Polluted air is flowing into the filter at a rate of 20 cubic meters per minute. There is half a gram of uranium dust per cubic meter of air. 30% of the dust escapes the stack, since the filter has an efficiency rating of 70%. You need to calculate the daily uranium emissions from the stack in kilograms per day and determine if the plant is in compliance. Here's how to do it (though this is not the only way):

1. (20m³ of air) x (0.5g uranium) = 10 grams of uranium per minute (or 10gU/min)

2. (10gU/min) x (60 mins) = 600gU/hour

3. (600gU/hr.) x (24 hours) = 14,400gU/day

4. 30% escapes from stack, so 0.3 x 14,400 = 4,320 gU/day

5. Regulations for emissions are given in kilograms (kg) per day, so the 4,320 grams per day must be converted to kilograms per day. One kg = 1,000g, so by converting you get:

6. 4,320 / 1,000 = 4.32kgU/day

7. Regulations say emissions should be below 1.5kgU/day, so the plant is out of compliance, as Gamma suspected!

Advanced Question:

This question required that you calculate the radioactivity in the uranium that has escaped the stack to see if it violates the standard for air quality at the plant boundary. The standard is expressed in picocuries per liter, and your information is expressed in grams per cubic meter of air. Here's how Gamma figured it:

1. We know that 30% of the 0.5gU per cubic meter is escaping the stack, so:

2. We know this amount is diluted by 10,000 by the time it reaches the plant boundary:

3. Now we need to convert cubic meters (m³) to liters. We know that 1 cubic meter = 1,000 liters (10³ liters - see table of unit conversions), so convert cubic meters to liters this way:

4. We now need to convert 1.5x10^-8gU / liter to picocuries per liter. We can calculate this since we know the specific activity of natural uranium is 0.67 microcuries/gram (0.67µCi/g).

5. We now need to convert microcuries (µCi) to picocuries (pCi). If you have visited the Online Classroom, you know that a microcurie is a millionth of a curie and a picocurie is a trillionth of a curie. (Another way to think of it is that a million picocuries equals one microcurie.) So to convert to picocuries, do the following (remember 10⁶pCi / 1µCi = 1):

6. Does the plant violate the air quality standard of 0.005pCi/liter?

So Gamma was right again, the plant does violate the air quality standard!