Problem 121 - Chemistry and Chemical Reactivity

Question

A drop of water has a volume of about 0.050 mL. How many molecules of water are in a drop of water? (Assume water has a density of \(1.00 \mathrm{g} / \mathrm{cm}^{3} .\) )

Step-by-Step Solution

Answer

1.67 × 10^{20} molecules.

Step 1: Convert Volume from mL to cm³

Since the density of water is given in \( \mathrm{g/cm}^3 \,\) and the volume is given in mL, we first need to convert the volume to cm³. We know that 1 mL = 1 cm³, so: \[ 0.050 \, \text{mL} = 0.050 \, \text{cm}^3 \]

Step 2: Calculate Mass of the Water Drop

Using the density formula \,\( \text{Density} = \frac{\text{Mass}}{\text{Volume}}\), calculate the mass. Since the density of water is \,\( 1.00 \, \text{g/cm}^3 \,\), the mass is: \[ \text{Mass} = \text{Density} \times \text{Volume} = 1.00 \, \text{g/cm}^3 \times 0.050 \, \text{cm}^3 = 0.050 \, \text{g} \]

Step 3: Convert Mass to Moles

Water (H₂O) has a molar mass of approximately 18.015 \, \text{g/mol}. Convert the mass of the water drop to moles: \[ \text{Moles of H}_2\text{O} = \frac{0.050 \, \text{g}}{18.015 \, \text{g/mol}} \approx 0.002776 \, \text{mol} \]

Step 4: Calculate Number of Water Molecules

Use Avogadro's number \,\( 6.022 \times 10^{23} \, \text{molecules/mol}\), to convert moles to molecules: \[ \text{Number of Molecules} = 0.002776 \, \text{mol} \times 6.022 \times 10^{23} \, \text{molecules/mol} \approx 1.67 \times 10^{20} \, \text{molecules} \]

Key Concepts

Density

Density is a measure of how much mass is contained in a given volume. It is calculated using the formula: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]This concept is useful when you want to find out how heavy something is for its size.

• If a substance is denser, it means it has more mass in a particular volume.
• Water, for instance, has a density of 1.00 g/cm extsuperscript{3}, meaning 1 cm extsuperscript{3} of water weighs exactly 1 gram.
• This property helps in understanding buoyancy and floatation as it determines whether an object will float or sink in a fluid.

When solving problems involving density, it is important to ensure that mass and volume are in compatible units, like grams and cubic centimeters for metric measurements.

Molar Mass

The molar mass of a substance is the mass of one mole of that substance. It is an essential concept when converting between the mass of a substance and the amount in moles.

• For water (H\textsubscript{2}O), the molar mass is about 18.015 g/mol.
• This value is calculated by adding the atomic masses of two hydrogen atoms and one oxygen atom: \[(2 \times 1.008) + 16.00 = 18.016 \, \text{g/mol}\]
• Molar mass allows us to interconvert between grams and moles, which is necessary when working in chemical reactions or physiological calculations.

It is typically expressed in units of grams per mole (g/mol) and varies depending on the makeup of the compound under consideration.

Avogadro's Number

Avogadro's number is a constant used to describe the number of particles, like atoms or molecules, in a mole of a substance.It has a value of \( 6.022 \times 10^{23} \text{ particles/mol} \). This number enables scientists and students to translate between the macroscopic scale, what we can see and measure, and the atomic scale, which is too small for direct observation.

• Named after Amedeo Avogadro, it is one of the fundamental constants in chemistry.
• Using this number, you can easily determine how many molecules are present in a given amount of a substance when its mass and molar mass are known.
• For instance, in our exercise, it helps calculate how many water molecules are in a drop by multiplying the moles by Avogadro's number.

Understanding Avogadro's number is crucial for comprehending not only chemical reactions but also the scale and magnitude of atomic interactions.

Conversion of Units

Conversion of units involves changing measurements from one set of units to another, ensuring consistency and accuracy in calculations. In scientific problems, this often means converting between metric units like milliliters (mL) and cubic centimeters (cm extsuperscript{3}), which is straightforward since 1 mL = 1 cm extsuperscript{3}.

• When working with volumes and densities, it is important to ensure units are consistent to avoid calculation errors.
• It's common to convert units to match those used in density calculations, which are often in g/cm extsuperscript{3}.
• This consistency is crucial because it allows for clear communication and understanding of the magnitude of a measurement.

To ensure accuracy, always double-check units and convert when necessary. This practice is fundamental in scientific communication and precision.