A typical home uses approximately 1.0⨉10³ kWh of energy per month. If the energy came from a nuclear reaction, what mass would have to be converted to energy per year to meet the energy needs of the home?

Verified step by step guidance

1. Identify the total energy consumption per year by multiplying the monthly energy usage by 12 months: \(1.0 \times 10^3 \text{kWh/month} \times 12 \text{ months/year}\).
2. Convert the energy from kilowatt-hours (kWh) to joules (J) using the conversion factor: \(1 \text{ kWh} = 3.6 \times 10^6 \text{ J}\).
3. Use Einstein's mass-energy equivalence formula \(E = mc^2\) to find the mass \(m\) that would be converted to energy, where \(E\) is the energy in joules and \(c\) is the speed of light \(3.0 \times 10^8 \text{ m/s}\).
4. Rearrange the formula to solve for mass \(m\): \(m = \frac{E}{c^2}\).
5. Substitute the calculated energy \(E\) and the speed of light \(c\) into the equation to find the mass \(m\) in kilograms.

Key Concepts

Energy-Mass Equivalence

Energy-mass equivalence is a principle derived from Einstein's theory of relativity, expressed by the equation E=mc². This equation states that energy (E) and mass (m) are interchangeable; a small amount of mass can be converted into a large amount of energy due to the speed of light squared (c²) being a very large number. This concept is crucial for understanding how mass can be converted to meet energy needs.

Nuclear Energy

Nuclear energy is the energy released during nuclear reactions, such as fission or fusion. In fission, heavy atomic nuclei split into smaller parts, releasing energy, while fusion involves light nuclei combining to form a heavier nucleus. Understanding nuclear energy is essential for calculating how much mass must be converted to meet energy demands, as it provides a context for the energy output from nuclear reactions.

Energy Consumption Calculations

Energy consumption calculations involve determining the total energy used over a specific period, often measured in kilowatt-hours (kWh). In this scenario, the monthly energy usage of 1.0⨉10³ kWh must be converted to an annual figure to assess the total energy requirement. This calculation is fundamental for quantifying the energy needs of a household and subsequently relating it to the mass of fuel required for energy production.