Question

Number of \(\text{H}_2\text{O}\) (water) molecules in a drop of water weighing 0.5 g is :

• A. \(1.5 \times 10^{23}\)
• B. \(1.672 \times 10^{22}\)
• C. \(1.5 \times 10^{21}\)
• D. \(6.022 \times 10^{22}\)

Hint:

1. Find molar mass of \(\text{H}_2\text{O}\): \(2(1) + 16 = 18\) g/mol. 2. Find moles: Moles = Mass / Molar mass = \(0.5 \text{ g} / 18 \text{ g/mol} = \frac{0.5}{18} \text{ mol}\). 3. Find number of molecules: Molecules = Moles \(\times\) Avogadro's number (\(N_A\)). Molecules = \(\frac{0.5}{18} \times 6.022 \times 10^{23}\). Molecules = \(\frac{3.011 \times 10^{23}}{18}\). Molecules \(\approx 0.16727... \times 10^{23} = 1.6727... \times 10^{22}\). Approximately \(1.672 \times 10^{22}\) molecules.

Answer 1

The Correct Option is B

Concept: To find the number of molecules in a given mass of a substance, we need to: 1. Calculate the molar mass of the substance (\(\text{H}_2\text{O}\) in this case). 2. Calculate the number of moles of the substance using the formula: Moles = Mass / Molar mass. 3. Calculate the number of molecules using Avogadro's number (\(N_A \approx 6.022 \times 10^{23}\) molecules/mol): Number of molecules = Moles \(\times N_A\). Step 1: Calculate the molar mass of water (\(\text{H}_2\text{O}\)) Atomic mass of Hydrogen (H) \(\approx 1 \text{ g/mol}\) Atomic mass of Oxygen (O) \(\approx 16 \text{ g/mol}\) Molar mass of \(\text{H}_2\text{O}\) = \(2 \times (\text{mass of H}) + 1 \times (\text{mass of O})\) Molar mass of \(\text{H}_2\text{O}\) = \(2 \times 1 + 16 = 2 + 16 = 18 \text{ g/mol}\). Step 2: Calculate the number of moles of water Given mass of water = 0.5 g. Number of moles (\(n\)) = \(\frac{\text{Given mass}}{\text{Molar mass}}\) \[ n = \frac{0.5 \text{ g}}{18 \text{ g/mol}} \] \[ n = \frac{1/2}{18} = \frac{1}{36} \text{ mol} \] Step 3: Calculate the number of water molecules Number of molecules = Number of moles \(\times\) Avogadro's number (\(N_A\)) Number of molecules = \(\frac{1}{36} \times 6.022 \times 10^{23}\) \[ \text{Number of molecules} = \frac{6.022 \times 10^{23}}{36} \] Let's perform the division: \(\frac{6.022}{36}\). \(\frac{6.022}{36} \approx 0.16727... \) So, Number of molecules \(\approx 0.16727 \times 10^{23}\). Step 4: Express the result in standard scientific notation To express \(0.16727 \times 10^{23}\) in standard scientific notation (one non-zero digit before the decimal point), we move the decimal point one place to the right, and decrease the exponent of 10 by 1: \(0.16727 \times 10^{23} = 1.6727 \times 10^{22}\). Rounding to three decimal places (to match option 2), we get \(1.672 \times 10^{22}\) or \(1.673 \times 10^{22}\) depending on the exact rounding of the division. The option uses \(1.672 \times 10^{22}\). This matches option (2). The handwritten calculation \(\frac{0.5}{18} \times 6.022 \times 10^{23}\) on the image is correct.