# Energy and power of batteries

In this article, Riccardo Ruffo (University of Milano Bicocca) introduces fundamental terms used to describe the performance of batteries.

The energy (E) of a battery depends not only on the potential difference (V), which is an intensive property of electrodes but also on stored electrical charge (Q), which is an extensive property: (E=Vtimes Q).

The theoretical amount of electrical charge stored in one electrode can be calculated from Faraday’s law:

• (Q=moltimes nF)

where mol is the moles of the substance generating n electrons and F is the Faraday constant, that is, the charge of one mole of electrons. For example, the oxidation of 1 kg of Zn, follows the equation:

• (Znrightarrow Zn^{2+}+2e)

This can deliver an electrical charge (Q) of:

• (Q=frac{1000 g}{65.4 g mol^{-1}}times 2 times 96485 C mol^{-1}times frac{1 h}{3600 s}=819.6 hspace{0.1 cm} Ah)

In this equation, the first term consists of the moles of Zn in 1 kg weight, 2 is the number of electrons exchanged in the preceding equation, the third term is the Faraday constant, and the last term is the conversion from Coulombs to Ampere-hours. In the example above, 1 kg of metallic Zn has a capacity of 819.6 Ah, that is, metallic Zn has a gravimetric specific capacity of (819.6 hspace{0.1 cm} Ah hspace{0.1 cm} kg^{-1}).

The gravimetric specific capacity is thus the intensive quantity that we can use both to evaluate an electrode material and to calculate the energy of a battery when the electrodes are known. Of course, we will have to take gravimetric specific capacity into account to properly balance the electrical charges of the electrodes in a full cell. For example, we need a substantial amount (in kilograms) of material with a low gravimetric specific capacity to compensate for a few kilograms of material with a high gravimetric specific capacity. Thus, once the complete cell is assembled, we can calculate the energy contained through:

1. the potential difference between the electrode reactions
2. the amount of electrode materials
3. their specific capacities.

Having some available energy does not necessarily mean that this energy can be used. In fact, we have to discharge the cell, and discharging is done by delivering an external current (i), which is proportional to the power (P):

• (P=i times V)

One of the problems with electrochemical cells is that the potential difference in a cell (V) is a function of the delivered current (i). Furthermore, in the specific area of batteries, the potential difference of a cell is a function of the state of charge, so the energy is calculated as follows:

• (E=int_0^Q V(q)dq)

where q is the delivered charge at a given time and Q is the cell capacity.

© Riccardo Ruffo-University of Milano Bicocca