Page 55 - Geothermal Energy Renewable Energy and The Environment

38 Geothermal Energy: Renewable Energy and the Environment

T = T i T = T i
V = V i V = V 1
P = P i P = P 1
S = S i 1 S = S 1

4 2
T

T = T 2 3 T = T 2
V = V 3 V = V 2
P = P 3 P = P 2
S = S i S = S 1

S
FIGUre 3.4 Temperature versus entropy graph for the same pathways indicated in the pressure versus vol-
ume graph in Figure 3.3.

GIbbs FUncTIon and GIbbs enerGy (ΔG)
As noted above, when one defines the initial state of the Carnot cycle there is no need to specify
how that initial state was achieved. Indeed, it would be impossible to deduce such a history from
the basic information that is contained in such a system. As a result, it is not possible to determine
the actual, absolute internal energy of a substance or system. What can be deduced, however, is
whether or not two systems (whether they be minerals, rocks, liquid and gas, or any other pair of
materials or substances) are in equilibrium, and what the absolute differences are in their respective
heat contents. Two systems that have the same temperature, and are therefore in thermodynamic
equilibrium, are incapable of doing work without some external action being taken. If they are not at
the same temperature, they can be a source of energy. To determine the amount of available energy
that can be extracted for useful work, the heat contents of the systems must be evaluated.
To compare states, we must have a means of comparing the energy contained in one substance
or system with that in another. Consider, again, the Carnot cycle and how the internal energy
changes. It is evident that there are three fundamental attributes that contribute to the energy in
that system at any point along the cycle: the energy that exists in the system at the initial set of
conditions before the cycle begins, the energy the system acquires or gives up along an isothermal
path, and the energy it acquires or gives up along an adiabatic path. In 1876, J. Willard Gibbs
defined a function that mathematically described the energy contained in such a system and how
that energy is affected by changes in temperature and pressure. The Gibbs function follows from
the discussions above regarding the First and Second Laws of Thermodynamics. Gibbs showed
that the internal energy of a substance at some specified pressure (P) and temperature (T) is fully
described through the following relationship,

T T P
Δ G = ΔH – T × ΔS P ,T + ∫ ΔC dT – T × ∫ (ΔCp/T) dT + ∫ ΔV dP, (3.15)

P ,T
P,T
1 1 1 1 T P T P
1 1 1
where ΔG is the Gibbs energy at P and T, ΔH P ,T is the enthalpy at some standard state, which
P,T
1 1
is usually selected to be 1 bar (0.1 MPa) pressure and 25°C (298 K), ΔS P ,T is the entropy at the
1 1
standard state, ΔC is the constant pressure heat capacity and ΔV is the change in volume. The
P
Gibbs energy of a substance is defined in terms of some reference components and compounds