Problem 105 - Chemistry: The Central Science in SI Units, Global Edition

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 If we assume that the voltage declines linearly as current is withdrawn, estimate the total maximum electrical work the battery could perform during discharge.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) The unit of milliamp-hours (mAh) represents the capacity of a battery, expressing 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 battery life. (b) The total maximum electrical work the lithium-ion battery can perform during discharge is estimated to be 0.8 Joules.

Step 1: a) Meaning of milliamp-hours (mAh)

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.

Step 2: b) Estimate the total maximum electrical work.

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.

Step 1: Convert the capacity to charge.

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

Step 2: Calculate the voltage difference.

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

Step 3: Calculate the electrical work.

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.

Key Concepts

Milliamp-hours (mAh)

Milliamp-hours, often abbreviated as mAh, is a vital unit that describes the capacity of a battery. Simply put, it measures how long a battery can continuously provide a specific current before it is completely depleted. For instance, a battery rated at 2000 mAh can deliver a current of 2000 milliamperes for one hour, 1000 milliamperes for two hours, and so on.
It's a crucial indicator when choosing batteries for devices since it helps gauge how long your device can operate on a single charge.

• Higher mAh ratings mean longer usage times.
• Helps in comparing battery lifespan across different devices.

Understanding mAh is particularly important when selecting batteries for energy-demanding gadgets like smartphones, where battery life significantly impacts user experience.

Lithium-ion Battery

Lithium-ion batteries are a popular choice for many portable electronics due to their high energy density and lightweight design. These batteries can store a substantial amount of energy in a compact size, making them ideal for today's tech-savvy world.
Unlike other types of batteries, lithium-ion batteries have some distinct characteristics:

• They have a slow loss of charge when not in use, ensuring longer standby times.
• Minimal memory effect, which means they don’t require complete discharging before recharging.
• Have a nominal cell voltage around 3.6-3.7 volts, providing consistent power.

The starting voltage of a lithium-ion battery may begin at 3.60 V and can decrease to around 3.20 V as the battery discharges. Monitoring this voltage drop is essential as it relates directly to the battery's health and ability to deliver its rated capacity.

Electrical Work Calculation

The concept of electrical work involves determining how much energy a battery can provide. Using the formula:\[\text{Electrical Work} = \text{Charge} \times \text{Voltage Difference}\]allows us to measure the total work done by a battery in terms of Joules.
To estimate this electrical work:

• First, convert the battery capacity from mAh to Ampere-hours and then to Coulombs (since 1 Coulomb = 1 Ampere-hour).
• Next, identify the voltage difference during discharge. It's the starting voltage minus the ending voltage.
• Finally, multiply the total charge by the voltage difference to get the electrical work.

For example, using a capacity of 2000 mAh and a voltage difference from 3.60 V to 3.20 V, we calculated an electrical work of 0.8 Joules. This calculation is crucial for determining how efficiently a battery-powered device can perform its operations over time.