Problem 100 - A Complete Resource Book in Chemistry for JEE Main

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) \(0.06 \mathrm{~cm}^{3}\)

Step-by-Step Solution

Answer

(c) 0.6 cm³

Step 1: Understand the problem

We need to find the volume occupied by the actual water molecules in 1 litre (1000 cm³) of steam at 100°C. Given the density of water vapor, we can determine how much mass is present, and then calculate the volume these molecules would occupy if they were in a liquid state.

Step 2: Calculate the mass of water in 1 litre of steam

The density of water vapor is given as 0.0006 g/cm³. With a volume of 1000 cm³ (1 litre), the mass can be calculated using the formula: \[ \text{mass} = \text{density} \times \text{volume} \]Substitute the values:\[ \text{mass} = 0.0006 \text{ g/cm}^3 \times 1000 \text{ cm}^3 = 0.6 \text{ g} \]

Step 3: Calculate the volume in liquid state

In liquid water, the density is given as 1.0 g/cm³. We can find the equivalent volume in liquid state using the formula: \[ \text{volume} = \frac{\text{mass}}{\text{density}} \]Substitute the values:\[ \text{volume} = \frac{0.6 \text{ g}}{1.0 \text{ g/cm}^3} = 0.6 \text{ cm}^3 \]

Step 4: Conclusion from calculations

The volume occupied by the water molecules in their liquid state in 1 litre of steam at 100°C is 0.6 cm³. Therefore, the correct answer is (c) 0.6 cm³.

Key Concepts

Density

Density is a measure of how much mass is contained in a given volume. It is expressed in units such as grams per cubic centimeter (g/cm³). In our exercise, we address the densities of both liquid water and water vapor. Knowing the density of a substance can help us understand and predict its behavior under different physical conditions.
Density = Mass/Volume.
The formula tells us that if you know the mass and volume of an object, you can calculate its density. In the context of our exercise, the density of liquid water is 1.0 g/cm³, and the density of water vapor is significantly lower at 0.0006 g/cm³.
This stark difference highlights how much more spaced apart the water molecules are in the gaseous state compared to the liquid state. Understanding density is crucial in solving problems such as calculating the volume occupied by water molecules in a steam system.

Phase Change

A phase change refers to the transformation of a substance from one state of matter to another, such as from solid to liquid, liquid to gas, or vice versa. Water can exist in three main states: solid (ice), liquid (water), and gas (vapor). In our problem, we focus on the change from liquid water to water vapor.
When water boils at 100°C under 1 atm pressure, it undergoes a phase change from liquid to gas. This change involves energy, specifically the breaking of intermolecular bonds, allowing the molecules to move more freely and occupy a larger volume. This is why the density of water vapor is far less than that of liquid water.
Understanding phase changes is essential for evaluating how substances behave under various temperature and pressure conditions and is particularly useful in stoichiometry calculations and practical scenarios like this one.

Stoichiometry

Stoichiometry is a branch of chemistry involved with the quantitative relationships between reactants and products in chemical reactions. It includes calculations that relate the masses, moles, and volumes of reactants and products.
In this exercise, we aren’t dealing directly with a chemical reaction but with a conceptual stoichiometric calculation involving density and phase change. We calculate the mass of water vapor in 1 litre of steam and convert it to its equivalent volume in the liquid phase. This uses the principles of stoichiometry because it requires a precise, numerical understanding of how substances convert and coexist in different states.
The skills learned through stoichiometry are broadly applicable in chemistry and necessary for determining the outcomes of reactions, as well as for understanding processes, such as how much liquid water can form from a given mass of water vapor.