If two elements can combine to form more than one compound, the masses of one element that combine with a fixe

NCERT Exemplar · Q29

Q. If two elements can combine to form more than one compound, the masses of one element that combine with a fixed mass of the other element, are in whole number ratio.

(a) Is this statement true?

(b) If yes, according to which law?

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Concept understanding — Unit Conversion

Unit Conversion: Why 1 Metre and 100 Centimetres Are the Same Thing

Imagine you're measuring the length of your desk. You pull out a ruler marked in centimetres and find it's 120 cm long. Your friend, using a metre stick, says it's 1.2 m. You're both right — you've just used different units to describe the same physical length.

That's the core idea: unit conversion is the process of changing how you express a quantity without changing the quantity itself.

The Intuition: Same Quantity, Different Labels

Think of a pizza. Whether you call it "one pizza" or "8 slices," the amount of pizza hasn't changed. You've just used a different unit (pizza vs. slice) to describe it.

Similarly, 1 metre and 100 centimetres are the same length — just like 1 pizza and 8 slices are the same amount. The number changes (1 becomes 100, or 1 becomes 8), but the actual thing being measured stays identical.

Note

This is the most important idea to hold onto: conversion changes the number, not the quantity. If you ever feel like the quantity has changed, you've made a mistake.

The Precise Statement

Unit conversion is the multiplication of a quantity by a conversion factor — a fraction equal to 1 — that cancels the old unit and introduces the new one.

A conversion factor looks like this:

\frac{\text{new unit}}{\text{old unit}} = 1

For length: $\frac{1 \text{ m}}{100 \text{ cm}} = 1$ and $\frac{100 \text{ cm}}{1 \text{ m}} = 1$ .

Why are these fractions equal to 1? Because 1 metre is 100 centimetres. The numerator and denominator describe the same physical length, so their ratio is exactly 1.

How to Convert: The Only Rule You Need

Multiply by a conversion factor that cancels the unit you have and leaves the unit you want.

Let's convert 120 cm to metres:

Start with what you have: $120 \text{ cm}$
Choose the conversion factor that has "cm" in the denominator (to cancel it) and "m" in the numerator: $\frac{1 \text{ m}}{100 \text{ cm}}$
Multiply:

120 \text{ cm} \times \frac{1 \text{ m}}{100 \text{ cm}} = \frac{120}{100} \text{ m} = 1.2 \text{ m}

The "cm" units cancel just like numbers do: $\frac{\text{cm}}{\text{cm}} = 1$ .

Tip

Always write the units explicitly. If the units don't cancel correctly, you've used the wrong conversion factor. This catches 90% of conversion mistakes.

The Reverse: Metres to Centimetres

Now convert 1.2 m to cm. This time, you want "cm" to remain and "m" to cancel. Use $\frac{100 \text{ cm}}{1 \text{ m}}$ :

1.2 \text{ m} \times \frac{100 \text{ cm}}{1 \text{ m}} = 1.2 \times 100 \text{ cm} = 120 \text{ cm}

Notice: when going from a larger unit (m) to a smaller unit (cm), the number gets larger (1.2 → 120). When going from smaller to larger, the number gets smaller (120 → 1.2). This is a useful sanity check.

Common Conversion Factors You'll Use

Quantity	Relationship	Conversion Factors
Length	1 m = 100 cm	$\frac{1 \text{ m}}{100 \text{ cm}}$ , $\frac{100 \text{ cm}}{1 \text{ m}}$
Mass	1 kg = 1000 g	$\frac{1 \text{ kg}}{1000 \text{ g}}$ , $\frac{1000 \text{ g}}{1 \text{ kg}}$
Time	1 h = 60 min	$\frac{1 \text{ h}}{60 \text{ min}}$ , $\frac{60 \text{ min}}{1 \text{ h}}$
Speed	1 km/h = $\frac{1000}{3600}$ m/s	$\frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}}$

Why this formula?

Dimensional Analysis: Why the Key Principles Hold

Dimensional Analysis is a powerful tool in physics and engineering that lets us check the consistency of equations, derive relationships, and convert units. But why does it work? Let's build the reasoning from the ground up.

