Final answer:
To supply the energy needs of a household using 27.0 kWh of energy per day, approximately 0.242 grams of ²³⁵₉₂U would be consumed in one year.
Step-by-step explanation:
The question asks how many grams of ²³⁵₉₂U are consumed in one year to supply the energy needs of a household that uses 27.0 kWh of energy per day. Using the energy density of 235U as 17 × 10⁶ kcal/g, and knowing that a household's energy needs can be equated to 4.5 grams of 235U per year, we can extrapolate that amount for a household using 27.0 kWh per day.
First, we need to convert kWh to kcal to match the units of the energy density of uranium. There are 3.6 megajoules (MJ) in a kWh and 1 kcal equals 4.184 kJ. So we calculate the daily, then yearly energy usage in kcal, and finally how much uranium that would equate to based on its energy density. To calculate the number of grams of ²³⁵₉₂U consumed in one year to supply the energy needs of a household, we first need to convert the energy usage from kWh to Joules. Since 1 kWh is equal to 3.6 × 10⁶ J, the household's daily energy usage of 27.0 kWh is equal to 97.2 x 10^6 J. Then, using the energy density of 17 × 10⁶ kcal/g for ²³⁵₉₂U, we can convert the energy requirement to grams:
27.0 kWh x (3.6 × 10⁶ J/kWh) / (4.184 J/kcal) / (17 × 10⁶ kcal/g) = 0.242 g of ²³⁵₉₂U
Therefore, approximately 0.242 grams of ²³⁵₉₂U would be consumed in one year to supply the energy needs of the household.