Question

A battery has a specific capacity of \(55 \frac{\mathrm{mA} \cdot \mathrm{h}}{\mathrm{g}}\) and a nominal voltage of \(2.4 \mathrm{~V}\). The battery has a mass of \(165 \mathrm{~g}\). Find the energy stored in the battery in \(\mathrm{J}\).

Short answer

The energy stored in the battery is 78,384 J.

Step-by-step solution

Understanding the Formula

The amount of energy stored in a battery can be calculated using the formula: \( E = C \times V \), where \( E \) is the energy in watt-hours (Wh), \( C \) is the capacity in ampere-hours (Ah), and \( V \) is the voltage in volts (V). Since we need the energy in joules (J), we will later convert it from watt-hours using the conversion factor 1 Wh = 3600 J.

Calculate Battery's Total Capacity

The specific capacity of the battery is given as \(55 \frac{\mathrm{mA} \cdot \mathrm{h}}{\mathrm{g}}\), and the mass of the battery is \(165 \mathrm{~g}\). To find the total capacity (\(C_{total}\)), we multiply the specific capacity by the mass: \[ C_{total} = 55 \frac{\mathrm{mA} \cdot \mathrm{h}}{\mathrm{g}} \times 165 \mathrm{~g} = 9075 \mathrm{mA} \cdot \mathrm{h} \]. Convert this to amp-hours (Ah) by dividing by 1000: \[ C_{total} = 9.075 \mathrm{Ah} \].

Calculate Energy in Watt-hours

Using the capacity in amp-hours and the nominal voltage, calculate the energy in watt-hours (Wh) using \( E = C \times V \): \[ E = 9.075 \mathrm{Ah} \times 2.4 \mathrm{~V} = 21.78 \mathrm{Wh} \].

Convert Energy to Joules

Convert the energy from watt-hours to joules. Since 1 watt-hour is equal to 3600 joules, \[ E = 21.78 \mathrm{Wh} \times 3600 \mathrm{J/Wh} = 78384 \mathrm{J} \].

Key concepts

Specific Capacity

Specific capacity is a measure of the amount of electric charge a battery can store per unit of its mass. It is often expressed in milliampere-hours per gram (mAh/g). This parameter is significant because it tells us how efficient a battery is in storing energy relative to its weight. For example, a specific capacity of \( 55 \, \frac{\mathrm{mA} \cdot \mathrm{h}}{\mathrm{g}} \) implies the battery stores 55 milliampere-hours of charge for every gram of battery material.

When calculating the total capacity of a battery, we multiply the specific capacity by the mass of the battery. This gives the overall storage capacity in milliampere-hours (mAh), which can then be converted to ampere-hours (Ah) by dividing by 1000. This conversion is crucial for further calculations of the battery's energy.

Nominal Voltage

Nominal voltage refers to the designated voltage of a battery under standard conditions. It is generally expressed in volts (V). This value is fundamental in understanding how much electric force a battery can exert.

The nominal voltage is used together with the capacity to determine the total energy a battery can deliver. For instance, if a battery has a nominal voltage of \(2.4 \, \mathrm{V}\), it means that under normal operations, the battery can supply power at this voltage level. It is important to note that the actual voltage might vary slightly during battery usage, but the nominal voltage is what we use for calculations.

Energy Conversion

Energy conversion in the context of batteries typically involves converting electrical energy into other forms or vice versa. When calculating stored energy in terms of watts or joules, understanding the conversion process is necessary.

In the given problem, we first calculate energy in watt-hours using specific capacity and nominal voltage. The next step is converting watt-hours to joules. This conversion is done by using the factor: 1 watt-hour (Wh) is equal to 3600 joules (J). Hence, the stored energy in joules can be calculated by multiplying the energy in watt-hours by 3600. This conversion is crucial for expressing the energy in terms of a standard unit, as joules are widely accepted in scientific calculations and technologies.

Ampere-hours

Ampere-hours, abbreviated as Ah, is a unit of electric charge and represents the amount of charge a battery can supply in one hour. This measure helps in understanding the duration a battery can last while delivering a particular current.

To find ampere-hours from milliampere-hours, we need to divide the mAh value by 1000, because 1 Ah equals 1000 mAh. In battery energy calculations, ampere-hours are used to determine the total energy stored and thereby its efficacy. Knowing the Ah capacity allows users to estimate how long a device can run before needing a recharge, which is critical in both consumer electronics and industrial applications. Understanding and accurately calculating ampere-hours is essential in ensuring optimum battery performance.