1. The Core Idea: Physical Quantities Have Dimensions

Every physical quantity (like length, time, mass) can be expressed in terms of fundamental dimensions. The most common set in mechanics is:

L = Length
M = Mass
T = Time

For example:

Speed has dimensions $[L T^{-1}]$
Force has dimensions $[M L T^{-2}]$
Energy has dimensions $[M L^2 T^{-2}]$

Why this matters: Two quantities can only be meaningfully compared or equated if they have the same dimensions. You cannot add apples to oranges — and you cannot add length to time.

2. The Principle of Dimensional Homogeneity

The key formula that underpins everything is:

Every valid physical equation must be dimensionally homogeneous.

This means: the dimensions on the left-hand side must equal the dimensions on the right-hand side.

Why must this hold?

Consider an equation like:

v = u + at

Left side: $[v] = L T^{-1}$
Right side: $[u] = L T^{-1}$ , $[at] = (L T^{-2})(T) = L T^{-1}$

Both sides have dimensions $L T^{-1}$ . If they didn't match, the equation would be physically meaningless — you'd be comparing quantities that cannot be equal in any real experiment.

Reasoning: Physical laws describe relationships between measurable quantities. If the dimensions don't match, the equation cannot represent a real physical relationship, because the numerical value would depend on the arbitrary choice of units.

3. The Buckingham Pi Theorem: Why We Can Derive Relationships

This is the deeper mathematical reason. The Buckingham Pi Theorem states:

If a physical problem involves $n$ variables and $k$ fundamental dimensions, then it can be reduced to $n-k$ independent dimensionless groups (called $\pi$ groups).

Why does this work?

Imagine you have a relationship:

f(Q_1, Q_2, \dots, Q_n) = 0

where each $Q_i$ has dimensions. Because the equation must be dimensionally homogeneous, we can rearrange it into a function of dimensionless products only:

F(\pi_1, \pi_2, \dots, \pi_{n-k}) = 0

The reasoning: Dimensions act as constraints. Each fundamental dimension (M, L, T) gives one constraint. So if you have $n$ variables and $k$ constraints, you only have $n-k$ independent dimensionless combinations.

Example: For a simple pendulum, the period $T$ depends on length $L$ , mass $m$ , and gravity $g$ . That's 4 variables with 3 dimensions (M, L, T). So $4-3 = 1$ dimensionless group: $\pi = \frac{T^2 g}{L}$ . This tells us $T \propto \sqrt{L/g}$ without solving any differential equation.

4. Why We Can Convert Units Using Dimensional Analysis

The conversion factor formula:

\text{Value in new unit} = \text{Value in old unit} \times \left(\frac{\text{old unit}}{\text{new unit}}\right)^{\text{dimension exponent}}

Why this works: …

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When two elements form multiple compounds, the masses of one element combining with a fixed mass of the other are in simple whole-number ratios — this is the Law of Multiple Proportions, illustrated by carbon oxides: CO and CO₂.

The statement captures one of the foundational patterns in chemistry that helped establish atomic theory. Let me walk you through why this happens, which law governs it, and a concrete example.

(a) Is the statement true?

Yes, the statement is true. When two elements combine to form more than one compound, the different masses of one element that unite with a constant mass of the other element stand in a ratio of small whole numbers. This isn't a coincidence — it reflects the discrete, atomic nature of matter.

(b) The governing law

This observation is codified as the Law of Multiple Proportions, formulated by John Dalton in 1803.

A

The law emerged from careful quantitative analysis of compounds and provided strong evidence that matter is composed of indivisible atoms combining in fixed numerical ratios. If atoms combine in ratios like 1:1, 1:2, 2:3, etc., then the mass ratios must also be simple whole numbers (scaled by the atomic masses).

Tip

The Law of Multiple Proportions is distinct from the Law of Definite Proportions (constant composition). The former compares different compounds of the same two elements; the latter states that a single compound always has the same elemental ratio.

