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Ch 20: The Micro/Macro Connection
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 20: The Micro/Macro ConnectionProblema 67
Capitolo 20, Problema 67

The 2010 Nobel Prize in Physics was awarded for the discovery of graphene, a two-dimensional form of carbon in which the atoms form a two-dimensional crystal-lattice sheet only one atom thick. Predict the molar specific heat of graphene. Give your answer as a multiple of R.

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Understand the concept of molar specific heat: Molar specific heat is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius. For solids, the molar specific heat can be predicted using the Dulong-Petit law or quantum mechanical models like the Einstein or Debye models.
Recognize that graphene is a two-dimensional material: Graphene is a single layer of carbon atoms arranged in a hexagonal lattice. Its specific heat behavior is influenced by its dimensionality and the vibrational modes of its atoms.
Identify the vibrational modes in graphene: In a two-dimensional material like graphene, there are three degrees of freedom per atom (two in-plane vibrations and one out-of-plane vibration). Each degree of freedom contributes to the molar specific heat.
Apply the equipartition theorem: At high temperatures, the equipartition theorem states that each degree of freedom contributes \( \frac{1}{2} R \) to the molar specific heat. Since graphene has three degrees of freedom per atom, the molar specific heat is \( 3 \times \frac{1}{2} R = \frac{3}{2} R \).
Conclude that the molar specific heat of graphene is \( \frac{3}{2} R \): This result is valid at high temperatures where all vibrational modes are fully excited. At lower temperatures, quantum effects may reduce the specific heat due to incomplete excitation of vibrational modes.

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Graphene Structure

Graphene is a single layer of carbon atoms arranged in a two-dimensional honeycomb lattice. This unique structure gives graphene remarkable mechanical, electrical, and thermal properties, making it a subject of extensive research in materials science and physics.
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Molar Specific Heat

Molar specific heat is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius. For two-dimensional materials like graphene, the specific heat can differ from three-dimensional materials due to their unique phonon modes and reduced dimensionality.
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R (Universal Gas Constant)

R is the universal gas constant, which is used in various equations in thermodynamics and physical chemistry. It has a value of approximately 8.314 J/(mol·K) and serves as a reference point for calculating molar specific heats and other thermodynamic properties in relation to ideal gases.
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