Uranium ore has two main isotopes, mostly U-238 with just a trace amount of U-235. In a sample of Uranium ore, 99.85% of the atoms are U-238 atoms and 0.15% are U-235 atoms.
Before the Uranium can be used in a nuclear power plant, the proportion of U-235 must be increased to 15% (thus reducing the proportion of U-238 to 85%). This is done by a process called gas diffusion. The ratio of the masses of these two isotopes is U-238 to U-235 = 1.013. Each cycle of the gas diffusion process decreases U-238 by 1.3%. How many cycles are required to reduce the U-238 to 85%?
You want to go from 99.85% U-238 to 85%.
Each cycle of the process decreases the U-238 fraction by a ratio 1-0.013 = 0.987. Let the number of processing cycles required be N.
0.850/0.9985 = 0.8513 = 0.987^N
Solve for N.
N = Log0.8513/Log0.987 = 12.3
Call it 13, for a margin of safety
This is not a calculus problem. Precalc, maybe.
178
To find out how many cycles are required to reduce the U-238 to 85%, we can set up an equation and solve for the number of cycles.
Let's assume we start with 100 grams of Uranium ore.
Step 1: Calculate the initial amount of U-238 and U-235:
- U-238: 99.85% of 100 grams = 99.85 grams
- U-235: 0.15% of 100 grams = 0.15 grams
Step 2: Calculate the target amount of U-238 and U-235 after the gas diffusion process:
- U-238: 85% of 100 grams = 85 grams
- U-235: 15% of 100 grams = 15 grams
Step 3: Calculate the mass ratio between U-238 and U-235:
- Mass ratio = U-238 / U-235 = 1.013
Step 4: Determine the decrease in U-238 per cycle:
- Decrease in U-238 per cycle = 1.3% of the current amount of U-238
Step 5: Set up the equation and solve for the number of cycles:
Current amount of U-238 - (Decrease in U-238 per cycle * Number of cycles) = Target amount of U-238
99.85 - (0.013 * Number of cycles) = 85
Simplifying the equation:
99.85 - 0.013 * Number of cycles = 85
Subtract 99.85 from both sides:
-0.013 * Number of cycles = -14.85
Divide by -0.013:
Number of cycles = 1142.31
Since we can't have a fraction of a cycle, we round up to the nearest whole number:
Number of cycles required = 1143
Therefore, 1143 cycles are required to reduce the proportion of U-238 to 85% in the uranium ore.
To find out how many cycles are required to reduce the proportion of U-238 to 85%, we can use a step-by-step approach:
Step 1: Calculate the initial proportion of U-238 in the Uranium ore.
Given that 99.85% of the atoms in the sample are U-238, we can express this as 99.85/100 or 0.9985.
Step 2: Calculate the target proportion of U-238.
The target proportion of U-238 is 85%. This can be expressed as 85/100 or 0.85.
Step 3: Calculate the difference between the initial and target proportions of U-238.
Subtracting the target proportion from the initial proportion, we get: 0.9985 - 0.85 = 0.1485.
Step 4: Determine the decrease in U-238 per cycle.
Since each cycle of the gas diffusion process decreases U-238 by 1.3%, we can express this as 1.3/100 or 0.013.
Step 5: Divide the difference from Step 3 by the decrease per cycle.
Dividing 0.1485 (the difference from Step 3) by 0.013 (the decrease per cycle), we get: 0.1485 / 0.013 ≈ 11.42.
Step 6: Round up the result to the next whole number.
Since you cannot have a fraction of a cycle, we need to round up to the nearest whole number. Therefore, we need 12 cycles to reduce the proportion of U-238 to 85%.
So, you would need approximately 12 cycles of the gas diffusion process to achieve the desired proportion of U-238 (85%) in the Uranium ore.