(c) A classic example: Carbon and Oxygen

Carbon and oxygen form two well-known compounds: carbon monoxide (CO) and carbon dioxide (CO₂). Let's see the law in action.

Step-by-step analysis:

Fix the mass of carbon.

Take $12 \, \text{g}$ of carbon (one mole, for convenience).
Determine oxygen mass in CO.

In carbon monoxide, one carbon atom bonds with one oxygen atom.

Molar mass of O = $16 \, \text{g/mol}$ .

So $12 \, \text{g}$ of C combines with $16 \, \text{g}$ of O.
Determine oxygen mass in CO₂.

In carbon dioxide, one carbon atom bonds with two oxygen atoms.

So $12 \, \text{g}$ of C combines with $2 \times 16 = 32 \, \text{g}$ of O. …

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Concept: Law of Multiple Proportions

This is one of the fundamental laws of chemical combination, discovered by John Dalton (1803).

(a) Is the statement true?

Yes, the statement is true.

(b) According to which law?

This is the Law of Multiple Proportions.

Statement: When two elements combine to form more than one compound, the masses of one element that combine with a fixed mass of the other element are in a simple whole number ratio.

(c) Example

Example: Carbon and Oxygen

Carbon and oxygen combine to form two compounds:

Compound	Formula	Mass of C (fixed)	Mass of O combining
Carbon monoxide	$CO$	12 g	16 g
Carbon dioxide	$CO_2$	12 g	32 g

Step-by-step reasoning:

Fix the mass of carbon = 12 g in both compounds.
Mass of oxygen in $CO$ = 16 g
Mass of oxygen in $CO_2$ = 32 g
Ratio of oxygen masses:

16 : 32 = 1 : 2

This is a simple whole number ratio (1:2), confirming the Law of Multiple Proportions.

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(a) Is the statement true?

Yes, the statement is true.

It describes the Law of Multiple Proportions.

(b) According to which law?

This is the Law of Multiple Proportions, given by John Dalton (1803).

Law of Multiple Proportions:

When two elements combine to form more than one compound, the masses of one element that combine with a fixed mass of the other element are in the ratio of small whole numbers.

(c) Example

Take carbon (C) and oxygen (O):

Carbon monoxide (CO):

12 g of C combines with 16 g of O.
Carbon dioxide (CO₂):

12 g of C combines with 32 g of O.

Here, the fixed mass of carbon is 12 g.

The masses of oxygen that combine with it are 16 g and 32 g.

Ratio of oxygen masses:

16 : 32 = 1 : 2

This is a simple whole number ratio, confirming the law.

Common Mistakes Students Make & How to Avoid Them

✗ Mistake 1: Confusing with Law of Definite Proportions

What students do: They think this statement is about the Law of Definite Proportions (which says a compound always has the same elements in the same mass ratio).
How to avoid: Remember:
- Definite Proportions → one compound, fixed ratio.
- Multiple Proportions → two or more compounds from same elements, different whole-number ratios.

✗ Mistake 2: Forgetting "fixed mass of the other element"

What students do: They compare masses of both elements freely, without fixing one.
How to avoid: Always fix the mass of one element (usually the heavier or the one common to both compounds) and then compare the masses of the other.

✗ Mistake 3: Using wrong example

What students do: They give an example where the ratio is not a simple whole number (e.g., H₂O and H₂O₂ — but here oxygen mass ratio is 1:2, which is correct; but sometimes they pick compounds like NO and NO₂ incorrectly).
How to avoid: Use classic textbook examples:
- CO and CO₂ (C fixed, O ratio 1:2)
- H₂O and H₂O₂ (H fixed, O ratio 1:2)
- SO₂ and SO₃ (S fixed, O ratio 2:3)

✗ Mistake 4: Writing the ratio incorrectly

What students do: They write the ratio as 16:32 = 1:2 but forget to mention which element's mass is fixed.
How to avoid: Always state: …

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Q.If two elements can combine to form more than one compound, the masses of one element that combine with a fixed mass of the other element, are in whole number ratio.

(a) Is the statement true?

(b) The governing law

(c) A classic example: Carbon and Oxygen

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