In the context of carbon, which of the following is a correct pair of allotropes of the same element?

Introduction / Context:
This question is about allotropy, a concept in chemistry where an element can exist in two or more different structural forms in the same physical state. These different forms, called allotropes, have the same kind of atoms but arranged differently, leading to distinct physical properties. Carbon is a classic example, with several allotropes such as diamond, graphite, and amorphous carbon. Understanding which pairs of substances are true allotropes is important for both conceptual clarity and for solving objective questions in general chemistry and materials science.

Given Data / Assumptions:
- The question asks for a pair of allotropes, meaning two forms of the same element in the same physical state.
- Options include combinations of carbon isotopes, diamond, graphite, and carbon dioxide.
- It is assumed you can distinguish between allotropes (different structural forms) and isotopes (atoms with different mass numbers), and between elements and compounds.

Concept / Approach:
Allotropes must satisfy two conditions: they must be forms of the same element, and they must differ in how the atoms are bonded or arranged. Diamond and graphite each consist entirely of carbon atoms, but diamond has a three dimensional tetrahedral network, making it extremely hard and transparent, while graphite has planar layers of hexagonally arranged carbon atoms, making it soft and conductive along the layers. These are classic allotropes of carbon. By contrast, carbon-16 and carbon-12 are isotopes, differing in neutron number, not in structure. Carbon dioxide is a compound containing carbon and oxygen; it is not an allotrope of carbon. Thus, only the pair diamond and graphite satisfies the definition of allotropes.

Step-by-Step Solution:
Step 1: Recall that allotropes are structurally different forms of the same element in the same physical state. Step 2: Examine option B, diamond and graphite. Both substances are made entirely of carbon atoms but with different bonding and crystal structures. Step 3: Recognise that diamond is a rigid three dimensional network solid, while graphite consists of layered sheets, which is exactly the type of structural difference characteristic of allotropy. Step 4: Identify that carbon-16 and carbon-12 in option A are isotopes, not allotropes, because they differ in neutron number, not structure. Step 5: Note that carbon dioxide in options C and D is a compound of carbon and oxygen, not a pure form of the element carbon, and therefore cannot be an allotrope of carbon. Step 6: Conclude that diamond and graphite form the only correct pair of allotropes among the given options.

Verification / Alternative check:
Standard chemistry textbooks list diamond and graphite as the most familiar allotropes of carbon, often alongside amorphous forms like charcoal and more modern forms like fullerenes and graphene. They also explain that isotopes such as carbon-12 and carbon-14 are different nuclei of the same element, not different structures of the pure substance. Diagrams and descriptions of carbon dioxide clearly show it as a molecular compound containing oxygen, disqualifying it as an allotrope. These references confirm that only diamond and graphite meet the criteria for allotropy in this question.

Why Other Options Are Wrong:
Carbon-16 and carbon-12 are isotopes with different numbers of neutrons, so they differ in mass but not in their chemical structure as pure carbon; they are not allotropes. Carbon dioxide and graphite pair a compound with an element, which violates the definition of allotropy, since allotropes must be pure forms of the same element. Diamond and carbon dioxide have the same problem, because carbon dioxide is not a pure carbon form but a compound of carbon and oxygen. Therefore, options A, C, and D do not describe pairs of allotropes.

Common Pitfalls:
A common confusion is between isotopes and allotropes. Isotopes differ in nuclear composition, whereas allotropes differ in how atoms are bound to each other in a solid or gaseous structure. Another pitfall is forgetting that compounds like CO2 contain more than one element and therefore cannot be allotropes of any single element. To avoid these mistakes, always check whether both members of the pair are pure forms of the same element and whether they differ in bonding or crystal structure, not in atomic mass or composition.

Final Answer:
The correct pair of allotropes is diamond and graphite.