The capacity of batteries such as a lithium-ion battery is expressed in units of milliamp-hours (mAh). A typical battery of this type yields a nominal capacity of \(2000 \mathrm{mAh}\). (a) What quantity of interest to the consumer is being expressed by the units of \(\mathrm{mAh}\) ? (b) The starting voltage of a fresh lithium-ion battery is \(3.60 \mathrm{~V}\). The voltage decreases during discharge and is \(3.20 \mathrm{~V}\) when the battery has delivered its rated capacity. If we assume that the voltage declines linearly as current is withdrawn, estimate the total maximum electrical work the battery could perform during discharge.

The unit of milliamp-hours (mAh) is used to express the capacity of a battery. It represents the amount of current (in milliamperes or mA) that a battery can deliver for one hour before it gets discharged. The higher the mAh value, the longer the batteriy life.

To find the total maximum electrical work the battery can perform, we can use the formula: Electrical Work = Charge × Voltage Difference We are given the nominal capacity of the battery (2000 mAh), and the starting and ending voltages (3.60 V and 3.20 V, respectively). We first need to find the total charge and the voltage difference. Step 1: Convert the capacity (2000 mAh) to charge (in Coulombs). Since the current is given in milliamperes (mAh), we need to convert it to amperes (Ah) first, and then multiply it by the time (1 hour) to get the charge. 1 Ah = 1000 mAh Charge (in Ah) = Capacity (in Ah) × Time (in hours) Step 2: Calculate the voltage difference. Voltage Difference = Starting Voltage - Ending Voltage Step 3: Calculate the electrical work. Electrical work = Charge × Voltage Difference. Now, we'll plug in the values and solve the equations.

First, let's convert the capacity (2000 mAh) to Ah: Capacity = 2000 mAh ÷ 1000 = 2 Ah Now, we'll find the charge using the given time (1 hour): Charge = Capacity × Time = 2 Ah × 1 hour = 2 Coulombs

We'll calculate the voltage difference using the starting and ending voltages. Voltage Difference = Starting Voltage - Ending Voltage = 3.60 V - 3.20 V = 0.40 V

Now, we have the charge (2 Coulombs) and the voltage difference (0.40 V). We'll use the formula to calculate the electrical work: Electrical work = Charge × Voltage Difference = 2 Coulombs × 0.40 V = 0.8 Joules The total maximum electrical work the lithium-ion battery can perform during discharge is estimated to be 0.8 Joules.