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 \(\mathrm{J} / \mathrm{nol} \cdot \mathrm{K}\) . (b) 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

(a) The molar heat capacity is \\(R \, \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\\). (b) At very low temperatures, it will be less than \\(R\\).

Step 1: Understanding the Degrees of Freedom

In a two-dimensional crystalline solid, atoms can move in two directions: along the x-axis and y-axis, within the plane of the crystal. This gives each atom 2 translational degrees of freedom. According to the equipartition theorem, each degree of freedom contributes \((1/2)kT\) to the energy per atom, where \(k\) is Boltzmann's constant and \(T\) is the temperature.

Step 2: Calculating Energy per Mole

Since we are dealing with a molar quantity, the energy for one mole is given by \(N_AkT\) for each degree of freedom, where \(N_A\) is Avogadro's number. Therefore, for 2 degrees of freedom, the energy per mole becomes \(2 \cdot (1/2)N_AkT = N_AkT\).

Step 3: Calculating Molar Heat Capacity

The heat capacity \(C\) is defined as the derivative of energy \(U\) with respect to temperature: \(C = \frac{dU}{dT}\). Here, since \(U = N_AkT\), we find that \(C = N_Ak = R\), where \(R = N_Ak\) is the ideal gas constant. Therefore, the molar heat capacity \(C_m = R\).

Step 4: Interpretation of Molar Heat Capacity in Units

Since we've established that the molar heat capacity \(C_m = R\), the heat capacity in terms of units will be \(R = 8.314 \, \mathrm{J} \, \mathrm{mol}^{-1} \, \mathrm{K}^{-1}\). Therefore, the heat capacity of the crystal near room temperature is \(R \, \mathrm{J} \, \mathrm{mol}^{-1} \, \mathrm{K}^{-1}\).

Step 5: Exploring Low-Temperature Behavior

At very low temperatures, atoms have significantly reduced thermal energy and, thus, are more restricted in their movement, leading to reduced heat capacity. In this context, quantum mechanical effects become significant. According to the third law of thermodynamics, the heat capacity of a system approaches zero as the temperature approaches absolute zero, which means the molar heat capacity will be less than \(R\) at very low temperatures.

Key Concepts

Molar Heat Capacity

Molar heat capacity is an important concept in thermodynamics, representing the amount of heat energy required to raise the temperature of one mole of a substance by one degree Kelvin. In the context of a two-dimensional crystal, it reflects the crystal's response to temperature changes.

According to the textbook solution, at room temperature, the molar heat capacity for a two-dimensional crystal is equal to the ideal gas constant, denoted as \( R \). This is because, as derived, each atom in the crystal has translational energy contributions due to their movement, and these movements are limited within the 2D plane of the crystal.

This simplicity allows for a clearer understanding of how heat impacts the vibrational modes of atoms specifically in a confined plane. In this specific scenario, the molar heat capacity simplifies to \( R = 8.314 \, \mathrm{J} \, \mathrm{mol}^{-1} \, \mathrm{K}^{-1} \)."

Degrees of Freedom

Degrees of freedom refer to the number of independent ways in which the atoms of a system can move. For a two-dimensional crystal, atoms can move only in two directions—along the x-axis and the y-axis—within the plane of the crystal.

• This results in 2 translational degrees of freedom for each atom.
• Each degree of freedom contributes to the energy of the system.

A full understanding of this has more profound implications, as it helps determine the energy of the system using the equipartition theorem, which directly affects the calculation of molar heat capacity. This simplification to two degrees is what limits the molar heat capacity to \( R \), under typical conditions.

Recognizing the degrees of freedom is crucial for predicting how the heat energy is absorbed and distributed within the crystal.

Equipartition Theorem

The equipartition theorem is a foundational principle in physics, specifically in thermodynamics and statistical mechanics. It states that for a system at thermal equilibrium, energy is equally distributed among the degrees of freedom of the atoms.

In the case of two-dimensional crystals, each degree of freedom contributes \( \frac{1}{2} k T \) to the energy per atom, where \( k \) is Boltzmann's constant and \( T \) is the temperature.

• This principle helps in calculating the total energy of the system, which is essential in determining the molar heat capacity.
• For the two-dimensional crystal, the total energy per mole becomes \( N_A k T \), aligning with the result that the molar heat capacity \( C_m = R \).

This distribution of energy across different degrees directly explains how and why energy affects the motion of atoms within the crystal plane.

Low-Temperature Behavior

At very low temperatures, quantum mechanical effects become more pronounced. For two-dimensional crystals, this manifests as a significant drop in thermal motion and vibrational energy of atoms.

The third law of thermodynamics underscores this behavior, stating that as temperatures approach absolute zero, the heat capacity of a system approaches zero. This means that the molar heat capacity of the crystal will be less than the value found at room temperature (which is \( R \)).

• Atoms have reduced energy and thus contribute less to the overall heat capacity.
• Quantum effects further restrict movement, limiting heat absorption capacity.

Understanding low-temperature behavior is vital for designing systems and materials which operate at cryogenic temperatures.

Boltzmann's Constant

Boltzmann's constant \( (k) \) is a fundamental constant in thermodynamics. It serves as a bridge between macroscopic and microscopic physics.

With a value roughly equal to \(1.38 \times 10^{-23} \mathrm{J/K} \), it represents the amount of energy per degree of freedom per degree of temperature increase.

• In the context of two-dimensional crystals, \( k \) helps calculate the energy contribution of each degree of freedom with the equipartition theorem.
• It aids in translating microscopic behaviors into macroscopic realities, as seen in the formula for energy \(( \frac{1}{2} k T \)).

Boltzmann's constant is not just a number; it is a critical piece of understanding thermodynamics and the statistical behaviors of atoms in a crystal lattice.