Problem 75 - Chemistry: The Central Science in SI Units, Global Edition

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 the number of cancer cells in the dish is approximately \(4.52 \times 10^{6}\) to 1.

Step 1: Calculate moles of drug in \(1.00 \mathrm{~mL}\) of stock solution

First, we need to determine the amount (in moles) of the drug present in \(1.00 \mathrm{~mL}\) of stock solution. We can use the formula: moles of drug = concentration x volume Since the stock solution concentration is \(1.5 \times 10^{-9} \mathrm{M}\) and the volume is \(1.00 \mathrm{~mL}\): moles of drug = \( (1.5 \times 10^{-9} \mathrm{M}) \times (1.00 \times 10^{-3} \mathrm{L}) \) moles of drug = \(1.5 \times 10^{-12} \mathrm{mol}\)

Step 2: Calculate the number of drug molecules

Now that we have the moles of the drug, we can use Avogadro's number to find the number of drug molecules. Avogadro's number is \(6.022\times 10^{23}\) molecules/mol. number of drug molecules = moles of drug × Avogadro's number number of drug molecules = \( (1.5 \times 10^{-12} \mathrm{mol}) \times (6.022\times 10^{23} \mathrm{molecules/mol}) \) number of drug molecules = \(9.033\times 10^{11} \mathrm{molecules}\)

Step 3: Calculate the ratio of drug molecules to cancer cells

Now that we have the number of drug molecules, we need to determine the ratio of drug molecules to cancer cells in the dish. There are \(2.0 \times 10^{5}\) cancer cells in the dish. Ratio of drug molecules to cancer cells = \( \frac{9.033 \times 10^{11} \mathrm{molecules}}{2.0 \times 10^{5} \mathrm{cancer~cells}}\) Ratio of drug molecules to cancer cells = \(4.517 \times 10^{6}\) The ratio of drug molecules to the number of cancer cells in the dish is approximately \(4.52 \times 10^{6}\) to 1.