Problem 56 - Fundamentals of Physics

Question

The energy released in the explosion of \(1.00 \mathrm{~mol}\) of \(\mathrm{TNT}\) is \(3.40 \mathrm{MJ}\). The molar mass of TNT is \(0.227 \mathrm{~kg} / \mathrm {mol}\). What weight of TNT is needed for an explosive release of \(1.80 \times 10^{14} \mathrm{~J} ?\) (b) Can you carry that weight in a backpack, or is a truck or train required? (c) Suppose that in an cxplosion of a fission bomb, \(0.080 \%\) of the fissionable mass is converted to released energy. What weight of fissionable material is needed for an explosive release of \(1.80 \times 10^{14} \mathrm{~J} ?\) (d) Can you carry that weight in a backpack, or is a truck or train required?

Step-by-Step Solution

Answer

Truck/train needed for both TNT and fission material.

Step 1: Calculate Moles of TNT Required

The energy released per mole of TNT is given as \(3.40 \text{ MJ} = 3.40 \times 10^6 \text{ J}\). To release \(1.80 \times 10^{14} \text{ J}\), divide the total energy by the energy per mole of TNT: \(\text{moles of TNT} = \frac{1.80 \times 10^{14} \text{ J}}{3.40 \times 10^6 \text{ J/mol}}\). Thus, \(\text{moles of TNT} = 5.29 \times 10^7\text{ mol}.\)

Step 2: Calculate Mass of TNT Required

Using the molar mass of TNT, \(0.227 \text{ kg/mol}\), calculate the total mass required: \(\text{mass of TNT} = \text{moles of TNT} \times \text{molar mass of TNT}\). Substituting values gives \(\text{mass of TNT} = 5.29 \times 10^7\text{ mol} \times 0.227 \text{ kg/mol}\ = 1.2 \times 10^7 \text{ kg}.\)

Step 3: Evaluate Transport of TNT

We have \(1.2 \times 10^7 \text{ kg}\) of TNT, which is 12 million kg. A standard backpack can typically carry 10-20 kg. Therefore, a truck or train is required for transport.

Step 4: Convert Energy to Mass for Fission Material

For a fission bomb, \(0.080\%\) of the mass is converted to energy. We use \(E = mc^2\), where \(c\) is the speed of light \(3 \times 10^8 \text{ m/s}\). Rearrange to find mass: \(m = \frac{E}{c^2}\). Thus, \(m = \frac{1.80 \times 10^{14} \text{ J}}{(3 \times 10^8 \text{ m/s})^2}\ = 2 \text{ kg}\) (total mass conversion). Since only \(0.080\%\) is converted: \(2 \text{ kg}\ = 0.080\% \text{ of original mass}\), compute original mass as \(\text{original mass} = \frac{2 \text{ kg}}{0.0008}\ = 2.5 \times 10^3 \text{ kg}.\)

Step 5: Evaluate Transport of Fission Material

The total weight of the fissionable material needed is \(2500 \text{ kg}\). This weight is beyond the capacity of a backpack, requiring transport by a truck or train.

Key Concepts

Molar Mass

Molar mass is a simple yet fundamental concept in chemistry that reflects the mass of one mole of a substance. It is expressed in units of kilograms per mole (kg/mol) or grams per mole (g/mol). To find the molar mass of a compound, you need to sum up the atomic masses of all atoms in a molecule. This number gives a straightforward way to convert between the mass of a chemical substance and the amount in moles.
For TNT, the molar mass is given as 0.227 kg/mol. This means that each mole of TNT has a mass of 227 grams (because 1 kg = 1000 g). Knowing the molar mass of a substance is essential when performing calculations involving chemical reactions, such as determining how much material is needed to produce a specific amount of energy.

Fission Reaction

Fission reactions are a type of nuclear reaction where an atomic nucleus splits into smaller parts, releasing a significant amount of energy. These reactions are used in nuclear power stations and weapons. The energy in a fission reaction comes from the conversion of mass to energy, as described by Einstein's famous equation, \[ E = mc^2 \].
In the context of the problem, only 0.080% of the fissionable material's mass gets converted into energy. This percentage highlights how efficient nuclear reactions are at producing large energy with a small amount of fuel. Despite only converting a tiny fraction of its mass, the energy release is so great due to the enormous value of the speed of light squared, making even small masses significant energy sources.

Energy Release

Energy release in chemical and nuclear reactions is a critical measure of how much energy is produced from a given reaction. It is usually expressed in units like joules (J) or megajoules (MJ). Understanding energy release is crucial in applications ranging from simple chemistry experiments to large-scale industrial processes.

• In chemical reactions, like the explosion of TNT, energy is released through the breaking and forming of chemical bonds.
• In nuclear reactions, energy release occurs through changes in nuclear structure, such as fission, where atomic nuclei split apart.

For TNT, the energy released is 3.40 MJ per mole, which is used in the problem to determine how much TNT is needed to release a total energy of \(1.80 \times 10^{14} \text{ J}\). Calculating energy release accurately helps in efficient resource use and ensures safety in explosive applications.

Backpack Capacity

The capacity of a backpack usually ranges from 10 to 20 kilograms, suitable for carrying personal items or supplies. In contexts involving large masses, like transporting explosives or materials generated from nuclear reactions, understanding this capacity is essential to decide on the transportation method.
In our scenario, carrying the calculated mass of 12 million kilograms of TNT or 2500 kilograms of fissionable material clearly exceeds the capacity of a backpack. This context is crucial in physics problem-solving, as it highlights the practical aspects of transporting large quantities of substances, pointing towards the need for a truck or train instead of attempting the impractical task of using a backpack.

Physics Problem Solving

Physics problem-solving involves using principles, formulas, and logical reasoning to find solutions to complex queries. This particular problem combines concepts from chemistry and physics, including molar mass, energy release, and nuclear physics, demonstrating a multidisciplinary approach to problem-solving.

• First, use the known energy release per mole to find how many moles of TNT are needed for a specific energy output.
• Convert moles to mass using the molar mass.
• Assess practical implications, like transport needs and feasibility of available carrying methods.

By breaking down the problem into clear steps, physics problem-solving becomes less overwhelming and more manageable, encouraging students to build confidence in tackling real-world problems with a structured approach.