play a crucial role in thethermal weakening of materials, etc.
In thermal equilibrium, how many are there?
(determines how hot is safe for some structure to get)
We calculate the number N
I
of interstitials to minimize F
Let’s say that an atom can either go to the surface, making a‘normal’ site, or sit at an interstitial site, with extra energy
I
For big crystals, the surface energy and entropy changes arenegligible. so the changes to F from making an interstitial comefrom its own extra energy and entropy.
Interstitial
Physics 213: Lecture 11, Pg 7
Because it costs energy to sit at the interstitialsites (if energy were lowered, then the crystalstructure itself would rearrange itself tominimize the energy), at some sufficiently lowtemperature the entropy gain by sitting at thesite (more places to sit) will dominated by theenergy cost
the free energy will be
minimized by “staying at home”.We might expect a Boltzmann-like dependenceon temperature...
Physics 213: Lecture 11, Pg 8
Say we have
N atoms, N possible interstitial sites
We want to know
I
, the number of interstitials at temperature T.
method: minimize F(N
I
), Helmholtz free energy for N
i
interstitials
» Call F(0)=0 for convenience» Assume the vibrational entropy is not much changed by
making an interstitial but U is increased by an amount
Ι
!
(
)
(
)
(
)
ln
!(
)!
I
I
I
I
I
I
I
N
F N
U N
TS N
N
kT
N
N
N
=
−
=
∆ −
−
# ways to put N
I
identical
particles in N bins.
(single occupancy)
Need energy
ΙΙΙΙ
for each defect
Physics 213: Lecture 11, Pg 10
To find N
I
we
set the ‘derivative’ to zero:
0
(ln(
)
ln(
))
ln(
)
I
I
I
I
I
I
I
N
dF
kT
N
N
N
kT
dN
N
N
=
= ∆ +
−
−
= ∆
−
So:
ln(
)
I
I
I
I
kT
I
I
N
N
e
N
N
kT
N
N
∆
−
∆
= −
⇒
=
−
−
Usually, N
I
<< N, so:
I
I
kT
N
e
N
∆
−
=
This looks like a simple Boltzmann expression.
Exponential increaseas T increases
Physics 213: Lecture 11, Pg 11
Atoms can also go out to the surface and leave vacancies.
Following exactly the same math:
V
V
kT
V
N
e
N
N
∆
−
=
−
Usually, N
V
<< N, so:
V
V
kT
N
e
N
∆
−
=
vacancy
Again, surface energy/entropychanges in a BIG crystal are negligible,unlike in this little picture.
Physics 213: Lecture 11, Pg 13
V
V
kT
N
e
N
∆
−
=
5
ln(0.01)
0.5eV
4.6(8.
10
eV/K
1264 K
1
000 C
V
V
kT
V
e
kT
kT
T
∆
−
−
∆
=
⇒
= −
=
∆
=
⇒
=
×
=
≈
Physics 213: Lecture 11, Pg 14
Particle equilibrium: chemical potential Particle equilibrium: chemical potential
μ μ
μ μ μμμμ
These solid defects provide examples in which some
particle
number
can vary. We want a convenient way to treat such
problems—we invent a new property,
chemical potential (
, which
simplifies the business of maximizing total S via minimizing F insuch problems (just as “T” simplified the description of how todistribute energy to maximize total S). Using
μμμμ
lets us treat
problems where we can’t quite list all the states.
Roughly speaking,
μμμμ
is the amount the (free) energy of a system
changes if we add one particle.
Volume exchange
p
1
= p
2
mechanical equilibrium
Energy exchange
T
1
= T
2
thermal equilibrium
Particle exchange
μ μ
μ μ
1
=
μμμμ
2
chemical equilibrium
Physics 213: Lecture 11, Pg 16
Now let’s look at a solid which doesn’t have time for defects todiffuse to or from the surface (a common case)
Now making an interstitial leaves behind a vacancy, i.e., dN
V
= dN
I
If we
start
with a perfect crystal:
V
I
Note this is different than our first example—exchanging particlebetween two volumes— there dN
1
dN
2
Therefore
μ
I
μ
V
= 0
Interstitialvacancy
V
V
I
I
I
I
I
I
V
Physics 213: Lecture 11, Pg 17
In equilibrium:
I
V
, so:
(
)
2
2
2
ln(
)
ln(
)
0
2
ln(
)
0 since here
(
)
if
then
I
V
T
T
V
I
I
V
I
V
V
I
V
V
I
V
V
kT
kT
T
I
V
V
kT
V
V
I
N
N
kT
kT
N
N
N
N
N
kT
N
N
N
N
N
e
e
N
N
N
N
N
N
N e
∆ +∆
∆
−
−
∆
−
∆ +
∆
=
−
−
∆ + ∆
=
=
−
=
=
∆
≡ ∆ + ∆
−
=
=
≪
This looks almost like Boltzmann, except for
kT,
since
both
defects make entropy.
Physics 213: Lecture 11, Pg 19
I
V
I
V
2
T
I
V
kT
N N
e
N
∆
−
=
Physics 213: Lecture 11, Pg 20
I
V
I
V
2
T
I
V
kT
N N
e
N
∆
−
=
