Question

One liter of gasoline \((1,000 \mathrm{~mL})\) has a mass of about \(750 \mathrm{~g}\) and contains about \(9.7 \mathrm{kWh}\) of energy. Meanwhile, a typical AA battery occupies \(7.4 \mathrm{~mL}\) of volume at a mass of \(23 \mathrm{~g}\), while holding about \(0.003 \mathrm{kWh}\) of energy. How much volume and how heavy would a collection of AA batteries be in order to match the energy in a liter of gasoline, and by what factors (in volume and mass) is gasoline superior?

Short answer

Answer: To match the energy of 1 liter of gasoline, 3,234 AA batteries are required. Gasoline is approximately 23.93 times more volume-efficient and 99.17 times more mass-efficient than AA batteries.

Step-by-step solution

Determine the number of AA batteries to match 1 liter of gasoline

First, we need to find out how many AA batteries are required to match the energy of 1 liter of gasoline. To do this, divide the energy in 1 liter of gasoline by the energy in one AA battery: Number of AA batteries = \(\frac{9.7 \mathrm{kWh}}{0.003 \mathrm{kWh}}\) Number of AA batteries = \(3{,}233.33\) Since we cannot have a fraction of a battery, we will round up to the nearest whole number, so we will need 3,234 AA batteries.

Calculate the total volume of the AA batteries

Now, we need to calculate the total volume of 3,234 AA batteries. To do this, multiply the number of AA batteries by the volume of one AA battery: Total volume of AA batteries = \(3{,}234 \times 7.4 \mathrm{mL}\) Total volume of AA batteries = \(23{,}931.6 \mathrm{mL}\)

Calculate the total mass of the AA batteries

Next, we need to calculate the total mass of 3,234 AA batteries. Multiply the number of AA batteries by the mass of one AA battery: Total mass of AA batteries = \(3{,}234 \times 23 \mathrm{g}\) Total mass of AA batteries = \(74{,}382 \mathrm{g}\)

Find the factors by which gasoline is superior in volume and mass

Finally, we need to find the factors by which gasoline is superior in volume and mass compared to the AA batteries. To do this, divide the total volume and mass of the AA batteries by the volume and mass of 1 liter of gasoline, respectively: Volume factor = \(\frac{23{,}931.6 \mathrm{mL}}{1{,}000 \mathrm{mL}} = 23.93\) Gasoline is approximately 23.93 times more volume-efficient than AA batteries. Mass factor = \(\frac{74{,}382 \mathrm{g}}{750 \mathrm{g}} = 99.17\) Gasoline is approximately 99.17 times more mass-efficient than AA batteries. In conclusion, a collection of 3,234 AA batteries would be required to match the energy in a liter of gasoline. This would result in a total volume of 23,931.6 mL and a total mass of 74,382 g. Gasoline is approximately 23.93 times more volume-efficient and 99.17 times more mass-efficient than AA batteries.

Key concepts

Energy Conversion Efficiency

Understanding energy conversion efficiency is crucial when comparing different power sources like gasoline and AA batteries. It's a measure of how effectively energy can be converted from one form to another. In our daily lives, energy conversion happens all the time—when we start a car, the chemical energy in gasoline is converted into mechanical energy.

For instance, internal combustion engines in cars convert the chemical energy of gasoline into motion, but not all of that energy is used for movement. A substantial part is lost as heat. In contrast, batteries convert chemical energy directly into electrical energy, which can be used more efficiently in electronic devices.

However, the energy density, which is the amount of energy stored in a given system per unit volume, also plays a significant role. Gasoline has a higher energy density compared to batteries, meaning it contains more energy for the same amount of space, which leads to a direct impact on the efficiency of energy conversion in practical applications.

In the context of the exercise, when we're talking about needing over three thousand AA batteries to match the energy of one liter of gasoline, we're also looking at a drastic difference in conversion efficiency. Gasoline's high energy density contributes to a more effective and practical energy conversion process for powering vehicles.

Energy Storage

The concept of energy storage encompasses methods and technologies used to keep energy in a form that it can be used later. Energy can be stored in various forms, such as chemical, gravitational, mechanical, or electrical.

In our example, gasoline stores chemical energy, which is released through combustion in an engine. On the other hand, a AA battery uses an electrochemical process to store and deliver energy. The ability to efficiently store and release energy when demanded is a cornerstone of modern technology.

The energy storage capacity is directly linked to the energy density of the material. Gasoline’s superior energy density means that it can store a significant amount of energy in a small volume, making it particularly useful where space is at a premium, such as in vehicles.

Considering the massive volume and weight required for AA batteries to match the energy capacity of a liter of gasoline, we see that energy storage technology for batteries still has limitations, especially in applications that require high energy over extended periods, like driving a car over long distances.

Energy Content Measurement

When it comes to energy content measurement, we are referring to the quantification of the total amount of energy stored within a fuel or power source. It is typically measured in terms of energy per unit of mass (calorific value) or volume.

The most common units for measuring energy content are kilowatt-hours (kWh), joules (J), calories (cal), and British Thermal Units (BTU). In our exercise, the energy content was given in kilowatt-hours, which is a convenient unit for larger quantities of energy such as those in fuel combustion or battery storage.

One liter of gasoline holds approximately 9.7 kWh of energy. In contrast, one AA battery holds approximately 0.003 kWh. This discrepancy illustrates the significant difference in energy content between the two energy sources. It also shows why such a large number of AA batteries are needed to match the energy provided by a relatively small amount of gasoline.

Knowing the energy content of fuels and batteries helps in planning for their use in different applications, such as powering vehicles or electronic devices. It's pivotal for engineers, scientists, and even consumers to understand the energy content for effective and efficient use of power sources.