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Thermodynamics and Geothermal Systems 37

which requires that

dw = C × dT. (3.13)
V

enTropy
If we consider the Carnot cycle, we are confronted by the fact that we have started at some initial
condition for which there is no history. The process that brought it to the pressure, volume, and
temperature at which our cycle begins is not specified, nor does it matter for the overall evalua-
tion of how heat and work relate to each other. As we move through the cycle from beginning to
end, we add and subtract heat and the system does work or has work done on it. As a result, the
enthalpy of the gas is changing. All of the changes in temperature are a function of the heat capac-
ity of the gas in our cylinder, but the heat content at the beginning of our cycle is never involved in
doing work and we can never use it. Furthermore, since our Carnot cycle is an idealized, revers-
ible system that is unattainable in real life, there is a certain amount of heat that we simply cannot
access—moving through a real-life cycle, we will inevitably lose some heat through friction and
conduction that can never be used for work. A measure of this unattainable heat that is present
at the initial state of our system as well as that lost in the process of moving through the cycle is
called entropy.
The definition of entropy is

dS ≡ dq/T, (3.14)

which states that any differential change in the heat a system contains, at a given temperature, leads
to a change in the entropy of the system. One way to conceptualize this relationship is to consider
the temperature–entropy path our Carnot cycle followed (Figure 3.4). At the initial point in the
cycle, the temperature and entropy are fixed (just as are the pressure and volume). When heat is
reversibly added to the system as work is done at constant temperature, dq/T and, hence, entropy
increase. In step 2, as the gas adiabatically expands, the temperature drops. Since there is no change
in the heat content, dq is zero and there is no change in the entropy. Steps 3 and 4 are the exact
reverse of steps 1 and 2, respectively, and the system returns to the same entropy and temperature it
initially had. In this reversible process, the system is now poised to continue the cycle again. In this
case, the isolated Carnot engine has experienced no discernible change.
But, in reality, the entropy of the universe within which the engine exists has increased. This
is apparent if one considers the fact that the addition and removal of heat was done using external
heat reservoirs. The higher temperature reservoir loses some heat (even though it is imperceptible)
with each cycle, and the low temperature reservoir gains heat. If this process were carried out a
very large number of times, the two thermal reservoirs would approach the same temperature. Once
this happens no more work can be done because Δq/q approaches zero, which is the same thing
in
as saying the efficiency approaches zero. Once the two thermal reservoirs have reached the same
temperature, regardless of how that happens, they are no longer of any use for doing work, and the
entropy of the system has reached its maximum state.
A key lesson in this is that the entropy of a system can be manipulated, as is evident in Figure 3.4,
but only at the expense of increasing the overall entropy of the surrounding environment. The scale
upon which these changes occur can be large. If one considers, for example, carrying out the Carnot
cycle experiment on a laboratory bench, the various stages of the cycle would presumably be done
using electricity generated at a distant power station burning a fossil fuel. Although the entropy
remains constant during the first step in our cycle engine, the electrical power required to carry out
that step came from dramatically increasing the entropy of the fuel that was burned to generate the
power. It is for this reason that, when considering overall energy budgets, the energy generation and
consumption milieu must be taken into account.