Problem 105 - Chemistry The Central Science In Si Units

Question

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 If we assume that the voltage declines linearly as current is withdrawn, estimate the total maximum electrical work the battery could perform during discharge.

Step-by-Step Solution

Answer

(a) mAh measures battery capacity, indicating how long it can supply a current. (b) The total electrical work done is 24480 J.

Step 1: Understanding mAh for Battery Capacity

The unit mAh (milliamp-hours) is a measure of electric charge. For a battery, it represents the amount of charge the battery can deliver over time; specifically, it tells how many milliamps of current the battery can supply continuously for one hour.

Step 2: Calculate the Average Voltage

Given that the voltage starts at 3.60 V and declines linearly to 3.20 V, we find the average voltage during discharge. The average voltage \( V_{avg} \) is calculated as:\[V_{avg} = \frac{V_{start} + V_{end}}{2} = \frac{3.60 + 3.20}{2} = 3.40 \text{ V}\]

Step 3: Calculate the Total Charge Delivered

The battery's rated capacity is given as 2000 mAh. This is the total charge the battery can provide:\[Q = 2000 \, \text{mAh} = 2000 \, \text{mA} \times 3600 \, \text{s/h} = 7200 \, \text{C}\] (since 1 Ah = 3600 C).

Step 4: Calculate the Total Electrical Work

The total electrical work \( W \) done by the battery is given by the product of the average voltage and the total charge:\[W = V_{avg} \times Q = 3.40 \, \text{V} \times 7200 \, \text{C} = 24480 \, \text{J}\]

Key Concepts

milliamp-hours (mAh)

The concept of milliamp-hours, abbreviated as mAh, is a way to quantify the energy storage capacity of a battery. Think of mAh as a measure of how much "energy fuel" is stuffed inside a battery. When you see a battery rated at 2000 mAh, this indicates that the battery can deliver 2000 milliamps of current for exactly one hour. Alternatively, it could also supply a current of 1000 milliamps over two hours, or 500 milliamps for four hours.

• Energy Measurement: mAh is specifically about electric charge, not about the electrical energy or power consumed directly, but it helps to infer how long a battery might last under a specific load.
• Hourly Concept: The "hours" in milliamp-hours show that it is an average over time, which makes it a practical measure for daily devices like mobile phones.

Overall, knowing the mAh of a battery helps consumers estimate how long their devices will run before needing a recharge. It's a crucial criterion for comparing the longevity of different batteries.

lithium-ion battery

Lithium-ion batteries are a popular type of rechargeable battery. Their widespread use in devices ranging from smartphones and laptops to electric vehicles is a testament to their efficiency.

• Lightweight and Compact: These batteries pack a lot of power into a small space, making them perfect for portable devices.
• Voltage Characteristics: A fresh lithium-ion battery typically begins with a voltage of around 3.60 V. As it discharges, the voltage gradually drops until it reaches a lower value like 3.20 V. Unlike some other types of batteries, the voltage doesn't stay constant throughout its discharge cycle.
• Energy Density: Their superior energy density means they can store more energy than other battery types of the same size.

If you rely on gadgets like phones or tablets, you're likely using battery technology based on lithium-ion cells. Their rechargeability and efficient energy storage make them an ideal choice for modern technology.

electrical work in batteries

The idea of electrical work involved in a battery’s operation is about converting stored energy into usable electrical energy for devices. Electrical work is determined by how much electric charge moves through a potential difference, or voltage.

• Work Calculation: To find the electrical work a battery can do, calculate the total charge moved (in Coulombs) and multiply it by the average voltage across the discharge period. This gives the electrical work done in Joules, represented as: \[W = V_{avg} \times Q\]where \(W\) is work, \(V_{avg}\) is the average voltage, and \(Q\) is the charge.
• Average Voltage: Given that battery voltage drops linearly as it discharges, using an average voltage provides a reliable calculation base.
• Practical Implications: Understanding electrical work helps explore the efficiency and performance of batteries and how they provide power to your devices.

This conversion of chemical energy inside the battery to electrical work powers all electronics, making it a fundamental concept in understanding battery and device performance.