Question

Estimate the distance (in nanometers) 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 water molecule to be a sphere with a diameter of \(0.3 \mathrm{nm} .\) )

Short answer

In the water vapor state at \(100^{\circ} C\) and \(1.0 \mathrm{~atm}\), the estimated distance between water molecules is about \(0.344 \mathrm{~nm}\). In the liquid state at the same temperature, this distance is smaller, about \(0.287 \mathrm{~nm}\). This suggests that water molecules are more closely packed in the liquid state than in the vapor state.

Step-by-step solution

Use Ideal Gas Law for Water Vapor

The ideal gas law is given as \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant and T is the temperature. If we rearrange to find the volume per mole \(V_m = \frac{RT}{P}\), we can substitute values for \(R = 0.0821 \mathrm{~L~atm/mol~K}\), \(T = 100 + 273 = 373 \mathrm{~K}\), and \(P = 1.0 \mathrm{~atm}\) to find \(V_m = 30.6 \mathrm{L/mol}\). This is the volume occupied by one mole of water vapor molecules. Since 1 L = 1 dm\(^3\), we can convert this to \(\mathrm{cm}^3\), giving us \(V_m = 30.6 \times 10^3 \mathrm{cm}^3/mol\).

Estimate Distance between Water Vapor Molecules

We know that one mole of a substance contains Avogadro's number (\(6.022 \times 10^{23}\)) of molecules. So, \(V_m\) is also the volume that contains \(6.022 \times 10^{23}\) water vapor molecules. Hence the volume for one molecule \(V_1 = \frac{30.6 \times 10^3 \mathrm{cm}^3/mol}{6.022 \times 10^{23} \mathrm{mol^{-1}}}\). We assume molecules are packed in a cubic manner to estimate the distance between them. The volume for one molecule is the cube of the distance between molecules, therefore the distance \(d = \sqrt[3]{V_1}\). By calculating, we find \(d \approx 0.344 \mathrm{nm}\).

Estimate Volume of One Water Molecule in Liquid

Given density \(\rho = 0.96 \mathrm{g/cm}^3\), we can use the mass of one mole of water \(M = 18.015 \mathrm{g/mol}\) to find the volume per mole \(V_m = \frac{M}{\rho} = 18.8 \mathrm{cm}^3/mol\). Dividing by Avogadro's number again gives \(V_1 = \frac{18.8 \times 10^3 \mathrm{cm}^3/mol}{6.022 \times 10^{23} \mathrm{mol^{-1}}}\).

Estimate Distance between Water Molecules in Liquid

Similar to the vapor, we use \(d = \sqrt[3]{V_1}\) to find the distance between molecules in the liquid, which gives \(d \approx 0.287 \mathrm{nm}\). Notice that this is comparable to the diameter of the water molecule, suggesting closer packing in the liquid state compared to the vapor state.

Commentary

By comparing the results, we can observe that the distances between molecules are smaller in the liquid state than in the vapor state. This is consistent with our knowledge that molecules in a liquid are more closely packed together than in a gas.

Key concepts

The Ideal Gas Law and Molecular Distance

The Ideal Gas Law is foundational for understanding the behavior of gases and estimating the distance between molecules in the gaseous state. It's mathematically expressed as PV = nRT, where P stands for pressure, V for volume, n for moles, R for the ideal gas constant, and T for temperature.

In the provided exercise, we use this relationship to calculate the volume occupied by one mole of water in vapor form at a given temperature and pressure. This volume gives the spacing between the vapor molecules. By extracting the cube root of the volume per molecule, we calculate the average distance between them. In the case of water vapor at 100°C and 1 atm, the distance was estimated to be around 0.344 nm, emphasizing how gases occupy much more space than their liquid counterparts due to the kinetic energy that keeps the molecules apart.

Understanding the behavior of gases through the Ideal Gas Law not only aids in academic exercises but also in real-world applications in fields such as meteorology, engineering, and even respiratory medicine.

Avogadro's Number and Molecular Estimations

Avogadro's Number, denoted as \(6.022 \times 10^{23}\) molecules/mole, represents the number of units in one mole of any substance. It serves as a bridge between the macroscopic world we observe and the microscopic world of atoms and molecules.

When estimating molecular distances in both gas and liquid states, Avogadro's number is crucial, as it allows us to scale down from a mole, a quantity easy to measure in the laboratory, to the volume occupied by just a single molecule. With this powerful number, the exercise demonstrates how to convert the macroscopic volume of a mole of substance to the microscopic volume of a single molecule. Calculating the cube root of this volume reveals the average distance between molecules. In practical scenarios, this concept is vital in fields like chemistry and nanotechnology, where understanding the scale of atomic and molecular structures is fundamental.

Liquid and Gas States of Matter

The states of matter, particularly liquids and gases, show distinct molecular arrangements and distances. In the liquid state, molecules are tightly packed compared to gases, which is reflected in the calculation for liquid water where the estimated distance is approximately 0.287 nm, very close to the actual molecular diameter.

Understanding the transition between liquid and gas states, such as during evaporation or condensation, shines a light on how molecular distances change with states of matter. The densities and spacing of molecules dictate many properties of the material, from its flow characteristics to its ability to compress. These facets of material science clarify why water expands when it evaporates and why it has a definite volume that doesn't change shape like a gas. Recognizing these differences helps students appreciate the microscopic mechanisms behind everyday phenomena, including weather patterns, boiling water for cooking, or even the function of natural and artificial cooling systems.