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 \(\left(4.2 \times 10^{15} \mathrm{J}\right)\)

Einstein's formula is central to understanding how mass can be converted into energy. This famous equation is given by \[ E = mc^2 \]. It states that the energy ( \(E\) ) is equal to the mass ( \(m\) ) multiplied by the square of the speed of light (\(c\) ). The speed of light is a constant value, approximately \(3 \times 10^8 \) meters per second. Each time mass is entirely converted into energy, you can see that even a tiny amount of mass can result in a vast amount of energy. Let's break this down:

• Energy ( \(E\) ) is measured in joules.
• Mass ( \(m\) ) can be measured in kilograms.
• The speed of light ( \(c\) ), which is approximately \(3 \times 10^8 \) m/s.

l When calculated correctly, you will find that even a small amount of mass will yield a tremendous amount of energy. This principle forms the foundation for various applications, including nuclear reactions and the subject we are covering today.

In the case of matter-antimatter annihilation, a lot of energy is released because all the mass turns into energy. Using 1 gram of ordinary hydrogen and 1 gram of antimatter hydrogen results in a total mass of 2 grams or \(0.002 \ kg \). When this mass annihilates, it converts into pure energy. Here’s how we compute it:

1. Convert 2 grams to kilograms: \(2 \ g \times \frac{1 \text{ kg}}{1000 \text{ grams}} = 0.002 \text{ kg} \)
2. Plug this mass into Einstein's formula: \(E = 0.002 \text{ kg} \times (3 \times 10^8 \ \text{m/s})^2\)
3. Calculate the result: \(E = 0.002 \times 9 \times 10^{16} \ \text{J}\). This gives us \(E = 1.8 \times 10^{14} \ \text{J}\).

This massive energy result underscores just how much energy can be unlocked by converting mass directly into energy.

To better grasp how powerful matter-antimatter annihilation is, we compare it to something substantial: a hydrogen bomb. A 1-megaton hydrogen bomb releases \(4.2 \times 10^{15} \) joules of energy. When we compare it to the energy from the matter-antimatter annihilation:

1. Matter-antimatter annihilation releases \(1.8 \times 10^{14} \ \text{J} \)
2. Hydrogen bomb releases \(4.2 \times 10^{15} \ \text{J} \)
3. Now, let's create the ratio: \(\frac{1.8 \times 10^{14} \ \text{J}}{4.2 \times 10^{15} \ \text{J}} \approx 0.043 \)
4. This ratio translates to approximately 4.3 ) percent.

Therefore, the energy from the annihilation is around 4.3 percent of the energy produced by a 1-megaton hydrogen bomb. That strikes a chord on just how powerful antimatter can be, even in small amounts.

Mass-energy equivalence is the principle underlining Einstein's formula. It essentially states that mass and energy are two forms of the same thing. This is why we can convert mass into energy and vice versa. By annihilating matter and antimatter, you see the energy equivalence clearly. Here’s why this matters:

• Every particle of matter has an equal and opposite antimatter particle.
• When these particles meet, they annihilate each other and the mass is converted to energy.
• The potential energy from mass conversion can power numerous applications, such as nuclear reactions.

To conclude, understanding mass-energy equivalence helps us grasp why such massive energy is released during matter-antimatter annihilation. This allows us to see potential future applications in energy production as well as the inherent dangers.