Ch 20: The Micro/Macro Connection

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 20: The Micro/Macro Connection

Problem 67

Chapter 20, Problem 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.

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Related Practice

Textbook Question

n moles of a diatomic gas with C_v = 5/2 R has initial pressure p_i and volume V_i. The gas undergoes a process in which the pressure is directly proportional to the volume until the rms speed of the molecules has doubled. How much heat does this process require? Give your answer in terms of n, p_i and V_i.

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Textbook Question

A monatomic gas is adiabatically compressed to 1/8 of its initial volume. Does each of the following quantities change? If so, does it increase or decrease, and by what factor? If not, why not? The mean free path.

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Textbook Question

The rms speed of the molecules in 1.0 g of hydrogen gas is 1800 m/s. 500 J of work are done to compress the gas while, in the same process, 1200 J of heat energy are transferred from the gas to the environment. Afterward, what is the rms speed of the molecules?

A thin partition divides a container of volume V into two parts. One side contains n_A moles of gas A in a fraction f_A of the container; that is, V_A = f_AV. The other side contains nB moles of a different gas B at the same temperature in a fraction f_B of the container. The partition is removed, allowing the gases to mix. Find an expression for the change of entropy. This is called the entropy of mixing.

Textbook Question

n₁ moles of a monatomic gas and n₂ moles of a diatomic gas are mixed together in a container. Derive an expression for the molar specific heat at constant volume of the mixture.

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