Question

Somehow you have a two-dimensional solid. a sheet of atoms in a square lattice, each atom linked to its four closest neighbors by four springs oriented along the two perpendicular axes. (a) What would you expect the molar heat capacity to be at very low temperature and at very thigh temperature? (b) What quantity would determine, roughly, the line between low and high?

Short answer

At very low temperatures, the molar heat capacity will be zero. At very high temperatures, the molar heat capacity will be approximately 3R or 24.942 J/(mol·K). The line between low and high temperature is roughly determined by the Debye temperature.

Step-by-step solution

Determine Low Temperature Heat Capacity

At very low temperatures, following the third law of thermodynamics, the atoms stop vibrating and therefore, the expected molar heat capacity is zero.

Determine High Temperature Heat Capacity

At very high temperatures, by Dulong-Petit law, the molar heat capacity of solids tends to a constant value. So, the expected molar heat capacity is 3R, with R being the gas constant, R=8.314 J/(mol·K).

Determine the Line Between Low and High Temperature

The Debye temperature of the material can be used to roughly determine the line between low and high temperatures. Below the Debye temperature, heat capacity increases with temperature. Above the Debye temperature, heat capacity tends to be constant.

Key concepts

Dulong-Petit law

The Dulong-Petit law is a simple and classical rule in thermodynamics, related to the heat capacity of solids. It states that the molar heat capacity at constant volume (\(C_V\)) for most solid elements approaches \(3R\) at high temperatures.
Here, \(R\) is the universal gas constant, approximately equal to 8.314 J/(mol·K).
This principle simplifies the prediction of heat capacity in materials with metallic bonding and similar properties, especially at higher temperatures.

• The atoms in a solid vibrate more intensely as the temperature increases, transferring kinetic energy among themselves.
• The Dulong-Petit law captures this vibrational behavior, resulting in a straightforward heat capacity constant (\(3R\)).

However, it's essential to note that this law is less applicable for materials at low temperatures or with complex molecular structures, where quantum effects become significant.
Understanding the Dulong-Petit law helps in anticipating the behavior of solids under different temperature conditions, which has practical implications in material science and engineering.

Debye temperature

The concept of Debye temperature is pivotal in understanding the thermal properties of solids, especially their heat capacities.
The Debye temperature (\(\Theta_D\)) is a theoretical temperature associated with the material's vibrational properties and the highest frequency mode of vibration among its atoms.
Below the Debye temperature, quantum effects dominate the heat capacity of the material, often leading to a sharp increase as the temperature rises.

• It provides a guideline for distinguishing when to expect a transition in heat capacity behavior, from quantum-dominated low-temperature physics to classic behavior.
• The Debye model accommodates the quantum nature of atomic vibrations, explaining anomalies in heat capacity at lower temperatures which are not covered by the Dulong-Petit law.

The transition point marked by \(\Theta_D\) is significant in material science, indicating how atoms are arranged and interact within a solid.
Materials with high \(\Theta_D\) values tend to maintain solid characteristics at low temperatures, while those with lower values may exhibit more pronounced changes in heat capacity.

Third law of thermodynamics

The third law of thermodynamics is a foundational principle stating that as the temperature of a system approaches absolute zero (\(0 \, \text{K}\)), the entropy, or disorder, approaches a constant value.
In simple terms, any pure crystalline substance has zero entropy at absolute zero because the atoms are perfectly ordered, not vibrating, thus no energy is transferrable.

• At such low temperatures, the vibrational contribution to heat capacity becomes negligible, aligning with the idea that no energy is being exchanged.
• This law implies that the molar heat capacity of a solid also approaches zero as the temperature approaches absolute zero.

The physical significance of the third law relates to cryogenics and low-temperature physics, where precise control of thermal energy is crucial.
When combined with heat capacity calculations, it aids in predicting how materials behave under extreme conditions, contributing to various scientific fields such as nanotechnology and quantum computing.