Question

The U.S. Department of Energy sells an isotope of the transuranium element californium for approximately \(\$ 10\) per microgram to qualified researchers. What would a pound of the Cf nuclide cost?

Short answer

The cost of a pound of the Cf nuclide would be approximately \(\$45,359,200,000\).

Step-by-step solution

Understand the given information

We are given the cost of the Cf nuclide as $10 per microgram. We need to find out its cost per pound.

Convert micrograms to grams

We know that 1 gram = \(10^6\) micrograms. Therefore, 1 microgram = \(\frac{1}{10^6}\) grams.

Convert grams to pounds

We know that 1 pound = 453.592 grams. Therefore, 1 gram = \(\frac{1}{453.592}\) pounds.

Find the cost per pound

Now, we have the cost per microgram and we need to convert it to cost per pound by multiplying it with the conversion factors we calculated in the previous steps (converting micrograms to grams and grams to pounds). Cost per pound = \(\$10\) per microgram × \(\frac{1}{10^6}\) grams per microgram × \(\frac{1}{453.592}\) pounds per gram = \(\$10\) × \(\frac{1}{10^6}\) × \(\frac{1}{453.592}\) pounds = \(\$10\) × \(\frac{1}{10^6 * 453.592}\) pounds = \(\$10\) × \(\frac{1}{453592000}\) pounds

Calculate the cost

Now, we can calculate the cost of a pound of Cf nuclide by multiplying the cost per pound with the number of pounds we want to buy, which in this case is 1 pound. Cost = \(\$10\) × \(\frac{1}{453592000}\) pounds × 1 pound = \(\$10\) × \(\frac{1}{453592000}\) = \(\$0.000000022035\) per pound, approximately. So, the cost of a pound of the Cf nuclide would be approximately \(\$0.000000022035\). However, this result seems very unusual and counter-intuitive, so we might have made a mistake during the calculations. Let's go back to step 4 and reconsider the conversion.

(Revised): Find the cost per pound

Cost per pound = \(\$10\) per microgram × \(\frac{10^6}{1}\) micrograms per gram × \(\frac{453.592}{1}\) grams per pound = \(\$10\) × \(10^6\) × \(453.592\) pounds = \(\$10\) × \(4535920000\) pounds

(Revised): Calculate the cost

Now, we can calculate the cost of a pound of Cf nuclide by multiplying the cost per pound with the number of pounds we want to buy, which in this case is 1 pound. Cost = \(\$10\) × \(4535920000\) pounds × 1 pound = \(\$10\) × \(4535920000\) = \(\$45,359,200,000\) per pound, approximately. So, the cost of a pound of the Cf nuclide would be approximately \(\$45,359,200,000\).

Key concepts

Micrograms to Grams Conversion

When exploring unit conversions, particularly between micrograms and grams, it's essential to understand the scale at which these units operate. A microgram is very small, just a millionth of a gram. This means that to convert micrograms to grams, you take the number of micrograms and divide it by one million.To visualize:

• 1 gram equals 1,000,000 micrograms.
• Thus, 1 microgram equals \( \frac{1}{10^6} \) grams.

Grasping this conversion is crucial for precise measurements, like in scientific research, where even small errors can impact results significantly. Always ensure the accuracy of the numbers you're working with by double-checking conversions with tools or calculators if needed.

Grams to Pounds Conversion

Converting from grams to pounds adds another layer of conversion, as these units measure the same quantity (mass) but in different systems. The metric system uses grams, while the imperial system uses pounds. The pivotal conversion factor between these two is that 1 pound is equivalent to 453.592 grams.When converting:

• Divide the number of grams by 453.592 to get the equivalent weight in pounds.

For example, converting 500 grams to pounds would involve calculating \( \frac{500}{453.592} \), resulting in approximately 1.102 pounds. Accurate conversions are essential, especially in fields such as chemistry or cooking, where the weight of materials must be precise.

California Nuclide Price Calculation

Determining the cost of isotopes, like the californium nuclide, involves understanding both the price per unit and the necessary unit conversions. Here, californium costs about $10 per microgram. If you want to find the cost per pound, you'll need to use both micrograms-to-grams and grams-to-pounds conversions.Here's a simplified breakdown:

• First, convert dollars per microgram to dollars per gram by multiplying the microgram price by \(10^6\), since there are 1,000,000 micrograms in a gram.
• Next, convert the price in grams to price in pounds by using the conversion factor for grams to pounds (multiplying by 453.592).

Analyzing these conversions allows researchers to budget properly for their experiments, which could involve significant isotopic mass, leading to substantial expenses.

Unit Conversion Error Analysis

Error analysis is an essential part of ensuring accuracy in unit conversions. Missteps in conversion, such as reversing a conversion factor or miscalculation, can lead to significant inaccuracies, as seen in the drastic price change initially calculated for californium nuclide. Key practices to avoid errors:

• Always confirm the units and conversion factors before performing calculations.
• Re-evaluate calculations that produce unexpected results. For example, unexpected low or high cost results should prompt a re-examination of the steps involved.
• Use systematic approaches, such as writing out unit conversions step-by-step, to catch any potential errors.
• Consider utilizing software tools for complex conversions to minimize human error.

By following these practices, precision and reliability in scientific and practical applications can be significantly improved.