Problem 67 - University Physics with Modern Physics

Question

The explosive yield of the atomic bomb dropped on Hiroshima near the end of World War II was approximately 15.0 kilotons of TNT. One kiloton is about \(4.18 \cdot 10^{12} \mathrm{~J}\) of energy. Find the amount of mass that was converted into energy in this bomb.

Step-by-Step Solution

Answer

Answer: Approximately \(6.97 \cdot 10^{-4}\) kg of mass was converted into energy in the atomic bomb dropped on Hiroshima.

Step 1: Understand the given information

We are given the explosive yield of the atomic bomb dropped on Hiroshima as 15.0 kilotons of TNT, which is equivalent to 15.0 \(\times 4.18 \cdot 10^{12} \mathrm{~J}\) of energy. We will use this information to find the mass converted into energy.

Step 2: Convert the explosive yield in kilotons to Joules

To find the energy in Joules, we need to multiply the explosive yield in kilotons by the conversion rate provided. That is: Energy (J) = 15.0 kilotons × (\(4.18 \cdot 10^{12}\) J/kiloton) thus, Energy (J) = \(6.27 \cdot 10^{13}\) J

Step 3: Use Einstein's equation to find the mass

Einstein's equation states that the energy of an object is equal to its mass multiplied by the speed of light squared (\(E=mc^2\)). We have the energy (E) and need to find the mass (m). The speed of light (c) is approximately \(3.0 \cdot 10^8\ \mathrm{m/s}\) Rearranging the equation to solve for mass, we get: m = \(\frac{E}{c^2}\) Substituting the values we have m = \(\frac{6.27 \cdot 10^{13} \mathrm{~J}}{(3.0 \cdot 10^8\ \mathrm{m/s})^2}\)

Step 4: Calculate the mass

Now calculate the mass by dividing the energy by the speed of light squared: m = \(\frac{6.27 \cdot 10^{13} \mathrm{~J}}{(3.0 \cdot 10^8\ \mathrm{m/s})^2}\) m = \(\frac{6.27 \cdot 10^{13} \mathrm{~J}}{9.0 \cdot 10^{16}\ \mathrm{m^2/s^2}}\) m ≈ \(6.97 \cdot 10^{-4}\) kg So, about \(6.97 \cdot 10^{-4}\) kg of mass was converted into energy in the atomic bomb dropped on Hiroshima.

Key Concepts

Einstein's Equation

Einstein's equation, known as \( E = mc^2 \), is a landmark in scientific understanding. This famous equation tells us that energy (\( E \)) and mass (\( m \)) are interchangeable, linked through the speed of light (\( c \)). The speed of light is a large number, approximately \( 3.0 \times 10^8 \) meters per second, which means even a small amount of mass can produce a tremendous amount of energy.

To understand it simply:

• \( E \) is the energy measured in joules (J).
• \( m \) is the mass in kilograms (kg).
• \( c \) is the speed of light in meters per second (m/s).

This equation was revolutionary because it showed that mass could be transformed into energy, providing the foundational understanding for nuclear reactions. In these reactions, the transformation of a small amount of mass into energy is what enables powerful events like nuclear explosions or the energy we get from the sun.

Nuclear Energy

Nuclear energy is the energy released during nuclear reactions, either through fission or fusion. It's the process where an atom's nucleus splits or combines with another, releasing a large amount of energy.

Fission is when a heavy nucleus splits into smaller nuclei, releasing energy in the process. This is the reaction used in atomic bombs and nuclear power plants. Fusion, on the other hand, combines lighter nuclei into heavier ones, like what powers the sun.

Nuclear energy is powerful because:

• It can release millions of times more energy than chemical reactions.
• It doesn't produce greenhouse gases during the reaction itself.

However, managing nuclear energy poses challenges, including radioactive waste, risk of accidents, and nuclear weapons proliferation. These challenges make it a topic of significant scientific and policy interest.

Atomic Bomb

The atomic bomb is one of the most powerful applications of energy-mass equivalence and nuclear energy. The bomb dropped on Hiroshima during World War II serves as a grim reminder of the destructive potential of nuclear technology.

Atomic bombs work through nuclear fission, where uranium-235 or plutonium-239 nuclei split, releasing energy. This fission results in a chain reaction that produces a massive explosion.

The bomb released about 15 kilotons of TNT equivalent energy, as per Einstein's equation. This was achieved by converting a small amount of nuclear mass into energy. Just a few grams of matter can unleash catastrophic energy levels, enough to destroy an entire city.

• The Hiroshima bomb converted approximately \( 6.97 \times 10^{-4} \) kg of mass into explosive energy.
• This showcases the effectiveness yet the destructiveness of nuclear technology.

While atomic bombs have only been used twice in warfare, their presence has significantly influenced global politics and disarmament discussions. The goal has often been to find peaceful applications for nuclear technology, focusing on energy generation rather than destruction.