Question

Calicheamicin gamma-1, \(\mathrm{C}_{55} \mathrm{H}_{74} \mathrm{IN}_{3} \mathrm{O}_{21} \mathrm{S}_{4},\) is one of the most potent antibiotics known: one molecule kills one bacterial cell. Describe how you would (carefully!) prepare 25.00 \(\mathrm{mL} .\) of an aqueous calicheamicin gamma-1 solution that could kill \(1.0 \times 10^{8}\) bacteria, starting from a \(5.00 \times 10^{-9} M\) stock solution of the antibiotic.

Short answer

To prepare 25.00 mL of an aqueous calicheamicin gamma-1 solution potent enough to kill \(1.0 \times 10^8\) bacteria, first, calculate the number of moles needed: \(n = \frac{1.0 \times 10^8}{6.022 \times 10^{23}}\). Then, find the required concentration: \(C_2 = \frac{n}{0.02500}\). Next, use the dilution formula to find the necessary volume of the stock solution: \(V_1 = \frac{C_2V_2}{C_1}\). Finally, measure the needed volume of the stock solution, add it to a volumetric flask, and fill it up to the 25.00 mL mark with distilled water. Mix thoroughly to prepare the final solution.

Step-by-step solution

Find the concentration required for the 25.00 mL solution

Since one molecule of calicheamicin gamma-1 can kill one bacterial cell, we need to find the concentration required to kill 1.0 x 10^8 bacteria in the final 25.00 mL solution. First, we need to find the number of moles of calicheamicin gamma-1 needed in the 25.00 mL solution. Number of moles = number of bacteria / Avogadro's number \(n = \frac{1.0 \times 10^{8}}{6.022 \times 10^{23}}\) Now, we can calculate the concentration using the number of moles found above and the given solution volume. Concentration = \( \frac{n}{V} \) Where n is the number of moles and V = 25.00 mL = 0.02500 L

Calculate the volume of the stock solution needed to prepare the 25.00 mL solution

Now that we have the required concentration for the 25.00 mL solution, we can use the dilution formula to find the volume of the stock solution needed. The dilution formula is: C1V1 = C2V2 Where C1 = concentration of the stock solution (5.00 x 10^-9 M) V1 = volume of the stock solution needed C2 = concentration of the 25.00 mL solution calculated in step 1 V2 = final volume of the 25.00 mL solution (0.02500 L) Solving for V1: V1 = \( \frac{C2V2}{C1} \) After calculating V1, we will have the required volume of the stock solution needed to prepare the 25.00 mL solution.

Prepare the 25.00 mL solution

Once we have the volume of the stock solution, we can proceed to prepare the final solution: 1. Measure the needed volume of the stock solution calculated in step 2. 2. Add this measured volume to a clean, dry volumetric flask. 3. Fill the flask with distilled water up to the 25.00 mL mark. 4. Gently mix the solution thoroughly to ensure complete dilution. The prepared 25.00 mL aqueous calicheamicin gamma-1 solution should now be ready and potent enough to kill 1.0 x 10^8 bacteria.

Key concepts

Molarity and Dilution

Molarity, also denoted as M, is a measure of concentration in chemistry, defined as the number of moles of solute divided by the total volume of the solution in liters. It's a fundamental concept when preparing solutions, as it allows chemists to precisely control how much of a substance is present in a certain volume of liquid.

When it comes to dilution, it involves reducing the concentration of a solute in solution by adding more solvent. The key equation used here is the dilution formula: \( C_1V_1 = C_2V_2 \), where \( C_1 \) and \( C_2 \) are the molarities of the concentrated and diluted solutions, respectively, and \( V_1 \) and \( V_2 \) are the corresponding volumes. It's essential to remember that while the concentration changes, the amount of substance in moles remains the same before and after dilution.

For accurate dilution, it's imperative to measure the volume of the stock solution carefully using pipettes or syringes, and then to transfer it to a volumetric flask that's designed to hold a precise volume of liquid when filled to the mark. This ensures the molarity of the final solution is exactly as intended, a process often used in laboratory settings to create solutions for various experiments or treatments.

Avogadro's Number

Avogadro's number, \( 6.022 \times 10^{23} \), is a constant named after Amedeo Avogadro, which represents the number of atoms, ions, or molecules in one mole of a substance. In the realm of chemistry, it bridges the gap between the microscopic scale of atoms and molecules and the macroscopic scale that we can measure in the lab.

For the calicheamicin gamma-1 problem, knowing Avogadro's number allows us to understand the incredible potency of the antibiotic at the molecular level. Since each molecule of calicheamicin gamma-1 can kill a single bacterial cell, we use Avogadro's number to convert between the number of bacterial cells and the amount of calicheamicin needed in moles. As a fundamental constant in stoichiometry, it facilitates the conversion from a count of particles to a measurable quantity in the laboratory, enabling precise preparation of solutions with specific molecular requirements.

Stoichiometry

Stoichiometry involves the quantitative relationships between reactants and products in a chemical reaction. It's based on the law of conservation of mass, which states that matter cannot be created or destroyed in a closed system. By using balanced chemical equations, stoichiometry allows us to predict the amounts of substances consumed and produced in a reaction.

In the context of our antibiotic solution problem, stoichiometry isn't about a reaction but is instead applied to the preparation of a solution. It allows us to calculate the exact amount of calicheamicin gamma-1 required to kill a given number of bacteria. Once the necessary amount of calicheamicin is known in moles, stoichiometry principles are used to determine the corresponding mass or volume needed for the solution, ensuring that the potency of the antibiotic solution is exactly as desired.

Understanding stoichiometry is crucial for students in predicting the outcomes of chemical reactions and in tasks such as creating solutions with precise concentrations, like the antibiotic solution mentioned in the exercise. It is a core skill for chemists, pharmacists, and other professionals who work with chemical substances regularly.