Direct Energy, 2018a

286 12.5 Chemical Energy Conversion
12.5 Chemical Energy Conversion
Batteries and fuel cells store energy in the chemical bonds of atoms. These
devices were studied in Chapter 9. Table 12.8 details how to describe
the physics of these chemical energy storage devices using the language of
calculus of variations.
Sometimes chemists discuss macroscopic systems and describe charge
distribution in a material by charge density ρ ch in units m C . In other cases,
3
chemists study microscopic systems, where they are more interested in the
number of electrons N and the distribution of these electrons around an
atom. The second and third columns of Table 12.8 specify how to describe
the macroscopic systems in the language of calculus of variations while the
last two columns specify how to describe the microscopic systems.
In the second column of Table 12.8, the generalized path is ρ ch and the
generalized potential is the redox potential V rp in volts. There is a close
relationship between the choice of variables specied in the second column
of Table 12.8 and the choices specied in the second columns of Table 12.1
and 12.3. More specically, the generalized path described in the second
column of Table 12.1 is charge Q in coulombs, where charge is the integral
of the charge density with respect to volume.
ˆ
Q = ρ ch dV (12.25)
The generalized path described in the second column of Table 12.3 is displacement
ux density −→ D in units C m 2 . In the third column of Table 12.8,
the opposite choice is made with V rp for the generalized path and ρ ch for
the generalized potential. In Chapter 13, we consider a calculus of variations
problem with this choice of variables in more detail to solve for the
electron density around an atom.
Another way to apply the language of calculus of variations to chemical
energy storage systems is to choose the number of electrons N as the
generalized path and the chemical potential μ chem as the generalized potential
[172]. This situation is described in the fourth column of the Table
12.8. We could instead choose μ chem as the generalized path and N as the
generalized potential, and this situation is detailed in the last column of
Table 12.8. Reference [172] details using calculus of variations with this
choice of variables. Chemical potential is also known as the Fermi energy
at T =0K, and it was discussed in Sections 6.3 and 9.2.3. It represents
the average between the highest occupied and lowest unoccupied energy
levels. The quantity E g , which shows up in the fourth row of the table, is
the energy gap in joules.