Problem 56 - Fundamentals of Physics

Question

(a) The energy released in the explosion of \(1.00\) mol of 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 explosion 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

(a) About 12 million kg of TNT; needs trucks/trains. (c) 2500 kg of fissionable material; needs a small truck.

Step 1: Calculate the amount of moles of TNT needed

To find the number of moles of TNT required to release the specified energy, we use the relation: \[\text{Number of moles} = \frac{\text{Total Energy Required}}{\text{Energy per mole of TNT}}\]We plug in the values:\[\text{Number of moles} = \frac{1.80 \times 10^{14} \text{ J}}{3.40 \times 10^6 \text{ J/mol}}\]This calculation results in a number of moles approximately equal to 52941176.5 moles.

Step 2: Convert moles to mass for TNT

Using the molar mass of TNT, convert the calculated moles to mass:\[\text{Mass of TNT} = \text{Number of moles} \times \text{Molar mass of TNT}\]Substitute the values in:\[\text{Mass of TNT} = 52941176.5 \text{ moles} \times 0.227 \text{ kg/mol}\]This gives a mass of approximately 12,006,862 kg (or about 12 million kg).

Step 3: Determine transportability of TNT mass

Considering that 12 million kg is an immense weight, it is impossible to carry this in a backpack. Transportation of this quantity of TNT would require multiple trucks or a train series due to its massive size.

Step 4: Calculate mass of fissionable material needed

The energy-mass equivalence is given by:\[ \text{Mass converted} = \frac{\text{Energy released}}{c^2} \]Where \(c\) is the speed of light (\(3 \times 10^8 \text{ m/s}\)).First, find the mass equivalent for a 100% conversion:\[\text{Mass converted for 100%} = \frac{1.80 \times 10^{14} \text{ J}}{(3 \times 10^8 \text{ m/s})^2} = 2 \text{ kg}\]Since only 0.080% is converted, we need:\[\text{Required Fissionable Mass} = \frac{2 \text{ kg}}{0.0008} = 2500 \text{ kg}\]

Step 5: Determine transportability of fissionable material

A mass of 2500 kg is still too heavy to carry in a backpack. This would typically require a vehicle suitable for large transport, such as a small truck.

Key Concepts

Molar Mass

Molar mass is a fundamental concept in chemistry and plays a vital role in energy calculations. It represents the weight of one mole of a substance and is expressed in units of mass per mole (e.g., grams per mole or kilograms per mole). Determining the molar mass of a substance involves adding up the atomic masses of all atoms in its chemical formula. For instance, in the case of TNT (trinitrotoluene), you identify the atomic masses of its constituent atoms—hydrogen, carbon, nitrogen, and oxygen—and total them to find the molar mass.

In our problem, the molar mass of TNT is given as 0.227 kg/mol. This value is crucial because it allows conversion from moles to a measurable mass. Calculating how much TNT is needed to produce a certain energy release involves first finding out how many moles are needed and then converting that into a mass using the molar mass.

• Calculating moles: helps determine how much of the substance is involved in the reaction.
• Molar mass: allows conversion from theoretical moles to practical mass measurements.

Fissionable Material

Fissionable materials are substances capable of undergoing fission, a nuclear reaction in which an atomic nucleus splits into smaller parts, releasing a significant amount of energy. Common examples of fissionable materials include uranium-235 and plutonium-239. These materials are key in nuclear reactors and weapons, where their ability to release energy is harnessed.

In fission reactions, a minute quantity of the mass of the reacting material is converted into energy. This process is highly efficient at energy release compared to chemical reactions, such as those involving TNT. Our problem involves an explosive yield from a fissionable material, demonstrating how small amounts of these materials can release enormous energy. However, only a tiny fraction (0.080%) of the fissionable mass is typically converted into energy, requiring precise calculations to determine the needed quantity.

• Fission: involves splitting atomic nuclei, releasing vast energies.
• Efficiency: fission reactions transform mass into energy, operating on energy-mass equivalence.

Energy-mass Equivalence

Energy-mass equivalence is a principle derived from Einstein's famous equation, \(E=mc^2\), where \(E\) is energy, \(m\) is mass, and \(c\) is the speed of light in a vacuum \( (3 \, \times 10^8 \, \/ \mathrm{m/s}) \). This equation indicates that mass can be converted into energy and vice versa, with the speed of light squared acting as the conversion factor.

In practical contexts, such as nuclear fission, a small amount of mass loss results in a substantial energy release. For our problem, calculating the mass equivalent of the energy released helps determine how much material is needed for a given energy output. When only 0.080% of the fissionable material is converted, this equation is used to calculate how much total mass is necessary to reach the desired energy outcome.

• \(E=mc^2\): suggests mass and energy are interchangeable.
• Significance: this principle is utilized in nuclear energy and provides context for the mass-energy conversions in the problem.

Moles to Mass Conversion

Converting moles to mass is an essential process in chemistry, allowing scientists to translate the abstract concept of moles into practical, measurable quantities. The conversion involves using the molar mass of a substance, defined as the mass of one mole of a substance.

To convert from moles to mass, multiply the number of moles by the molar mass of the substance. In our example with TNT, after calculating the required number of moles for the desired energy release, you multiply these moles by 0.227 kg/mol (the molar mass of TNT) to determine the required mass. This conversion is critical for efficiently calculating and managing materials in both experimental and industrial settings.

• Conversion process: involves using molar mass for translating moles to mass.
• Application: helps in determining material quantities for reactions and manufacturing, ensuring accurate measurements.