Problem 11 - A Complete Resource Book in Chemistry for JEE Main

Question

The correct relationship between molecular mass and vapour density is (a) V.D. \(=2 \mathrm{M}\) (b) V.D. \(=\frac{\mathrm{M}}{2}\) (c) \(\mathrm{M}=(\mathrm{V} \cdot \mathrm{D} .)^{\frac{1}{2}}\) (d) \(\mathrm{VD}=\mathrm{M}^{2}\)

Step-by-Step Solution

Answer

The correct relationship is given by option (b): V.D. \( = \frac{M}{2} \).

Step 1: Understanding Vapour Density

Vapour density (V.D.) is defined as the mass of a certain volume of a gas compared to the mass of an equal volume of hydrogen gas at the same temperature and pressure. Mathematically, vapour density is half of the molecular mass when compared to hydrogen.

Step 2: Relationship Formula

The formula relating molecular mass (M) and vapour density (V.D.) is given by:\[ V.D. = \frac{M}{2} \]

Step 3: Identify Correct Option

Look for the option that has the form \( V.D. = \frac{M}{2} \). From the given options, option (b) matches this formula.

Key Concepts

Molecular Mass

Molecular mass is an essential concept in chemistry. It refers to the mass of a single molecule of a chemical compound, and it's often expressed in atomic mass units (amu).
Molecular mass is calculated by summing the atomic masses of all atoms in a molecule. For example, water (H₂O) consists of two hydrogen atoms and one oxygen atom. Each hydrogen atom has an atomic mass of about 1 amu, and oxygen has an atomic mass of 16 amu. Therefore, the molecular mass of water is approximately 18 amu.

Understanding molecular mass is crucial because it helps in determining how a molecule interacts chemically and physically with other molecules.

• It allows chemists to calculate the exact amount of reactants and products in a chemical reaction (stoichiometry).
• It helps predict physical properties like boiling and melting points and vapor density of gases.

Gas Laws

Gas laws describe the behavior of gases and their relationships between volume, temperature, and pressure. These laws are vital when studying gases' physical characteristics and behaviors.

Some well-known gas laws include:

• Boyle's Law: This law states that the pressure of a gas is inversely proportional to its volume when temperature is kept constant - expressed as \( P \times V = ext{constant} \).
• Charles's Law: It states that the volume of a gas is directly proportional to its absolute temperature when pressure is constant - \( V \propto T \).
• Ideal Gas Law: It combines several gas laws into one and is represented as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is amount in moles, \( R \) is the ideal gas constant, and \( T \) is temperature.

Understanding these laws helps explain how gases expand, contract, and interact with their environments.

Hydrogen Comparison

Hydrogen comparison plays a crucial role in determining the vapour density of a gas. Vapour density is a measure of a gas's density compared to hydrogen, which is considered the lightest element.

The importance of hydrogen in this context lies in a standard frame of reference for comparing densities. So, vapor density can be determined using the formula:

• Vapour Density (V.D.) = \( \frac{\text{Mass of a certain volume of gas}}{\text{Mass of an equal volume of hydrogen}} \)
• Hydrogen gas, being simple and light, is used to provide an easily comparable basis for gases of different molecular masses.

By understanding hydrogen comparison, one can easily determine how much heavier or lighter a gas is relative to hydrogen, which can be particularly useful in predicting how gases will behave under different conditions.

Formula Derivation

The derivation of formulae is a fundamental process in understanding relationships between different quantities in chemistry. Let's look at how the formula relating molecular mass (M) and vapour density (V.D.) comes about.

Given:

• Vapour density (V.D) is defined as half of the molecular mass when compared to hydrogen.
• This leads us to the relationship: \( V.D. = \frac{M}{2} \)

To understand why this formula holds:

• Consider a gas's mass as a certain volume and compare it against the mass of the same volume of hydrogen gas.
• Since hydrogen, the reference point, has a defined light mass, other gases' mass are heavier by a factor of their molecular mass.
• So, the molecular mass of the gas divided by two gives the vapour density compared to hydrogen.

This formula is instrumental for chemists in describing the relative properties and characteristics of gases.