Question

Consider three 10-g samples of water: one as ice, one as liquid, and one as vapor. How do the volumes of these three samples compare with one another? How is this difference in volume related to the physical state involved?

Short answer

The volumes of the three 10-g samples of water in different physical states compare as follows: ice has a volume of \( 10.87 cm^3 \) (slightly larger than liquid water), liquid water has a volume of \( 10 cm^3 \), and vapor has a volume of \( 16,666.67 cm^3 \) (much larger than ice and liquid). These differences in volume are related to the varying densities and the arrangement and interaction of the water molecules in each physical state. In the solid state (ice), the water molecules are arranged in a crystalline structure, leading to a lower density and a slightly larger volume than liquid water. In the gaseous state (vapor), the water molecules are farther apart and more energetic, resulting in a much lower density and a significantly larger volume compared to ice and liquid water.

Step-by-step solution

Understand the physical states of water and their densities

The density of a substance is the mass per unit volume, given by ρ = m/V, where ρ is the density, m is the mass, and V is the volume. Water has different densities in different physical states due to the arrangement and interaction between the water molecules in each state. For water: - Ice (solid state): Water molecules are arranged in a crystalline structure, with a lower density compared to liquid water. The density of ice is approximately \( 0.92 \frac{g}{cm^3} \). - Liquid (liquid state): Water molecules are closer together and free to move around, resulting in a higher density compared to ice. The density of liquid water is approximately \( 1.00 \frac{g}{cm^3} \). - Vapor (gaseous state): Water molecules are much farther apart and more energetic in the gaseous state, resulting in a much lower density compared to ice and liquid water. The density of water vapor at 100°C and 1 atm is approximately \( 0.60 \frac{g}{L} \) or \( 0.0006 \frac{g}{cm^3} \).

Calculate the volume of each sample using their densities

Since we know the mass (10g) and the densities of each physical state, we can calculate the volume using the formula V = m/ρ. For each state, calculate the volume: 1. Ice: \( V_{ice} = \frac{m}{\rho_{ice}} = \frac{10g}{0.92\frac{g}{cm^3}} \approx 10.87 cm^3 \) 2. Liquid: \( V_{liquid} = \frac{m}{\rho_{liquid}} = \frac{10g}{1.00\frac{g}{cm^3}} = 10 cm^3 \) 3. Vapor: \( V_{vapor} = \frac{m}{\rho_{vapor}} = \frac{10g}{0.0006\frac{g}{cm^3}} \approx 16,666.67 cm^3 \)

Compare the volumes and relate to physical states

Now that we have calculated the volumes of each 10-g sample, we can compare the values: - Ice: \( 10.87 cm^3 \) (slightly larger than liquid water) - Liquid: \( 10 cm^3 \) - Vapor: \( 16,666.67 cm^3 \) (much larger than ice and liquid) The differences in volume can be related to the physical states as follows: 1. Between ice and liquid water, the larger volume of ice is due to the spacing and arrangement of water molecules in the solid state, leading to a lower density. 2. Between liquid water and vapor, the difference is much more significant, with water vapor having a much larger volume. The increased distance between water molecules in the gaseous state and the increased energy of the molecules lead to the much lower density of the vapor state.

Key concepts

Density

Density plays a critical role in understanding how substances behave in different states of matter. Density is defined as the mass of a substance per unit volume, expressed with the formula \( \rho = \frac{m}{V} \). This relationship means substances with higher densities will have more mass in a smaller volume compared to those with lower densities.

In the case of water, density varies with its physical state:

• Ice: In the solid state, water molecules form a crystalline structure. The unique hexagonal lattice makes ice less dense than liquid water, allowing it to float.
• Liquid water: Here, molecules are more densely packed and freely moving, resulting in higher density than ice.
• Water vapor: In gaseous form, molecules spread out significantly, creating a very low density.

Understanding density is key to explaining the differing characteristics between ice, water, and steam.

Volume Calculations

Volume calculations help to quantify how much space a substance occupies. By using the relationship \( V = \frac{m}{\rho} \), where \( V \) is volume, \( m \) is mass, and \( \rho \) is density, you can calculate the volume of a substance given its mass and density.

In our exercise, 10g samples of water are considered in three states:

• Ice: With a density of about 0.92 \( \frac{g}{cm^3} \), the ice sample occupies approximately 10.87 \( cm^3 \).
• Liquid: At a density of 1.00 \( \frac{g}{cm^3} \), the water sample occupies 10 \( cm^3 \).
• Vapor: With a very low density of 0.0006 \( \frac{g}{cm^3} \), the vapor occupies an enormous volume of about 16,666.67 \( cm^3 \).

These calculations show the drastic volume change as water transitions from solid to liquid to gas.

Water Molecules

Water molecules are fascinating units that showcase unique properties in different states. Composed of two hydrogen atoms bonded to one oxygen atom, they form a polar molecule with a bent shape. This polarity is crucial for the behavior of water in its various states.

• In ice: Water molecules align in a structured lattice due to hydrogen bonding, which keeps them at a fixed distance, making ice less dense.
• In liquid water: The molecules remain close but have enough energy to move freely. This allows liquid water to adapt to the shape of its container.
• In vapor: Molecules have gained sufficient energy to break free from sticking together, scattering far apart.

Each state reflects the balance between the kinetic energy of the molecules and intermolecular forces.

Gas Laws

Gas laws describe the behaviors of gases and are crucial for understanding changes in water vapor. These laws include Boyle’s Law, Charles’s Law, and the Ideal Gas Law, serving to explain various relationships in gaseous substances.

• Boyle’s Law: This states that pressure and volume of a gas are inversely proportional at constant temperature; \( P_1 V_1 = P_2 V_2 \).
• Charles’s Law: States volume and temperature have a direct proportionality at constant pressure; \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \).
• Ideal Gas Law: Integrates Boyle’s, Charles’s, and Avogadro’s laws with formula \( PV = nRT \), linking pressure, volume, and temperature with gas quantity.

In water vapor, these gas laws help us understand why vapor occupies significantly more space. The molecules are full of energy and minimal interactions, following these gas laws closely.