A visual proof of entropy production during thermalization with a heat reservoir
Ramandeep S. Johal



  
 [e-mail: ]rsjohal@iisermohali.ac.in
  Department of Physical Sciences, 
 Indian Institute of Science Education and Research Mohali,  
 Sector 81, S.A.S. Nagar, Manauli PO 140306, Punjab, India 
 § ABSTRACT 
In this note, the equilibrium curve of a thermodynamic
system is used to depict entropy production in the process
of thermalization with a reservoir. For the given 
initial and final equilibrium states of the system, the entropy
production is reduced when work is also 
extracted during thermalization. 
The case of maximum work extraction corresponds to a reversible
process. For less than optimal work extraction, 
the lost available work is shown to be directly
proportional to the entropy produced. 
 
 A visual proof of entropy production 
during thermalization with a heat reservoir
    Ramandeep S. Johal
    September 9, 2024
==================================================================================

Entropy production is the core concept underlying
the Second law which states that 
irreversible or spontaneous processes always
increase the entropy of the universe. 
An example is the flow of heat across a temperature
gradient such as when 
a system thermalizes with a heat reservoir. Here,
even if the heat is transferred in a 
quasi-static manner, there is a net increase
in the total entropy of the system plus
reservoir <cit.>. 
For the case when  the reservoir 
is initially at a higher (lower) temperature than the system,
it implies that the increase (decrease) in the entropy
of the system is more than 
the decrease (increase) in the entropy of the reservoir.
Now, the change in the entropy
of the system depends on its nature, unlike  
for the reservoir. Assuming an ideal gas
system, the increase of total entropy
may be easily demonstrated using 
the well-known logarithm inequality. Visual demonstrations
of the Second law for such irreversible processes 
have also assumed an ideal-gas type behavior
for the system <cit.>. 
In this paper, we present a diagram 
using equilibrium curve of the 
system which overcomes these limitations.
Since visual proofs often 
help in easy comprehension of abstract
concepts, a demonstration of entropy
production based on a generic thermodynamic system 
is desirable. 
  

For a given amount of a
thermodynamic 
system, the equilibrium state is described in terms of its internal energy U(S,V) as a function 
 of its entropy S and volume V  <cit.>. 
Then, the temperature of the 
system is defined as T = (∂ U/ ∂ S)_V^. One of the fundamental attributes 
of the equilibrium state is that U(S,V) is a convex function of S at 
constant V, which implies that the heat capacity
at constant volume is positive (C_V > 0).  
Now, with an initial state at energy U_1, entropy S_1 and 
temperature T_1, the system is placed in thermal contact
with a heat reservoir at temperature T_2 > T_1.
Heat flows from the reservoir to the system
till its temperature rises to T_2, corresponding
to a final energy U_2 and entropy S_2.
As the temperature is defined to be positive, 
U_2 > U_1 implies S_2 > S_1, for a fixed volume.

Thus, the system is in thermodynamic equilibrium 
in the 
 initial and the final state of the process. 
The entropy of the system increases by Δ S = S_2-S_1,  while 
the energy increases by  Δ U = U_2 - U_1 which
equals the heat exchanged with the reservoir, 
Q_2= Q_1 = Δ U. On the other hand, 
the decrease in the entropy of the reservoir is:
Δ S_ res = Q_2/T_2. 
Thus, the net or total change in the entropy of 
system plus reservoir is given by:

    Δ S_ tot = Δ S - Δ S_ res.

The standard evaluation of the above quantities goes as follows.   
As the process happens  at a fixed 
system volume V, we can write
Δ U = ∫_T_1^T_2 C_V dT and  
Δ S = ∫_T_1^T_2 (C_V / T) dT. 
Without assuming a specific form for
the function C_V(T) >0, 
 a general proof showing 
 Δ S_ tot >0 is 
as follows. From the explicit expressions
given above, we can 
write Eq. (<ref>) as

    Δ S_ tot = ∫_T_1^T_2( 1/T - 1/T_2)
     C_V   dT.

Since T_2 > T_1, the integrand above is positive and 
so is the value of the integral. A similar proof can be 
constructed for the case T_2 < T_1 i.e. when 
the system is cooled by the reservoir.



