Problem 85 - Chemistry The Central Science In Si Units

Question

The average energy released in the fission of a single uranium- 235 nucleus is about \(3 \times 10^{-11} \mathrm{~J}\). If the conversion of this energy to electricity in a nuclear power plant is \(40 \%\) efficient, what mass of uranium- 235 undergoes fission in a year in a plant that produces 1000 megawatts? Recall that a watt is \(1 \mathrm{~J} / \mathrm{s}\).

Step-by-Step Solution

Answer

\( \approx 7,000 \) kg of uranium-235 undergoes fission in a year.

Step 1: Calculate Total Energy Produced in a Year

First, calculate the total energy produced by the power plant in one year. Given that the power plant produces 1000 megawatts, we convert this to joules per second. Since 1 megawatt equals \(10^6\) watts, the power produced is \(1000 \times 10^6 = 10^9\) joules per second. In one year, there are \(365 \times 24 \times 3600\) seconds. Thus, the total energy produced in a year is \(10^9 \times 365 \times 24 \times 3600\) joules.

Step 2: Calculate Energy from Uranium-235 Fission

Since the energy conversion efficiency is 40%, the total energy from fission is higher than the energy produced. Divide the total energy computed by the efficiency: \( \text{Total Energy from Uranium} = \frac{10^9 \times 365 \times 24 \times 3600}{0.4} \).

Step 3: Calculate Number of Fissions Required

The average energy released per fission is given as \(3 \times 10^{-11}\) joules. The total number of uranium fissions needed is the total energy from fission divided by the energy per fission: \( \text{Number of Fissions} = \frac{10^9 \times 365 \times 24 \times 3600}{0.4 \times 3 \times 10^{-11}} \).

Step 4: Calculate Mass of Uranium-235

To find the mass of uranium-235, we need to multiply the number of fissions by the mass of a single uranium nucleus. The molar mass of uranium-235 is 235 grams/mole, and Avogadro's number (\(6.022 \times 10^{23}\) nuclei/mole) gives the number of nuclei per mole. Thus, the mass for the total number of fissions is: \( \text{Mass} = \frac{\text{Number of Fissions}}{6.022 \times 10^{23}} \times 235 \).

Key Concepts

uranium-235

Uranium-235 is a vital component in nuclear fission reactions. It is a heavy isotope of uranium, with an atomic number of 92, and a mass number of 235. Uranium-235 is particularly important because it can easily undergo fission, making it a key fuel in nuclear reactors and atomic bombs.
In a fission reaction, a uranium-235 nucleus absorbs a slow-moving neutron. This absorption makes the nucleus unstable, causing it to split into smaller particles, along with releasing a significant amount of energy. This energy is what powers nuclear reactors.
When it comes to nuclear power plants, Uranium-235 is often used in the form of pellets which are packed into fuel rods. As the uranium undergoes fission, the energy released is used to heat water, producing steam that drives turbines. This process ultimately generates electricity.

• Most naturally occurring uranium is Uranium-238, with Uranium-235 making up about 0.7%.
• Enrichment processes are required to increase the concentration of Uranium-235 for use in reactors.
• It's the nucleus of choice for many reactors due to its ability to sustain a chain reaction.

energy conversion efficiency

Energy conversion efficiency is a measure of how well a power plant converts the energy produced during nuclear fission into usable electrical energy. It is pivotal because it directly impacts the effectiveness and sustainability of a power plant.
In the context of a nuclear power plant using uranium-235, conversion efficiency indicates the proportion of energy released from uranium fission that is transformed into electricity. Efficiency differs across power plants; in this example, it is given as 40%. This means that only 40% of the energy created during fission is utilized as electrical power.
Understanding energy conversion efficiency helps in optimizing the energy output and assessing the operational costs of a power plant. Improving efficiency is beneficial for reducing waste and using resources more effectively.

• Efficiency can be reduced by heat losses during conversion processes, primarily from engineering and mechanical limits.
• Research in materials and technology aims to enhance efficiency.
• Higher efficiency results in less fuel use for the same energy output.

joules per second

Joules per second is a unit of power, which is crucial in understanding how quickly energy is being released or consumed. It is commonly known as a watt (W), named after James Watt. 1 watt is equivalent to 1 joule of energy per second.
In the case of the exercise, the power plant produces energy at a rate of 1000 megawatts, where 1 megawatt equals 1 million watts (or joules per second). Therefore, understanding this unit enables one to comprehend the scale of energy production in nuclear power plants.
A nuclear plant's power output is often expressed in watts to indicate the energy produced over time, providing a clear understanding of the plant's capacity. This measure is vital for planning energy supply and demand.

• Power expressed in watts allows for quick comparisons between power sources.
• Large-scale energy production is often measured in megawatts or gigawatts.
• It helps in calculating energy production and requirements accurately.

molar mass

Molar mass is a key concept in chemistry when dealing with quantities of atoms or molecules. It measures the mass of a given substance divided by the amount of substance, typically in grams per mole (g/mol).
In nuclear reactions, such as those involving uranium-235, knowing the molar mass is necessary to quantify the quantity of uranium used in fission reactions. Uranium-235 has a molar mass of 235 grams/mole.
The molar mass helps convert between the number of fissions or atoms and the mass. This is crucial for calculating how much uranium-235 is needed to generate a specific amount of energy, as seen in the exercise.

• It provides a bridge between microscopic atomic levels and macroscopic quantities.
• Molar mass is utilized alongside Avogadro's number for such conversions.
• Ensures consistent and accurate calculations regarding chemical and nuclear reactions.