Problem 4 - Chemical Principles in the Laboratory

Question

In this experiment it takes about 10 micro-liters of solution to produce a spot \(1 \mathrm{cm}\) in diameter. If the \(\mathrm{Bi}\left(\mathrm{NO}_{3}\right)_{3}\) solution contains about \(21 \mathrm{g} \mathrm{Bi}^{3+}\) per liter, how many micrograms of \(\mathrm{Bi}^{3+}\) ion are there in one spot?

Step-by-Step Solution

Answer

There are 210 micrograms of \(\mathrm{Bi}^{3+}\) ion in one spot.

Step 1: Calculate the Volume of One Spot in Liters

We know that each spot requires 10 micro-liters of solution. To convert to liters, remember that 1 liter equals 1,000,000 micro-liters. Thus, the volume of one spot in liters is:\[\text{Volume} = \frac{10 \text{ micro-liters}}{1,000,000} = 0.00001 \text{ liters}\]

Step 2: Determine the Mass of \( \mathrm{Bi}^{3+} \) in One Spot

Given that the solution contains 21 grams of \( \mathrm{Bi}^{3+} \) per liter, we find the mass of \( \mathrm{Bi}^{3+} \) in the 0.00001 liters by using the proportion:\[\text{Mass of } \mathrm{Bi}^{3+} = 21 \text{ grams/liter} \times 0.00001 \text{ liters} = 0.00021 \text{ grams}\]

Step 3: Convert the Mass from Grams to Micrograms

To convert from grams to micrograms, use the conversion factor that 1 gram equals 1,000,000 micrograms. Thus, the mass of \( \mathrm{Bi}^{3+} \) in one spot in micrograms is:\[\text{Mass in micrograms} = 0.00021 \text{ grams} \times 1,000,000 \text{ micrograms/gram} = 210 \text{ micrograms}\]

Key Concepts

Micro-liter to Liter Conversion

Converting micro-liters into liters is often necessary when working with tiny volumes typically found in chemistry labs. Micro-liter stands for one-millionth of a liter, or simply, the prefix 'micro' represents a factor of \(10^{-6}\). Thus, converting a given number of micro-liters to liters is straightforward. You divide the number of micro-liters by 1,000,000.

For example, if you have 10 micro-liters, you convert it to liters by calculating \( \frac{10}{1,000,000} = 0.00001 \) liters.
This conversion is essential for calculations that require input in standard units like liters as commonly used in solution concentration.

• 1 micro-liter \(= 10^{-6}\) liters.
• Division by 1,000,000 to convert to liters.
• Key for balancing chemical equations or preparing solutions.

By understanding the conversion, it's easier to measure small amounts accurately, which is vital in precise scientific experiments.

Mass Calculation

Calculating the mass of a solute in a solution involves understanding both the concentration of the solute and the volume of the solution used. In the given problem, the concentration of \( \mathrm{Bi}^{3+} \) ions is provided as 21 grams per liter. To find out how much mass of the ion is contained in a smaller volume, you multiply the concentration by that volume in liters.

For a volume of 0.00001 liters, this means the mass of \( \mathrm{Bi}^{3+} \) is calculated as:\[\text{Mass} = 21 \text{ grams/liter} \times 0.00001 \text{ liters} = 0.00021 \text{ grams}\]

• Mass \(= \text{Concentration} \times \text{Volume}\).
• Essential for determining how much of a chemical is in a given volume.
• Helps ensure precise dosages and reactions in a lab setting.

This calculation is critical in chemistry for scaling formulas and ensuring the correct amounts of reactants are mixed.

Unit Conversion

Understanding how to convert units is crucial in science, especially converting between measurement systems like grams and micrograms. The prefix 'micro' implies a factor of \(10^{-6}\). Therefore, when converting from grams to micrograms, you multiply by 1,000,000, since one gram equals one million micrograms.

For calculating the mass of \( \mathrm{Bi}^{3+} \) in micrograms, given the mass in grams:\[\text{Mass in micrograms} = 0.00021 \text{ grams} \times 1,000,000 = 210 \text{ micrograms}\]

• 1 gram \(= 1,000,000\) micrograms.
• Multiply by 1,000,000 for conversion to micrograms.
• Vital for making precise measurements in minute quantities.

By mastering these conversions, you can handle small and intricate measurements easily, a common necessity in chemical and pharmaceutical applications.