Question

The world uses energy at a rate of \(18 \mathrm{TW}\), amounting to almost \(6 \times 10^{20}\) J per year. What is the mass-equivalent \(^{83}\) of this amount of annual energy? What context can you provide for this amount of mass?

Short answer

Answer: The mass-equivalent of the world's annual energy consumption is approximately \(6.67\times 10^6\) kg. To provide context, this is around the mass of roughly 1000 average-sized cars, assuming an average car mass of about 1500 kg.

Step-by-step solution

Use the energy-mass conversion formula, E=mc^2

To find the mass-equivalent of the annual energy consumption, we need to use the formula E=mc^2, where E is energy, m is mass, and c is the speed of light in a vacuum (\(3\times 10^8\) m/s). Rearranging the formula to find mass, we get m = E/c^2.

Substitute the given energy consumption

Now we can substitute the given annual energy consumption of \(6\times 10^{20}\) J into the formula: m = \((6\times 10^{20}\ \text{J})/((3\times 10^8\ \text{m/s})^2)\).

Calculate the mass-equivalent

Using our calculator, we find the mass-equivalent for the given energy consumption: m = \((6\times 10^{20}\ \text{J})/((3\times 10^8\ \text{m/s})^2) \approx 6.67\times 10^6 \) kg.

Provide context for the amount of mass

To provide context for the amount of mass, compare it to a real-world object. The mass-equivalent we found is approximately \(6.67\times 10^6\) kg, which is around the mass of roughly 1000 average-sized cars (assuming an average car mass of about 1500 kg). This shows the enormous amount of energy the world consumes, as it is equivalent to the mass of 1000 cars in terms of energy.

Key concepts

Energy-Mass Equivalence

Understanding the relationship between energy and mass is pivotal in physics, and it's beautifully encapsulated in the formula \( E=mc^2 \). Coined by Albert Einstein as part of his Theory of Special Relativity, this equation tells us that energy (E) and mass (m) are interchangeable; they are different forms of the same thing. To clarify, the 'c' in the formula represents the speed of light in a vacuum, which is approximately \(3 \times 10^8 \text{m/s}\). Since the speed of light is a very large number, even a small amount of mass can be converted into a huge amount of energy, and vice versa.

This concept not only revolutionized physics but also provided a new comprehension of the universe. It implies that the mass of an object is a measure of its energy content. When it comes to practical applications, this energy-mass equivalence has implications in fields ranging from cosmology to energy production, as well as in the understanding of nuclear reactions and the principles behind atomic bombs and nuclear power plants.

To delve deeper, when energy is released in the form of light, as with the burning of a candle, a tiny amount of mass is lost. The same principle is at work in nuclear reactions, where a considerable amount of energy is produced from a relatively small mass due to the speed of light squared acting as a multiplier. This underlying principle is fundamental in grasping the importance of conserving mass in chemical reactions and the significance of energy resources for our planet.

Annual Energy Consumption

The annual energy consumption of the world is a measure of how much energy humanity uses over the course of a year. Expressed commonly in joules (J), terawatt-hours (TWh), or quads, this figure captures all energy utilized across various sectors such as transportation, residential, commercial, and industrial.

To put this into perspective, the world uses energy at a staggering rate of \(18 \text{TW}\) (terawatts), which, when tallied over a year, amounts to nearly \(6 \times 10^{20} \text{J}\). Seeing energy consumption in numbers this large helps to understand the scale at which our civilization operates and the energy needs of an ever-growing population. It also sheds light on the importance of sustainable energy usage and the impact of human activities on the planet.

The vast amount of energy consumed requires a constant supply and has led to an increased emphasis on more sustainable and renewable energy sources. Debates and research into the efficiency, cost, and environmental impact of different energy sources are central to planning for a sustainable future. The switch to renewables like solar, wind, and hydropower is not just an environmental concern; it's also about ensuring that the world's energy needs continue to be met without depleting resources or causing irreversible harm to our ecosystem.

Energy to Mass Conversion

Converting energy into mass might seem like a concept out of science fiction, but it's a real process explained by Einstein's equation \(E=mc^2\). Although we do not convert energy into mass in our day-to-day experiences, the universe does it all the time. For instance, in the heart of the sun through the process of nuclear fusion, mass is created from energy.

However, for practical and illustrative purposes, such as understanding our massive energy consumption, we can use this formula to determine the mass-equivalent of a given amount of energy. To calculate this, as seen in the exercise, we rearrange Einstein's equation to solve for mass (\(m\)) given energy (\(E\)), hence \({m = \frac{E}{c^2}}\). Substituting our world's annual energy consumption in joules into the equation provides us with a mass figure.

When we realize that the mass-equivalent of the annual energy consumption is approximately \(6.67 \times 10^6 \text{kg}\), it's mind-boggling! It is comparable to the combined mass of about 1000 average-sized cars. Such comparisons help us comprehend the scale and significance of the energy we use each year and why advances in energy efficiency and sustainability are crucial. Although we don't directly turn energy into mass in our daily lives or industries, understanding this conversion helps us appreciate the interplay between mass and energy that governs cosmic phenomena, enabling us to better understand the physical underpinnings of our universe.