Problem 16 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

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 is (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

The volume occupied by water molecules in 1 litre of steam is 0.6 cm³.

Step 1: Understand the Problem

We need to find the volume occupied by water molecules in 1 litre of steam at 100°C, given the densities of liquid water and water vapour. We know the density is the mass per unit volume.

Step 2: Calculate the Mass of Steam

The density of water vapour is given as \(0.0006 \mathrm{~g/cm}^{3}\). Therefore, in 1 litre (which is 1000 cm³) of steam, the mass is \(0.0006 \times 1000 = 0.6 \mathrm{~g}\).

Step 3: Find Volume of Water at 100°C

The density of liquid water at 100°C is \(1.0 \mathrm{~g/cm}^{3}\). This means that 1 g of water occupies 1 cm³. For the mass of 0.6 g (calculated in Step 2), the volume occupied by the water molecules would be \(0.6 \times 1 = 0.6 \mathrm{~cm}^{3}\).

Key Concepts

Density Calculations

Density is a crucial concept in understanding the behavior of substances in different states. It is defined as the mass of a substance per unit volume, typically expressed in units of grams per cubic centimeter (g/cm³). In the exercise, two densities are given: the density of liquid water, which is 1.0 g/cm³, and the density of water vapor, which is significantly less at 0.0006 g/cm³. This huge difference illustrates how molecules in a gas state are much more spread out than in a liquid state.

To calculate density, you can use the formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]

For practical applications, if you know the density and the volume of a substance, you can find its mass by rearranging the formula:
\[ \text{Mass} = \text{Density} \times \text{Volume} \]

This calculation can help understand how much of a substance is contained within a certain volume, essential for solving the provided exercise.

Water Vapour

Water vapor is the gaseous state of water and occurs when liquid water evaporates. In this form, the water molecules have more energy and spread apart, which results in a much lower density compared to liquid water.

At 100°C and 1 atm pressure, water vapor has a density of 0.0006 g/cm³. This low density means that for the same mass, water vapor occupies a much larger volume than liquid water. This is a critical point in the exercise, as it highlights the transformation and behavior of water molecules when they transition from a liquid to a gaseous state.

Understanding water vapor is important in many real-world applications, such as in weather patterns, the water cycle, and industrial processes, where controlling and predicting vapor behavior is crucial.

Liquid Water

Liquid water, at 100°C, has a density of 1.0 g/cm³. This means that each gram of water takes up exactly one cubic centimeter of volume. This property of water is due to the strong hydrogen bonding between molecules that keeps them relatively close to each other compared to gases.

During heating, while the temperature increases to reach the boiling point, water remains in the liquid form, with molecules vibrating faster but still closely packed. This density value allows us to determine how the same mass of water will behave, both in liquid and vapor forms, which is essential for the calculations made in the exercise.

Knowing the density of liquid water helps predict how much space a certain mass of water occupies, which is a foundational aspect of chemistry and physics.

Steam Volume

When water is heated to 100°C, it transitions from liquid to steam, or water vapor. This transformation involves a significant increase in volume because the density of steam is much lower than that of liquid water.

In the given exercise, we are tasked with determining the volume occupied by the water molecules within 1 liter of steam. We first calculate the mass of the steam using its density, and then find out how much space this mass would occupy if it were in a liquid state.

The process involved:

• Calculate the mass of steam: Using the density formula at the vapor state.
• Convert the mass into the volume of liquid: Since the density of liquid water shows 1 g occupies 1 cm³, translating mass directly into volume is straightforward.

This change shows how molecules spread out immensely during vaporization, demonstrating principles of pressure and volume relationships in gases.