Problem 47 - Fundamentals of Physics

Question

A 5.00 -grain aspirin tablet has a mass of \(320 \mathrm{mg}\). Forhow many kilometers would the encrgy cquivalent of this mass power an automobile? Assume \(12.75 \mathrm{~km} / \mathrm{L}\) and a heat of combustion of \(3.65 \times 10^{7} \mathrm{~J} / \mathrm{L}\). for the gasoline used in the automobile.

Step-by-Step Solution

Answer

The car can be powered for approximately \(1.01 \times 10^{7}\) kilometers.

Step 1: Convert tablet mass to kilograms

First, convert the aspirin tablet mass from milligrams to kilograms. Since there are 1000 mg in a gram and 1000 grams in a kilogram, we have:\[ 320 \text{ mg} = 320 \times 10^{-6} \text{ kg} = 0.00032 \text{ kg} \]

Step 2: Calculate energy equivalence of mass using E=mc^2

Use Einstein's equation \(E = mc^2\) to calculate the energy equivalent of the aspirin tablet's mass. The speed of light \(c = 3 \times 10^8 \text{ m/s}\):\[ E = 0.00032 \times (3 \times 10^8)^2 = 2.88 \times 10^{13} \text{ J} \]

Step 3: Calculate equivalent energy in liters of gasoline

Using the heat of combustion of gasoline given as \(3.65 \times 10^{7} \text{ J/L}\), calculate how many liters of gasoline the energy \(E\) is equivalent to:\[ \text{Liters of gasoline} = \frac{2.88 \times 10^{13} \text{ J}}{3.65 \times 10^{7} \text{ J/L}} \approx 7.89 \times 10^{5} \text{ L} \]

Step 4: Calculate distance in kilometers

With the gasoline efficiency at \(12.75 \text{ km/L}\), calculate the distance that can be powered using the equivalent liters of gasoline:\[ \text{Distance} = 7.89 \times 10^{5} \text{ L} \times 12.75 \text{ km/L} \approx 1.01 \times 10^{7} \text{ km} \]

Key Concepts

Einstein's Equation

Einstein's equation, represented as \( E = mc^2 \), is a groundbreaking formula in physics that shows the relationship between a body's mass \( m \) and its energy \( E \). In this equation, \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \text{ m/s} \). The formula implies a profound equivalence of mass and energy, suggesting that mass can be converted into energy and vice versa.
To put it simply, even a tiny amount of mass can produce a tremendous amount of energy due to the square of the speed of light being such a large number. This equivalence is the principle behind nuclear power and atomic reactions, where small amounts of mass are transformed into tremendous energy.
By applying this equation to the problem of the aspirin tablet, we calculate how much energy would be released if the tablet's mass could be entirely converted into energy. This theoretical scenario demonstrates the powerful idea that all mass has an energy equivalent, even something as small as an aspirin tablet.

Mass-Energy Conversion

The concept of mass-energy conversion highlights the possibility of transforming mass into energy. Based on Einstein's equation, this transformation is not only theoretical but has practical implications. Such conversion is an observable phenomenon in nuclear reactions, where atomic nuclei undergo changes resulting in the release of energy.
In nuclear reactions, like fission and fusion, small amounts of mass are converted into energy. This released energy is significantly larger than that from chemical reactions, making nuclear reactions a potent source of energy. The sun, for instance, powers itself through nuclear fusion, converting hydrogen into helium and releasing energy.
Returning to our aspirin example, although we can't physically convert the aspirin's mass to energy practically, understanding the theoretical output demonstrates the immense energy potential contained in even minute masses. If you could do so, a mere aspirin tablet could power a car to travel over millions of kilometers, highlighting the sheer magnitude of energy from mass-energy conversion.

Heat of Combustion

Heat of combustion is a concept describing how much energy is released when a substance is burned. In our exercise, it's quantified as \(3.65 \times 10^7 \text{ J/L}\). This measurement helps us estimate how efficiently a fuel can generate energy for consumption.
The practical application of heat of combustion is vital in evaluating different fuels for efficiency. Gasoline, for example, has a specific heat of combustion value meaning that's how much energy it can deliver per liter when burned. Higher heat of combustion values indicate that a fuel is capable of more power and efficiency.
In the example given, once the aspirin's mass is converted to energy, we equate that energy to the amount of gasoline required to enact an equivalent level of power. By comparing these energy outputs, we see the incredible energy potential residing in mass, should it undergo a theoretical complete conversion into usable energy.