ESCI 341 –
Atmospheric Thermodynamics
Lesson 10 – The
Energy Minimum Principle
References: Thermodynamics
and an Introduction to Thermostatistics, Callen
Physical Chemistry,
Levine
THE ENTROPY MAXIMUM PRINCIPLE
l We’ve
seen that a reversible process achieves maximum thermodynamic efficiency.
l Imagine
that I am adding heat to a system (dQin), converting some of
it to work (dW), and discarding the remaining heat (dQout).
l We’ve
already demonstrated that dQout cannot be zero (this would
violate the second law).
l We
also have shown that dQout will be a minimum for a
reversible process. This means that the
differential of heat for the system (dQ
= dQin - dQout) will be a maximum for the
reversible process,
.
l The
change in entropy of the system, dS, is defined as dQrev/T. We can thus write the inequality
,
or
l For
an isolated system (dQ = 0) the inequality becomes
,
which is just
a restatement of the second law of thermodynamics.
l We
can divide this expression by dt and find that
.
l For
an unconstrained, isolated system irreversible process will operate and
increase the total entropy of the system until equilibrium is reached. Once equilibrium is reached,
,
and the
irreversible processes will cease.
l This
is the entropy maximum principle.
It states that
o
For an isolated, unconstrained system, the
equilibrium state will be that state which has the maximum entropy for a given
internal energy.
l An
example of application of the entropy maximum principle is an isolated metal
bar that is initially hotter at one end than another. The temperature distribution that maximizes
the entropy is one that is uniform (isothermal). Therefore, if left alone, the temperature
will spontaneously adjust to an isothermal distribution.
l Some
authors have mistakenly applied the entropy maximum principle in an attempt to
explain why a well-mixed layer of air has an adiabatic temperature
profile. However, it turn out that an
isothermal layer has greater entropy than any other temperature profile for the
same static energy. This was shown by J.
Willard Gibbs in 1928 (Collected Works of
J. Willard Gibbs, Vol I: Thermodynamics, pp. 145).
THE ENERGY MINIMUM PRINCIPLE
l The
first law for can be written as
,
so that we
arrive at another useful inequality,
,
or
.
l This
inequality tells us that, for a system at constant entropy and volume,
.
o
This means that the system held at constant
entropy and volume will continue to change until it reaches a point where dU/dt
= 0. At this point, the internal energy,
U, will be a minimum and the system will be in material equilibrium.
l This
principle is known as the energy minimum principle. It states that
o
For an unconstrained system at constant
volume and entropy, the equilibrium state will be that state which has the
minimum internal energy.
l The
energy minimum principle was derived from the entropy maximization
principle. The energy minimum
principle and the entropy maximum principle are complimentary.
ENTHALPY
l Beginning
with
and writing
we get
.
l We’ve
seen the combination U + pV before…it is another state function
called enthalpy (H). Therefore,
.
l For
processes occurring at constant entropy and pressure,
.
l An
unconstrained system at constant entropy and pressure will spontaneously evolve
into a state that minimizes enthalpy.
l This principle is known as the enthalpy
minimum principle. It states that
o
For an unconstrained system at constant
pressure and entropy, the equilibrium state will be that state which has the
minimum enthalpy.
HELMHOLTZ FREE ENERGY
l The
energy and enthalpy minimum principles aren’t often very handy for solving
problems, since we don’t really work with many processes that occur at constant
entropy. It would be nice, therefore, to
have some way of expressing a minimum principle in terms of processes where
entropy is allowed to change.
l We
do this by starting with
.
o
From the product rule for differentiation we
have
.
o
This means we can write the inequality as
.
l The
quantity (U - TS) is another state variable, and we name it the Helmholtz
free energy (F º U - TS).
l For
processes that occur under constant temperature and volume,
.
l This
principle is known as the Helmholtz free energy minimum principle. It states that
o
For an unconstrained system at constant
temperature and volume, the equilibrium state will be that state which has the
minimum Helmholtz free energy.
l Notice
that for a reversible process, if temperature is held constant, then dF = -pdV. We can therefore interpret F as the
work that can be extracted from a constant temperature process.
GIBBS FREE ENERGY
l The
Helmholtz free energy is handy for processes that occur at constant temperature
and volume. In the atmosphere, however,
we frequently are dealing with processes that occur at constant temperature and
pressure. It would be nice to have
another form of the energy minimum principle that we can apply to these
processes.
l If
we start with the enthalpy form
and substitute
we get
.
l The
quantity (H - TS) is another state variable, and we name it the Gibbs
free energy (G º H - TS).
l For
processes that occur under constant temperature and pressure,
.
l This
principle is known as the Gibbs free energy minimum principle. It states that
o
For an unconstrained system at constant
temperature and pressure, the equilibrium state will be that state which has
the minimum Gibbs free energy.
l U,
H, F, and G are referred to as thermodynamic
potentials. The expressions for each are
summarized below
MAXWELL
RELATIONS
l Under
equilibrium conditions, the thermodynamic potentials are minimized. We therefore can write the following
equations, known as the Gibbs equations, for equilibrium states
Gibbs equations
l The
Gibbs equations are exact differentials (because they are differentials of
state functions). One property of exact
differentials is that if df(x,y) = Mdx +Ndy, then
. Euler
reciprocity relation
This is because
so that
and partial
derivatives commute, so that
.
l This
allows us to write, from the Gibbs equations,
