Problem 81 - University Physics with Modern Physics

Question

It is possible to make crystalline solids that are only one layer of atoms thick. Such "two-dimensional" crystals can be created by depositing atoms on a very flat surface. (a) If the atoms in such a two-dimensional crystal can move only within the plane of the crystal, what will be its molar heat capacity near room temperature? Give your answer as a multiple of \(R\) and in \(\mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\) ) At very low temperatures, will the molar heat capacity of a two-dimensional crystal be greater than, less than, or equal to the result you found in part (a)? Explain why.

Step-by-Step Solution

Answer

The molar heat capacity near room temperature is R, at low temperatures, it's less.

Step 1: Understanding Molar Heat Capacity

Molar heat capacity is the amount of heat required to raise the temperature of one mole of a substance by one Kelvin. For a solid, this depends on the degrees of freedom available to the atoms.

Step 2: Degrees of Freedom in Two-Dimensional Crystal

In a two-dimensional crystal, each atom can move in two directions (x and y). Therefore, each atom has two translational degrees of freedom.

Step 3: Applying Equipartition Theorem

The equipartition theorem states that each degree of freedom contributes \(\frac{1}{2}R\) to the molar heat capacity. Thus, for two translational degrees of freedom, the contribution to the molar heat capacity is 2 \(\times\ \frac{1}{2}R = R\).

Step 4: Concluding Molar Heat Capacity at Room Temperature

Hence, the molar heat capacity of a two-dimensional crystal near room temperature is \(R\) when expressed as a multiple of \(R\), and \(R = 8.314 \, \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\).

Step 5: Heat Capacity at Very Low Temperatures

At very low temperatures, the heat capacity becomes negligible. This is due to the third law of thermodynamics where the heat capacity approaches zero as temperature approaches absolute zero.

Step 6: Comparing Heat Capacities at Different Temperatures

Thus, the molar heat capacity at very low temperatures is less than the result near room temperature.

Key Concepts

Two-Dimensional Crystals

Two-dimensional crystals are fascinating structures that consist of only a single layer of atoms. These crystals, unlike their three-dimensional counterparts, confine atom movement to just a flat plane. This attribute showcases a unique arrangement and dynamic behavior in molecular terms. Atoms in two-dimensional crystals, like those made from materials such as graphene, exhibit only in-plane movement due to the restriction of other directions (z-direction).
This constraint significantly affects properties such as thermal and electrical conductivity. In the context of heat capacity, since the atoms can only move within this plane, it directly influences the degrees of freedom available to the system. Understanding these limitations helps in calculating other thermodynamic properties of two-dimensional crystals.

Equipartition Theorem

The equipartition theorem is fundamental in statistical mechanics. It states that each degree of freedom for a molecule contributes an equal amount of energy, \( \frac{1}{2} kT \), to the molecule's total internal energy. Here, \(k\) represents Boltzmann’s constant, and \(T\) is the absolute temperature.
For a two-dimensional crystal, each atom has two translational degrees of freedom – moving along the x and y axes. According to the equipartition theorem, each of these freedoms contributes \( \frac{1}{2}R \) to the molar heat capacity, where \( R \) is the ideal gas constant. Totalizing, a two-dimensional crystal thus has a molar heat capacity of \( R \) as each atom can only move in two directions. This contribution is crucial for predicting how these materials behave under various thermal conditions.

Degrees of Freedom

Degrees of freedom refer to independent ways in which a system can possess energy. For an atom, these primarily include translational, rotational, and vibrational movements. However, in a two-dimensional system like a crystal, the degrees of freedom are limited to only translational. Atoms can only move in the plane of the x and y axes.
Due to this restriction, we consider only two degrees of freedom in our calculations of heat capacity. This limited movement affects how energy is distributed within the crystal. In this context, knowledge of degrees of freedom is essential to understand concepts like molar heat capacity, further supported by theories such as the equipartition theorem.

Low Temperature Heat Capacity

At very low temperatures, a system’s heat capacity tends to deviate from values observed at higher temperatures. This is particularly noticeable in two-dimensional crystals. As temperature decreases, the energy states available to the atoms reduce significantly.
According to the Debye model, as the temperature approaches absolute zero, contributions from vibrational energy modes diminish, leading to a drop in heat capacity. This behavior aligns with predictions made by the third law of thermodynamics. Due to diminishing thermal activity, the atoms settle into their lowest energy state, resulting in a negligible heat capacity. Hence, at these low temperatures, the molar heat capacity is much less than the calculated value at room temperature.

Third Law of Thermodynamics

The third law of thermodynamics is a guiding principle in understanding the behavior of systems at low temperatures. It states that as the temperature of a perfect crystal approaches absolute zero, its entropy approaches a constant minimum. A direct implication of this is that the system's heat capacity also approaches zero.
In the context of a two-dimensional crystal, this means that no matter how complex or simple the crystal is, as the temperature gets very low, the ability of the system to absorb heat diminishes. This behavior leads to the observed decrease in heat capacity at low temperatures, reinforcing the fact when extrapolating from room temperature results. The third law provides important insight into the thermal dynamics and energy distribution within low-temperature systems.