Fig. 1  shows the (convex) equilibrium curve U(S) 
of the system at a  given volume V. The entropy
changes involved in the process are 
depicted as certain line segments  
showing that Δ S > Δ S_ res, 
and hence Δ S_ tot>0 due to Eq. (<ref>). 
Further, the diagram is only based on 
two properties: i) positivity of the temperature
and ii) convexity of the function U(S).
A major difference of the present diagram from
the previous ones is that it is not restricted
to the ideal gas systems. 
The reader is invited 
to draw the corresponding diagram 
for the case where the reservoir is at a lower
temperature than the system (T_2 < T_1). 

Note that the net rise in total entropy does
not require a complete thermalization with
the reservoir. Any amount of heat flow 
across a finite temperature gradient 
increases the  total entropy. 
Following Fig. 1,
we can as well analyze the case of incomplete thermalization
where the final state of the system is some intermediate
state lying on the equilibrium curve in between 
the points (1) and (2). 
The temperature of the system T' (T_1 < T' < T_2)
is again given by the slope of the tangent at that point.
It is easily seen that 
Δ S_ tot >0 holds in this case too,
though the length of the segment denoting 
entropy production is smaller than in the case
of complete thermalization. Thus, 
we observe from the figure that the entropy
production attains its maximum value when the 
system reaches thermal equilibrium with the 
reservoir. 

 

Suppose that instead of making a thermal contact, we couple the reservoir
and the system by means of a heat engine 
for which these 
act as heat source and heat sink, respectively.
The engine runs by executing certain heat 
cycles—absorbing an amount of heat from the reservoir, converting
a part of it into work and rejecting the 
rest of the heat to the system. The engine 
produces useful work till the system comes to be
in thermal equilibrium with the reservoir. 
Thus, the initial and the final states
of the system are the same as in the case of 
thermalization above, yielding 
the amount of heat rejected to the sink as 
Q_1 = Δ U. Likewise, the change in system entropy
is equal to  Δ S. Now, suppose that W ≥ 0 amount 
of work is extracted by the end of this process.
Since the engine undergoes cycles,
the conservation of energy implies that 
Q_2 = W+Q_1 amount of heat is absorbed from 
the reservoir, which implies  Q_2 ≥ Q_1.
This process is depicted in Fig. 2. 
It is apparent that the entropy production here
is smaller in magnitude as compared to pure thermalization
where no work was extracted.  
Fig. 2 also suggests that the magnitude of
work can be enhanced till
Δ S_ res = Δ S_ i.e.  
when Δ S_ tot vanishes and 
the engine becomes a reversible one.
Thus, we observe that 
maximum work (W_ max)  
is extracted when 
thermalization proceeds as 
a reversible process—with no entropy
production. In general, we have
W ≤ W_ max. In fact, using the similarity property of triangles in Fig. 2,
we can show that W_ max - W = T_2Δ S_ tot,
where Δ S_ tot is the entropy produced in the 
process that extracts W amount of work. The difference 
W_ max - W, called the lost available work or 
the exergy destroyed, is directly proportional to 
the entropy produced. This relation is 
well known in the engineering parlance as 
the Gouy-Stodola theorem. 
Since, the initial and 
final states of the system remain the same
irrespective of the amount of work extracted, 
so it implies that Q_2 = W + Δ U increases 
in direct proportion to the work extracted, 
with its maximum value being T_2 Δ S
(see Fig. 2). 
Thus, we obtain W_ max  = T_2 Δ S - Δ U. 


Temperature-energy interaction diagrams, depicting 
heat and work flows in reversible as well as irreversible heat cycles,  
were introduced in the engineering
literature <cit.> and 
also reported in the physics literature 
<cit.>. 
As pointed out by Bejan <cit.>, 
these instances mirrored the almost parallel developments
in the techniques of “entropy generation minimization”
amongst the engineering community and that of 
“finite-time thermodynamics” within the physics community.
It is interesting to note that these earlier 
diagrams show changes in entropy by
angles, whereas  
the present diagram depicts such changes by 
line segments, while making use of the thermodynamic 
equilibrium curve of the finite-system involved. 




10

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