Problem 56 - University Physics with Modern Physics

Question

A nuclear bomb containing 8.00 \(\mathrm{kg}\) of plutonium explodes. The sum of the rest masses of the products of the explosion is less than the original rest mass by one part in \(10^{4} .\) (a) How much energy is released in the explosion? (b) If the explosion takes place in 4.00\(\mu \mathrm{s}\) , what is the average power developed by the bomb? (c) What mass of water could the released energy lift to a height of 1.00 \(\mathrm{km} ?\)

Step-by-Step Solution

Answer

(a) 7.2 x 10^13 J; (b) 1.8 x 10^19 W; (c) 7.35 x 10^9 kg of water.

Step 1: Calculate Energy Released

The rest mass of plutonium is reduced by 1 part in \(10^4\), meaning \(\Delta m = \frac{8.00 \mathrm{kg}}{10^4} = 0.0008 \mathrm{kg}\). Using Einstein’s mass-energy equivalence principle \(E=\Delta m c^2\), where \(c\) is the speed of light \(3.00 \times 10^8 \mathrm{m/s}\), the energy released is \[E = 0.0008 \times (3.00 \times 10^8)^2 = 7.2 \times 10^{13} \mathrm{J}.\]

Step 2: Calculate Average Power

Power is defined as energy divided by time. The energy from the explosion is \(7.2 \times 10^{13}\) J, and the explosion duration is \(4.00 \mu \mathrm{s} = 4.00 \times 10^{-6} \mathrm{s}\). Therefore, the average power developed by the bomb is \[\text{Power} = \frac{7.2 \times 10^{13} \mathrm{J}}{4.00 \times 10^{-6} \mathrm{s}} = 1.8 \times 10^{19} \mathrm{W}.\]

Step 3: Calculate Mass of Water Lifted

The energy required to lift a mass \(m\) to a height \(h\) against gravity is given by \(E = mgh\), where \(g = 9.8 \mathrm{m/s^2}\). Let's solve for \(m\) using the energy calculated: \[m = \frac{E}{gh} = \frac{7.2 \times 10^{13}}{9.8 \times 10^3} = 7.35 \times 10^{9} \mathrm{kg}.\] Therefore, \(7.35 \times 10^{9} \mathrm{kg}\) of water could be lifted to a height of 1.00 \(\mathrm{km}\).

Key Concepts

Mass-Energy Equivalence

Mass-energy equivalence is a core concept in nuclear physics, introduced by Albert Einstein. It states that mass and energy are interchangeable and are essentially two forms of the same thing. This concept is represented by the famous equation \(E = mc^2\). In this equation, \(E\) stands for energy, \(m\) represents mass, and \(c\) is the speed of light in a vacuum, approximately \(3 \times 10^8 \text{m/s}\).

If you ever wondered why nuclear reactions release so much energy, it's because they convert a small amount of mass into a huge amount of energy. The conversion factor \(c^2\) is immensely large, so a tiny bit of mass results in enormous energy. In the exercise, the reduction in the plutonium mass, though small \(0.0008 \text{kg}\), releases a tremendous amount of energy, calculated as \(7.2 \times 10^{13} \text{J}\). This is why nuclear energy is so powerful and efficient compared to chemical reactions.

• Mass-energy equivalence highlights that even the smallest mass changes in nuclear processes have significant energy effects.
• It has paved the way for nuclear technologies, including power generation and medical treatments.

Energy Conversion

Energy conversion is the process of changing one form of energy to another. In nuclear reactions, like those involving plutonium in a bomb, part of the nuclear mass is converted into energy. The released energy manifests in different forms such as heat, light, and kinetic energy.

Consider the explosion in the exercise, where the decrease in mass due to the transformation results in \(7.2 \times 10^{13} \text{J}\) of energy. This massive release of energy is an example of nuclear energy conversion, transforming the potential energy stored in the nuclear bonds into other usable forms.

• Nuclear power plants operate on a similar principle by controlling the energy conversion process to produce electricity.
• It highlights the potential for generating great amounts of energy from minimal resources, one of the reasons nuclear power is considered when seeking sustainable energy solutions.

Power Calculation

Power refers to the rate at which energy is transferred or converted. It is often measured in watts (W), where 1 watt equates to 1 joule per second. Power calculation is crucial in understanding how quickly an energy transfer occurs.

In the exercise, the average power developed by the bomb was determined by dividing the total energy released \(\text{(7.2} \times 10^{13} \text{J})\) by the time taken \(\text{(4.00} \times 10^{-6} \text{s})\). The calculation \[\text{Power} = \frac{7.2 \times 10^{13} \text{J}}{4.00 \times 10^{-6} \text{s}} = 1.8 \times 10^{19} \text{W}\]shows the bomb's massive power output during the incredibly brief explosion window.

• Power calculation helps in estimating the potential impact and efficiency of energy systems, such as nuclear bombs or power plants.
• In the context of nuclear explosions, it illustrates the tremendous power released in a short timeframe.

Nuclear Reactions

Nuclear reactions involve changes in an atom's nucleus and can release or absorb large amounts of energy due to changes in mass resulting from the reaction.

Two primary types of nuclear reactions are fission, splitting a heavy nucleus into lighter nuclei, and fusion, where lighter nuclei combine to form a heavier one. Both generate considerable energy due to mass conversion inherent in nuclear events.
In the exercise, a plutonium explosion signifies a fission reaction, leading to the conversion of a fraction of mass into energy. This understanding of nuclear reactions is pivotal not only for energy production (e.g., nuclear power plants) but also in broader applications such as medical and industrial uses.

• Nuclear reactions have immense energy potentials due to the binding energies involved in the nucleus.
• They necessitate a deep understanding of radiation safety and waste management due to the potential environmental implications.