Question

Estimate the distance (in \(\mathrm{nm}\) ) between molecules of water vapor at \(100^{\circ} \mathrm{C}\) and \(1.0 \mathrm{~atm} .\) Assume ideal behavior. Repeat the calculation for liquid water at \(100^{\circ} \mathrm{C},\) given that the density of water is \(0.96 \mathrm{~g} / \mathrm{cm}^{3}\) at that temperature. Comment on your results. (Assume each water molecule to be a sphere with a diameter of \(0.3 \mathrm{nm} .\) ) (Hint: First calculate the number density of water molecules. Next, convert the number density to linear density, that is, the number of molecules in one direction.)

Short answer

Distance between molecules: 3.7 nm (vapor), 0.3 nm (liquid). Liquid water is denser than vapor.

Step-by-step solution

Calculate Molar Volume of Water Vapor

For water vapor at 100°C and 1 atm, assume it behaves as an ideal gas. Use the ideal gas law: \[ PV = nRT \]For 1 mole, \( n = 1 \), and \( R = 0.0821 \text{ L atm/mol K} \). Convert the temperature: \( T = 273 + 100 = 373 \text{ K} \).\[ V = \frac{nRT}{P} = \frac{1 \times 0.0821 \times 373}{1} = 30.6 \text{ L} \]Convert to cubic centimeters: \( V = 30.6 \times 1000 = 30600 \text{ cm}^3 \). This is the volume occupied by 1 mole of water vapor.

Calculate Number Density of Water Vapor

Avogadro's number tells us there are \(6.022 \times 10^{23}\) molecules in a mole.Number density (molecules per cm³) is calculated by:\[ \frac{6.022 \times 10^{23}}{30600} \approx 1.97 \times 10^{19} \text{ molecules/cm}^3 \]

Calculate Linear Density for Water Vapor

Linear density is calculated by taking the cube root of the number density:\[ d = \left(1.97 \times 10^{19}\right)^{1/3} \approx 2.7 \times 10^6 \text{ molecules/cm} \]To find the distance between molecules, take the reciprocal and convert to nanometers:\[ \frac{1}{2.7 \times 10^6} \, \text{cm} \times 10^7 \, \text{nm/cm} \approx 3.7 \, \text{nm}\]

Calculate Number Density of Liquid Water

Given density of water is \(0.96 \, \text{g/cm}^3\).Molar mass of water is approximately \(18 \text{ g/mol}\).Number density in liquid form: \[ \frac{0.96}{18} \times 6.022 \times 10^{23} \approx 3.21 \times 10^{22} \text{ molecules/cm}^3 \]

Calculate Linear Density for Liquid Water

Find the linear density by taking the cube root of the number density:\[ d = \left(3.21 \times 10^{22}\right)^{1/3} \approx 3.2 \times 10^7 \, \text{molecules/cm} \]To find the distance between molecules, take the reciprocal and convert to nanometers:\[ \frac{1}{3.2 \times 10^7} \, \text{cm} \times 10^7 \, \text{nm/cm} \approx 0.3 \, \text{nm} \]

Compare Distances and Conclusion

The distance between water molecules in vapor form is approximately \(3.7 \, \text{nm}\), while in liquid form, it is about the diameter of a water molecule, \(0.3 \, \text{nm}\). This shows that molecules are much more closely packed in liquid water compared to water vapor.

Key concepts

Number Density

Number density is a crucial concept in understanding how molecules are distributed in a given volume. It refers to the number of molecules present per cubic centimeter. To calculate this, we utilize Avogadro's number, which tells us there are approximately \(6.022 \times 10^{23}\) molecules in a mole of a substance.

• For water vapor at \(100^{\circ} \text{C}\) and \(1.0\) atm, we first calculate the molar volume using the ideal gas law and find that the volume for one mole of water vapor is \(30600 \text{ cm}^3\).
• This allows us to calculate the number density as \(1.97 \times 10^{19} \text{ molecules/cm}^3\).

In the liquid state, given the density of water is \(0.96 \text{ g/cm}^3\), we find that the number density is substantially higher at \(3.21 \times 10^{22} \text{ molecules/cm}^3\). This demonstrates how molecules are more densely packed in the liquid state compared to the gaseous state.

Molecular Distance

Molecular distance refers to how far apart molecules are in a particular state of matter. In the gaseous phase, molecules are much further apart due to lower density.

• For water vapor, by taking the cube root of the number density, we determine the linear density, which gives us the distance between molecules as approximately \(3.7 \text{ nm}\).
• In the liquid phase, the distance is essentially just the size of the molecules themselves, around \(0.3 \text{ nm}\).

This vast difference in molecular distances between the gaseous and liquid phases emphasizes the close packing of molecules in the liquid phase, causing it to be more compact.

Density of Water

Density is the measure of how much mass is contained in a given volume. For water, the density plays a central role in determining its physical properties. At \(100^{\circ} \text{C}\), the density of liquid water is \(0.96 \text{ g/cm}^3\).

• This density information helps us derive the number density of water molecules, which is much higher compared to its gaseous state.
• Understanding density allows us to compare how packed molecules are in different phases, aiding in calculations like those involved in the Ideal Gas Law.

The density difference between liquid water and water vapor helps explain phenomena like buoyancy and phase changes.

Phase Changes

Phase changes involve the transformation of a substance from one state of matter to another, such as from liquid to gas.

• When water transforms from liquid to vapor, molecules gain energy, move further apart, and occupy more volume.
• The distance calculated between molecules in different phases (\(3.7 \text{ nm}\) in vapor vs. \(0.3 \text{ nm}\) in liquid) highlights how molecular arrangements alter during phase changes.

Understanding phase changes is essential when applying concepts like the Ideal Gas Law, as they provide context for how temperature and pressure influence molecular behavior. Phase changes underscore the dynamic nature of matter and its responsiveness to energy shifts.