The capacity is a rating of the charge or energy stored in the cell. This value is expressed in either ampere-hours [A·h] or watt-hours [W·h]. The first represents the capacity in terms of coulombs of charge available, the second in terms of energy available. The two are related simply by the average voltage of the cell, W·h = A·h × V_avg. Clearly, if one wishes to increase the capacity of the cell, more active material is added. Thus, the capacity is directly related to the amount of active material and is a good indication of the size of the cell.

Since the current in amperes has units of C·s⁻¹, the capacity or size of a cell expressed in terms of A·h is really a measure of the coulombs available from the cell (1 A·h = 3600 C). The theoretical capacity of an electrode for different chemistries can be determined from the electrode reactions and the mass of active material according to the following equation:

(7.6)

where m_i is the mass of active material and M_i is its molecular weight. This calculated charge in A·h is often nomalized by the mass of active material to create a theoretical specific capacity that can be expressed in units such as [A·h·kg⁻¹], [A·h·g⁻¹], or [mA·h·g⁻¹]. The theoretical value represents the maximum possible capacity from the active material itself, and does not include the mass of other electrode components such as the current collector, conductive additives, or packaging. Other practical issues related to extraction of the energy are also not taken into account in the theoretical capacity.

Similarly, the theoretical capacity can be calculated for the cell, rather than for just one electrode. In this calculation, the charge capacity of each of the two electrodes is set to the same value. The coulombs of charge are determined from the stoichiometery and normalized by the combined mass of the two electrodes. This calculation is shown in Illustration 7.1. Again, the masses of the other cell components are neglected. The useable capacity of a commercial cell may only be one-fourth of its theoretical capacity. In practice, the mass and volume of the other cell components must be considered, as well as the extent to which the active material can be converted according to its stoichiometry. A number of factors influence the extent of conversion, such as side reactions and limitations on the maximum and minimum potential of the cell. These real-world effects will be discussed over much of the remainder of this chapter and in Chapter 8.

ILLUSTRATION 7.1

1. Calculate the theoretical specific capacity of the following electrodes: 
2. The last example from part (a) was for the negative electrode of the silver–zinc cell. What is the theoretical specific capacity of this full cell?Using the stoichiometry for the silver–zinc cell from Table 7.1,The total active material mass (positive electrode and negative electrode) = 3.544 g +1.00 g = 4.544 g

There is a need to quantify the available capacity remaining in a cell while in use. It is common to refer to either the state of charge (SOC) or the state of discharge (SOD, or depth of discharge, DOD) of a battery to identify the amount of unconverted active material in the cell. This concept is illustrated in Figure 7.4, again for the silver–zinc battery. When the cell is fully charged or at 100% SOC, pure Zn and Ag₂O are present. As the reaction proceeds, charge flows through the cell converting the reactants to ZnO and Ag. If the reactants are present in stoichiometric amounts, then the fraction (or percentage) of either Zn or Ag₂O that is in the initial state rather than the product state corresponds to the state of charge.

Later we’ll see that a measure of the SOC is critical for managing a battery system, as will be discussed in Chapter 8. However, as you might imagine, once a battery is assembled and operated, determing the state of charge is not so easy. One well-known exception is the flooded lead–acid cell. Since the electrolyte is involved in the discharge reaction, its density changes with SOC. Where there is access to the electrolyte, its density can be measured and used to indicate the SOC. Unfortunately, most often we do not have a direct measure of active material in the battery. A further difficulty with most cells is that the two electrodes may have different capacities. Consequently, even if the amount of unconverted material in each electrode were known, we would still need to decide which electrode to use for the SOC. Finally, the full capacity of a battery based on the amount of active material is most often not accessible. If we take SOC as

(7.7)

we have two challenges. First, how is the total capacity defined? Total capacity can be expressed in A·h; therefore, a detailed knowledge of the mass of all the individual components in the battery is not required in order to specify its capacity. Still, several choices remain. For example, should the total capacity be the theoretical capacity, the nominal measured capacity for a new cell, or the capacity of the latest charge/discharge cycle? Second, the available capacity depends on temperature and rate of discharge as discussed below. How do these factors impact our definition of the SOC? As the previously mentioned challenges can be addressed in different ways, we must accept that the term SOC is not precise. For now, we’ll defer detailed discussion, and we take the SOC to be an indication of the remaining capacity of the cell. As mentioned previously, SOC is used commonly with the SOD where