Statistical Thermodynamics
Z. Suo
Electric potential
Electric charge. The net electric charge in the world is constant. If electric charge
increases in one place, electric charge must decrease by the same amount somewhere else.
Every electron has the same charge. Every proton has the same charge, which has the same
amount as, but an opposite sign to, that of each electron. The charge of each proton is called the
elementary charge. A system usually contains electrons and protons, so that the total charge in
the system is the elementary charge times the algebraic sum of the number of protons and
electrons in the system.
For example, a water molecule has 10 electrons and 10 protons, so that the molecule is
neutral. A glass of wine has a large number of electrons and protons. A large system is usually
nearly neutral: the difference in the number of electrons and the number of protons is much
smaller than the number of electrons. Thus, to obtain the net charge of a large system, counting
the two large numbers and then finding their difference is usually a bad method. There are
number of more effective methods to measure charge on a system, as described in textbooks of
electromagnetism.
Electric charge is commonly reported in two units. The first unit is just the number; for
example, if a drop of oil picks up 3 electrons, you say that the charge of the drop is − 3 . The
second unit is called Coulomb. Here is the conversion factor of the two units: the charge of a
proton is 1.60 × 10−19 Coulomb. Thus, the charge on the drop of oil is − 4.80 × 10−19 Coulomb.
Movements of charged particles. We distinguish two kinds of movements of charged
particles:
•
Conduction: charged particles move over macroscopic distances. Examples include
electrons moving in space, electrons moving in a metal line, ions moving in an
electrolyte, and holes moving in a semiconductor.
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Statistical Thermodynamics
•
Z. Suo
Polarization: charged particles move relative to each other by a small amount, and is
restrained by inter-particle forces. Examples include a small distortion of the electron
cloud of an atom, a small change in distance between a positive ion and a negative ion in
a dielectric.
Elastic dielectric. All materials contain electrons and protons.
In a dielectric, these
charged particles form bonds, and move relative to one another by short distances in response to
a voltage or a force. That is, all dielectrics are deformable. The notion of a rigid dielectric is as
fictitious as that of a rigid body: they are idealizations useful for some purposes, but misleading
for others.
Work done by a battery and by a weight. The following figure illustrates a system of
insulators and conductors, loaded by a field of weights and batteries, of which only one of each
is drawn. All batteries are connected to a common ground. We can measure the displacement
δl of the weight, and the amount of charge δQ pumped by the battery from the ground to the
electrode. There might be other weights dropping or rising and other batteries pumping charge
from or to the ground, but the work done by this particular weight is Pδl , and the work done by
this particular battery is ΦδQ . If we regard work, displacement and charge as primitive,
measurable quantities, the above statements of work define the force P supplied by the weight,
and the voltage Φ supplied by the battery. The force is said to be the work conjugate to the
displacement, and the voltage the work conjugate to the charge. We will use the word weight as
shorthand for all mechanisms (including inertia) that do work through displacements, and the
word battery as shorthand for all mechanisms that do work through flows of charge. We will
neglect the effects of magnetism and electromagnetic radiation.
The unit of electric potential is energy/charge. When the unit of charge is Coulomb, and
the unit of energy is Joule, the unit of electric potential is volt.
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Statistical Thermodynamics
Z. Suo
electrode
dielectric
δQ
Φ
ground
P
δl
Electromechanical coupling. Now imagine that the weight and battery are adjustable,
so that the force P and the voltage Φ can vary. When the displacement is held constant, a
change in the charge may cause the force to change. When the charge is held constant, a change
in the displacement may cause the voltage to change. These electromechanical coupling effects
are universal to all dielectrics, because all dielectrics have electrons and protons, and the charged
particles can move relative to one another.
Conservative system.
The two electromechanical coupling effects are linked for
conservative systems. A conservative system is one for which, under isothermal conditions, the
work done by the weight and the battery is fully stored as the Helmholtz free energy of the
system. That is, associated with small changes δl and δQ , the free energy of the system, F,
changes by
δF = Pδl + ΦδQ .
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Statistical Thermodynamics
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To this equation we should add the work done by all other weights and batteries. For simplicity,
however, here we assume that only one weight and one battery do work. This may be achieved
by removing all other weights and batteries, and making sure that every other part in the system
other than the particular electrode is either grounded or charge neutral. The temperature is held
constant, and we do not list it as a variable.
These idealizations ensure that the free energy of the system is a function of two
variables, F (l , Q ) . We only need to measure the difference in F, l and Q between the current
state and a reference state. For the conservative system, the force and the voltage are partial
derivatives:
P=
∂F (l , Q )
,
∂l
Φ=
∂F (l , Q )
.
∂Q
Associated with small changes δl and δQ , the force and the voltage change by
δP =
∂ 2 F (l , Q )
∂ 2 F (l , Q )
δ
δQ ,
+
l
∂l∂Q
∂l 2
δΦ =
∂ 2 F (l , Q )
∂ 2 F (l , Q )
δl +
δQ .
∂l∂Q
∂Q 2
We may call ∂ 2 F (l , Q ) / ∂l 2 the mechanical tangent stiffness of the system, and ∂ 2 F (l , Q ) / ∂Q 2
the electrical tangent stiffness of the system. The two electromechanical coupling effects are
both characterized by the same cross derivative, namely,
∂P(l , Q ) ∂ 2 F (l , Q ) ∂Φ(l , Q )
=
=
.
∂Q
∂l∂Q
∂l
Consequently, for a conservative system, the two electromechanical coupling effects
reciprocate.
Experimental determination of electric potential. Electric potential can be determined
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Statistical Thermodynamics
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by measurable quantities. For example, using an elastic dielectric, an adjustable weight and an
adjustable battery, we can measuring P for various l and Q.
That is, by experimental
measurement, we can determine the function P (l , Q ) . We then adjust the electric potential such
that the electric charge on the electrode is held constant, so that
δΦ =
∂P(l , Q )
δl .
∂l
This will determine the electric potential of the battery up to an additive constant.
Lagendre transformation. Define
Π = F (l , Q ) − ΦQ .
Its differential is
dΠ = Pdl − QdΦ ,
so that Π is a function of (l , Φ ) , and
P=
∂Π (l , Φ )
,
∂l
Q=−
∂Π (l , Φ )
.
∂Φ
Parallel-plate capacitor. As an illustration, consider the parallel-plate capacitor, loaded
by both the voltage and the weight. The two electrodes are separated by a vacuum. The
separation between the two electrodes l may vary, but the area of either electrode remains to be
A.
The battery maintains a charge + Q on one electrode and a charge − Q on the other
electrode. The charges cause the two plates to attract each other, and the attraction is balanced
by the weight. Let us say that we have done the experiment, or a more practical version of the
experiment, and found that that the charge on either electrode is linear in the electric potential:
Φ
Q
= ε0 .
A
l
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Statistical Thermodynamics
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The ε 0 is a constant characteristic of the vacuum known as the permittivity of vacuum. Our
experiment shows that ε 0 = 8.85 × 10−12 F / m if we use meter for length, volt for electric potential
and Coulomb for charge. Here F stands for Faraday, a derived the unit consistent with the above
expression.
Recall that Φ(Q, l ) = ∂F (Q, l ) / ∂Q . In this case, vacuum may be taken to have zero
entropy, and the Helmholtz function is identical to the internal energy. Integrating in Q gives
lQ 2
.
F (l , Q ) =
2ε 0 A
Recall that P (Q, l ) = ∂F (Q, l ) / ∂l . The weight needed to balance the electric attraction of
the two plates is
P(Q, l ) =
Q2
.
2 Aε 0
A
Φ
−Q
l
+Q
P
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