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Chapter 20: Problem 105

The capacity of batteries such as the typical AA alkaline battery is expressed in units of milliamp-hours (mAh). An AA alkaline battery yields a nominal capacity of 2850 mAh. (a) What quantity of interest to the consumer is being expressed by the units of mAh? (b) The starting voltage of a fresh alkaline battery is 1.55 V. The voltage decreases during discharge and is 0.80 \(\mathrm{V}\) 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 If we assume that the voltage declines linearly as current is withdrawn, estimate the total maximum electrical work the battery could perform during discharge.

Short Answer

Expert verified
a) The quantity of interest to the consumer expressed by the units of milliamp-hours (mAh) is the battery's capacity, which indicates how long a device powered by the battery will last. b) The maximum electrical work the AA alkaline battery could perform during discharge, considering the average voltage of 1.175 V, is approximately 3.35 watt-hours.

Step by step solution

01

Understanding the meaning of milliamp-hours (mAh)

The unit of milliamp-hours (mAh) is used to express the capacity of batteries. It describes the amount of current a battery can supply for a specific period. Specifically, it tells us how many milliamperes (mA) of current the battery can supply continuously for 1 hour. In this exercise, we are given that a typical AA alkaline battery has a capacity of 2850 mAh, which means the battery can supply a current of 2850 mA for 1 hour. This is of interest to the consumer because it indicates how long a device powered by this battery will last.
02

Calculate the average voltage during discharge

We are given that the starting voltage of the battery is 1.55 V, and the final voltage is 0.80 V when the battery has delivered its rated capacity. We can calculate the average voltage during discharge by finding the midpoint of the starting and final voltage: Average voltage = \(\frac{1.55\,\text{V} + 0.80\,\text{V}}{2}\) Average voltage = \(1.175\,\text{V}\)
03

Calculate the total charge delivered by the battery

To find the total charge (Q) delivered by the battery, we can use the formula: Q = capacity × time We know the capacity is 2850 mAh and since our battery is rated for 1 hour, the time is 1 hour. However, to make our units consistent, we need to convert the capacity from milliamperes to amperes: 2850 mA = 2.85 A Now we can calculate the total charge: Q = 2.85 A × 1 hour = 2.85 Ah (ampere-hours)
04

Calculate the maximum electrical work performed by the battery

Finally, to find the maximum electrical work (W) the battery can perform during discharge, we can use the formula for electrical work: W = Q × ΔV We know the total charge and the average voltage. So, we can calculate the electrical work: W = 2.85 Ah × 1.175 V W = 3.34875 Wh (watt-hours) The maximum electrical work the AA alkaline battery could perform during discharge is approximately 3.35 watt-hours.

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