Question

At \(100^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\), if the density of liquid water is \(1.0 \mathrm{~g} \mathrm{~cm}^{-3}\) and that of water vapour is \(0.0006 \mathrm{~g} \mathrm{~cm}^{-3}\), then the volume occupied by water molecules in 1 litre of steam at that temperature (a) \(6 \mathrm{~cm}^{3}\) (b) \(60 \mathrm{~cm}^{3}\) (c) \(0.6 \mathrm{~cm}^{3}\) (d) \(0.06 \mathrm{~cm}^{3}\)

Short answer

The volume occupied by water molecules is 0.6 cm³.

Step-by-step solution

Determine Mass of Water in 1 Litre of Steam

Since densities are given, we must first calculate the mass of the water in 1 litre of steam. Given that the density of water vapour is \(0.0006 \text{ g cm}^{-3}\), the mass can be calculated by multiplying the density by the volume \((1000 \text{ cm}^3)\). Thus, the mass of the water vapour is \(0.0006 \times 1000 = 0.6 \text{ g}\).

Calculate Volume Occupied by Water Molecules

Now that we have the mass of the water vapour, we need to calculate the volume that this mass of water would occupy in liquid form. The density of the liquid water is \(1.0 \text{ g cm}^{-3}\), so using the formula for density \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\), we solve for volume: \(\text{Volume} = \frac{\text{Mass}}{\text{Density}}\). Substituting the values gives: \(\text{Volume} = \frac{0.6}{1.0} = 0.6 \text{ cm}^3\).

Conclusion and Answer Verification

The calculations show that the water molecules in 1 litre of steam occupy a volume of \(0.6 \text{ cm}^3\) as liquid water. Therefore, the correct answer is option (c).

Key concepts

Volume and Mass Relationship

Understanding the relationship between volume and mass is crucial in density calculations. When you think of density, imagine it as a bridge connecting mass and volume. Density defines how much mass is packed into a specific volume of a substance. In this scenario, we're dealing with water vapour and liquid water, both of which have distinct densities.

Let's break it down: if a substance is denser, it means there is more mass packed into each unit of volume. Conversely, if it is less dense, like water vapour at high temperatures, there's less mass per unit volume. In simpler terms:

• Higher density = less volume occupied by the same mass.
• Lower density = more volume occupied by the same mass.

By understanding this relationship, you can convert between the mass of a substance and its occupied volume using the given density, which is exactly what we did when calculating the mass of water vapour in steam.

Density Formula

The density formula is a fundamental concept used in science to describe how mass and volume are related. It is expressed as: \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]The formula tells us that if you have the density of a substance and its volume, you can easily find the mass. The reverse is also true: with mass and density, you can determine volume.

In the given problem, we utilized this formula to find the mass of water vapour in one litre of steam. By multiplying the density of water vapour (a light, airy form of water) by the volume, we found the mass. Then, using the density formula in reverse, we discovered the volume that this mass of water would occupy in liquid form, where it's more compact.

Remember: knowing how to rearrange the density formula to solve for the desired variable—mass or volume—is a handy skill in tackling various problems involving substance properties.

Water Vapour Density

Water vapour density is a key element in understanding how water behaves in its gaseous state. At high temperatures, like at the boiling point of water, water transitions from a liquid to a gaseous state, forming steam. In the problem, the density of water vapour is given as \(0.0006 \text{ g cm}^{-3}\). This low density indicates that steam is much less dense than liquid water, highlighting the expanded nature of gas molecules.

This expansion happens because gas molecules move energetically and spread out to fill the available space. As a result, the mass of water vapour that fills a container, like 1 litre, weighs significantly less and occupies more space compared to its liquid counterpart.

• Water vapour: low density, occupies more space.
• Liquid water: high density, occupies less space.

Understanding this concept is essential for grasping why the volume of liquid water derived from steam is so small—the molecules are much more compact in the liquid state.