Problem 75 - Chemistry The Central Science In Si Units

Question

A medical lab is testing a new anticancer drug on cancer cells. The drug stock solution concentration is \(1.5 \times 10^{-9} \mathrm{M},\) and \(1.00 \mathrm{~mL}\) of this solution will be delivered to a dish containing \(2.0 \times 10^{5}\) cancer cells in \(5.00 \mathrm{~mL}\) of aqueous fluid. What is the ratio of drug molecules to the number of cancer cells in the dish?

Step-by-Step Solution

Answer

The ratio of drug molecules to cancer cells is approximately \(4.52 \times 10^{6}\).

Step 1: Calculate Moles of Drug in 1 mL

We start by calculating the number of moles of the drug in the 1 mL of the stock solution.The concentration of the stock solution is given as \(1.5 \times 10^{-9} \mathrm{M}\), which means there are \(1.5 \times 10^{-9} \text{ moles per liter.}\)Convert 1 mL to liters:\[1 \mathrm{~mL} = 0.001 \mathrm{~L}\]Then, calculate the moles of drug:\[\text{Moles of drug} = 1.5 \times 10^{-9} \mathrm{M} \times 0.001 \mathrm{~L} = 1.5 \times 10^{-12} \text{ moles}\]

Step 2: Calculate the Number of Drug Molecules

To find the number of drug molecules, we use Avogadro's number, \(6.022 \times 10^{23} \text{ molecules/mol}.\)Multiply the moles of the drug by Avogadro's number:\[\text{Number of drug molecules} = 1.5 \times 10^{-12} \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mole}\]\[= 9.033 \times 10^{11} \text{ molecules}\]

Step 3: Determine Ratio of Drug Molecules to Cancer Cells

Given \(2.0 \times 10^{5}\) cancer cells in the dish, the next step is to determine the ratio of the drug molecules to cancer cells.\[\text{Ratio} = \frac{9.033 \times 10^{11} \text{ drug molecules}}{2.0 \times 10^{5} \text{ cancer cells}}\]\[= 4.5165 \times 10^{6}\]

Step 4: Simplify the Ratio

So, approximately, for every cancer cell in the dish, there are:\[\approx 4.52 \times 10^{6} \text{ drug molecules}\]This means each cancer cell on average encounters 4.52 million drug molecules.

Key Concepts

Avogadro's number

Avogadro's number is a fundamental constant used to describe the quantity of particles, usually atoms or molecules, present in a mole of substance. It is denoted as approximately \(6.022 \times 10^{23}\) particles/mole.
This number bridges the gap between the atomic scale and macroscopic quantities involved in chemical reactions, making it a critical element in stoichiometry and conversion between moles and numbers of particles. If you have a mole of anything—carrots, stars, or even atoms—you have \(6.022 \times 10^{23}\) of that item.

• Enables counting of molecules in a given sample by relating macroscopic measurements to microscopic scales.
• Crucial for converting from moles (a measure of quantity) to number of atoms/molecules, enabling precise chemical equations and reactions.

In our context, Avogadro's number helps convert moles of the anticancer drug into the number of molecules, allowing us to understand how many drug molecules are present in the 1 mL of solution.

Concentration Calculations

Concentration calculations allow us to quantify the amount of a substance present within a certain volume of a solution. Concentration is often expressed in molarity (M), which is moles per liter (mol/L).
In the case of the anticancer drug test, the concentration of the drug is given as \(1.5 \times 10^{-9} \text{ M}\), indicating a very dilute solution. Here's a simple breakdown of the calculations we perform to leverage this information:

• Convert the volume from milliliters to liters, as molarity is based on liters of solution.
• Use the known concentration and the converted volume to find moles of drug: \(\text{Moles} = \text{Concentration} \times \text{Volume}\).

With the known concentration and calculated volume, you can easily find the number of moles in a sample, which is then multiplied by Avogadro's number to determine how many molecules are present.

Drug to Cell Ratio

The drug to cell ratio gives insight into how many drug molecules are available to interact with each cell. This is crucial in pharmacology for determining efficacy and potential dosage requirements.
To find this ratio, you follow these steps:

• Determine the number of drug molecules in the administered volume using both concentration and Avogadro's number.
• Count the number of cancer cells in the medium.
• Divide the total number of drug molecules by the number of cancer cells to find the ratio.

In the given problem, we find \(4.52 \times 10^{6}\) drug molecules per cell.
This means that each cancer cell is exposed to approximately 4.52 million drug molecules, which gives an idea of the drug concentration experienced at the cellular level and helps predict the potential impact of the therapy.