PHYS3410
BIOPHYSICS
II
Lecture Notes
Section
I:
Thermodynamics of Biological Systems
Lecture
3 :The uncompensated transformation: entropy production
The limitation
on the convertibility of heat to work is a fundamental limitation
in all natural processes. Thus universe as a whole can never
return to its initial state. The first and second laws were
best summarized by Clausius:
"The
energy of the universe is a constant.
The entropy of the universe approaches a maximum."
Clausius
described the change in entropy in the irreversible transformation:
 (3.1)
But textbooks
often fail to mention that in his 9th memoir Clausius
included the irreversible processes as integral part of the
Second Law.
 (3.2)
S entropy
of final state, S0 entropy of initial state dQ/T
entropy due to exchange of heat (gain or loss by the system,
compensated by equal gain or loss by the exterior).
Clausius
wrote: "the magnitude of N thus determines the uncompensated
transformation"
And he realised that it is always positive.
Perhaps
Clausius hoped to compute the entropy produced in irreversible
process, but this was not formulated until 20th
century. Theophile De Donder (1872 - 1957) incorporated "uncompensated
heat" of Clausius into the formalism of the Second Law
through the concept of affinity.
Prigogine
approaches the production of entropy by defining local
equilibrium.
Many systems, which are not in equilibrium, still have
well defined thermodynamic quantities. Intensive variables
(e.g. temperature and pressure) are measurable in each elemental
volume. Extensive variables (e.g. entropy and internal energy)
can be replaced by their corresponding densities.
The task
is to obtain explicit expressions for deS and diS
in terms of experimentally measurable quantities.
Irreversible
processes can be described in terms of thermodynamic forces F and thermodynamic
flows, dX.
diS
= FdX (3.3)
There
may be different thermodynamic forces and flows in a system,
so in general:
 (3.4)
For isolated
systems:
DeS
= 0, 
For closed
system energy is exchanged:
 (3.5)
For open
system (deS)matter adds to eqn. 3.5
If the
transformation is a reversible process, diS =0
and combining the First Law and eqn 2.7, we obtain
dU = TdS
+ dW = TdS + pdV (3.6)
Eqn. 3.4
allows us to calculate only the changes in entropy. However,
in 1906, Walther Nernst (1864 - 1941) formulated a law, which
stated that at the absolute zero of temperature the entropy
of every chemically homogeneous solid or liquid body has a
zero value. This has now become the Third Law of Thermodynamics.
The physical
basis of this law lies in the behaviour of matter at low temperature
that can only be explained by quantum theory. It is remarkable
that the theory of relativity gave us the means to define
absolute value of energy
and the quantum theory enables us to define absolute
value of entropy.
Examples
of Entropy Changes due to Irreversible Process
Heat
Conduction
The system is isolated (deS = 0) and consists of
two parts at T1 and T2. Irreversible
heat flow from higher temperature T1 to lower temperature
T2 results in the increase of entropy. The amount
of heat transferred in time dt is dQ. The volume of each part
is constant so dW = 0.
The energy
change in each part is solely due to the flow of heat:
dUi
= dQi, i = 1,2
The heat
gained by one part is equal to the heat lost by the other.
-dQ1
= dQ2 = dQ
 (3.7)
heat flow
JQ = dQ/dt
Rate of
heat flow was investigated by Jean-Baptiste-Joseph
Fourier (1768 – 1830).
JQ
= a (T1
- T2) (3.8)
a
coefficient of heat conductivity
 (3.9)
Due to
the flow of heat from the hot part to the cold part, the temperatures
eventually become equal, Teq, the driving force
vanishes and the flow stops. The non-equilibrium state evolves
to the equilibrium state through increase of entropy. Thus
at equilibrium the production of entropy is at minimum, but
the entropy is at maximum.
D
= T1 – T2
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