Let's analyze the given question which asks about the most thermodynamically stable allotrope of carbon. Allotropes are different structural modifications of an element; the atoms of the element are bonded together in a different manner.

Understanding Carbon Allotropes and Stability

Carbon exists in several allotropic forms. The question lists Graphite, Diamond, Fullerene, and Carbon-black. The thermodynamic stability of an allotrope is related to its standard enthalpy of formation. The allotrope that is considered the most stable at standard conditions (typically 298 K and 1 atm pressure) is assigned an enthalpy of formation of zero. All other allotropes will have positive enthalpies of formation relative to the most stable one, indicating they are less stable under standard conditions.

Why Graphite is the Most Stable Allotrope

Based on experimental data and thermodynamic principles, graphite is considered the most stable allotrope of carbon under standard conditions. The standard enthalpy of formation of graphite is defined as 0 kJ/mol. For other allotropes like diamond, the standard enthalpy of formation is positive, indicating they are less stable than graphite under these conditions.

• Graphite: It has a layered structure where carbon atoms are arranged in hexagonal rings. The layers are held together by weak van der Waals forces, allowing them to slide easily, which is why graphite is soft and used as a lubricant.
• Diamond: It has a rigid, three-dimensional tetrahedral structure where each carbon atom is bonded to four other carbon atoms. This structure makes diamond extremely hard. Diamond is stable at high pressures and temperatures, but under standard conditions, it is less stable than graphite. The conversion of diamond to graphite is thermodynamically favorable but kinetically very slow.
• Fullerene: Fullerenes, like Buckminsterfullerene (\text{C}_{60}), are spherical or elliptical molecules of carbon atoms. They are less stable than graphite.
• Carbon-black: Carbon-black is an amorphous form of carbon, consisting of fine particles. Amorphous forms generally have higher energy and are less stable than crystalline forms like graphite.

Therefore, among the given options, graphite is the thermodynamically most stable allotrope of carbon under standard conditions.

Revision Table: Carbon Allotropes

Additional Information on Carbon Allotropes and Thermodynamics

The stability of different allotropes can be compared using their standard free energy of formation (\Delta_f G^\circ). Since \Delta G = \Delta H - T\Delta S, both enthalpy and entropy play a role. However, under standard conditions, enthalpy differences are often dominant when comparing solid phases of the same element. The transformation of one allotrope to another often requires significant activation energy, which is why less stable allotropes like diamond can exist for long periods under standard conditions.

The concept of standard state defines the most stable form of an element under standard pressure (1 bar) and a specified temperature (usually 298.15 K). For carbon, graphite is defined as the standard state, meaning its standard enthalpy and free energy of formation are zero.