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Chapter 22: Problem 53

Suppose you brought together a gram of ordinary-matter hydrogen atoms (each composed of a proton and an electron) and a gram of antimatter hydrogen atoms (each composed of an antiproton and a positron). Keeping in mind that 2 grams is less than the mass of a dime, a. Calculate how much energy (in joules) would be released as the ordinary- matter and antimatter hydrogen atoms annihilated one another. b. Compare this amount of energy with the energy released by a 1-megaton hydrogen bomb \(\left(1.6 \times 10^{14} \mathrm{J}\right)\)

Short Answer

Expert verified
The annihilation of 1 gram of matter and 1 gram of antimatter hydrogen releases 1.8 × 10^{14} joules of energy, which is slightly more than the energy released by a 1-megaton hydrogen bomb.

Step by step solution

01

Understanding the concept of annihilation

When ordinary matter comes into contact with antimatter, they annihilate each other, releasing energy. According to Einstein's mass-energy equivalence principle, the energy released is given by the equation: E = mc^2 where E is energy, m is mass, and c is the speed of light ( c ≈ 3 × 10^8 meters/second).
02

Calculate the total mass

The problem involves 1 gram of ordinary hydrogen and 1 gram of antimatter hydrogen. The total mass m is: m = 1 gram + 1 gram = 2 grams Converting grams to kilograms (since SI units are in kilograms): 2 grams = 2 × 10^{-3} kilograms.
03

Calculate the energy released

Using the equation E = mc^2, we can calculate the energy released during annihilation. Plug in the values: E = (2 × 10^{-3} kg) × (3 × 10^8 m/s)^2 First, square the speed of light: (3 × 10^8 m/s)^2 = 9 × 10^16 m^2/s^2 Now, multiply by the mass: E = (2 × 10^{-3} kg) × (9 × 10^16 m^2/s^2) = 1.8 × 10^{14} joules.
04

Compare with the energy of a hydrogen bomb

The problem states that the energy released by a 1-megaton hydrogen bomb is 1.6 × 10^{14} joules Comparing the two energies: 1.8 × 10^{14} joules (annihilation energy) vs. 1.6 × 10^{14} joules (hydrogen bomb energy) The energy released by the annihilation of the matter and antimatter hydrogen atoms is slightly greater than the energy released by a 1-megaton hydrogen bomb.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Einstein's mass-energy equivalence
Einstein's mass-energy equivalence is a principle that shows the relationship between mass and energy. It tells us that mass can be converted into energy and vice versa. This concept is expressed by the famous equation \(E = mc^2\). This equation means that the energy (E) of an object is equal to its mass (m) times the speed of light (c) squared. The speed of light is a very large number (approximately 3 × 10^8 meters per second). Thus, even a small amount of mass can be converted into a huge amount of energy. This principle is the foundation of many phenomena in modern physics, including nuclear reactions and particle physics.
Energy Calculation
Calculating the energy released during the annihilation of matter and antimatter involves using Einstein's mass-energy equivalence principle. Let’s break down the steps for better understanding:
  • First, gather the total mass of the matter and antimatter being annihilated. In our exercise, we have 1 gram of matter and 1 gram of antimatter for a total of 2 grams.
  • Since the equation uses kilograms, convert grams to kilograms: 2 grams equals 2 × 10^{-3} kilograms.
  • Now, use the formula \(E = mc^2\). Substitute the values into the formula: \(E = (2 \times 10^{-3} \text{ kg}) \times (3 \times 10^8 \text{ m/s})^2\). First, square the speed of light: \( (3 \times 10^8 \text{ m/s})^2 = 9 \times 10^{16} \text{ m}^2/\text{s}^2\).
  • Finally, multiply by the mass: \(E = (2 \times 10^{-3} \text{ kg}) \times (9 \times 10^{16} \text{ m}^2/\text{s}^2) = 1.8 \times 10^{14} \text{ J}\).
The result is that 1.8 × 10^{14} joules of energy is released.
Comparison of Energies
Now let's compare the energy released from the annihilation process with the energy released by a 1-megaton hydrogen bomb.
  • The energy released by the annihilation of 1 gram of matter with 1 gram of antimatter is 1.8 × 10^{14} joules.
  • A 1-megaton hydrogen bomb releases about 1.6 × 10^{14} joules of energy.
Comparing these two energies, we see that the energy released during the matter and antimatter annihilation (1.8 × 10^{14} joules) is slightly more than that released by a 1-megaton hydrogen bomb (1.6 × 10^{14} joules). This comparison helps us understand the immense amount of energy produced when matter and antimatter meet and annihilate each other. Even a tiny mass, like 2 grams, can release an extraordinary burst of energy far surpassing conventional explosive devices.

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Most popular questions from this chapter

What is the flatness problem, and why has it created difficulties for cosmologists?

Of the four fundamental forces in nature, which one depends on electric charge? a. gravitational force b. electromagnetic force c. strong nuclear force d. weak nuclear force

Clips from a four-part episode of the public television series NOVA called "The Fabric of the cosmos" can be accessed on PBS's website (http://pbs.org/wgbh/nova/physics/fabric-ofcosmos.html and complete episodes can be viewed on the Top Documentary Films website (http://topdocumentaryfilms.com/the-fabric-of-the-cosmos). Watch the clips or at least one of the episodes. Are the arguments made in these programs compelling? Is the science explained in a way that makes sense to a general audience?

Go to the website for the Dark Energy Survey, an international project beginning in 2012 (https: \(/ /\) www.darkenergysurvey org/index.shtml). What observations will be made for this project? What will it tell scientists about dark energy? Click on "News." What is the status of this project? Are there any results yet?

If astronomers ignored any cosmological constant, the future of the universe could be determined solely from a. the mass of the universe. b. the volume of the universe. c. the amount of light in the universe. d. the density of the universe.
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