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Chapter 18: Problem 69

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 twodimensional 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 \(\mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\). (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.

Short Answer

Expert verified
a) The molar heat capacity near room temperature for a two-dimensional crystal is \(R\) J/mol.K. b) At very low temperatures, the molar heat capacity of a two-dimensional crystal will be less than the result found in part (a) because atoms will not have enough kinetic energy to move, hence their degrees of freedom reduce effectively to zero.

Step by step solution

01

Molar Heat Capacity at Room Temperature

At room temperature, each atom in the two-dimensional crystal has 2 degrees of freedom (they can move in two directions: up-down and left-right, but not out of the plane of the crystal). Using the equipartition theory formula, the molar heat capacity \(C\) at constant volume is given by \(C = f/2 * R\), where \(f\) is the degree of freedom which is 2, and \(R\) is the gas constant. Substitute these values to find the molar heat capacity.
02

Molar Heat Capacity at Very Low Temperatures

At very low temperatures, the vibrations of atoms decrease significantly. In other words, the degrees of freedom are effectively reduced to zero, as atoms will not have enough kinetic energy to move. As such, the molar heat capacity will also decrease to approximately zero, because the gas constant \(R\) still remains, but the degrees of freedom drop towards zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Two-Dimensional Crystals
Two-dimensional crystals are fascinating structures where atoms organize in a sheet that is only one atom thick. Unlike the more familiar three-dimensional crystals, these structures restrict atomic movement to just a plane. This is why they're often discussed in advanced physics and materials science.
Some essential properties of two-dimensional crystals include:
  • Atomic Arrangement: Atoms in a two-dimensional crystal are typically arranged in a very ordered pattern.
  • Movement Constraint: Because of their thinness, atoms can only move within the plane of the crystal, meaning they can't move up or down.
  • Applications: These materials have various applications, including electronics, due to their unique electrical properties.
Understanding two-dimensional crystals is crucial for exploring nanotechnology and the development of new materials.
Equipartition Theory
Equipartition theory is a foundational concept in thermodynamics that helps us understand energy distribution in systems at thermal equilibrium. According to this theory, energy is equally partitioned among all degrees of freedom available to particles in a system.
Imagine heating a bulk of particles. The particles can translate, rotate, and vibrate. For each direction of movement, they have degrees of freedom.
In the context of a two-dimensional crystal:
  • Translation: Atoms can only translate, or move along, the two axes of the plane.
  • Degrees of Freedom: Each atom in a two-dimensional sheet has two translational degrees of freedom (corresponding to the x and y directions in the plane).
  • Energy Contribution: The energy associated with each degree of freedom is given as \( \frac{1}{2} k T \), where \(k\) is the Boltzmann constant and \(T\) is temperature.
This leads us to calculate thermodynamic properties like the molar heat capacity discussed in textbooks and research.
Degrees of Freedom
Degrees of freedom in physics refer to the number of independent ways in which a system's components can move without violating any constraints. It's an incredibly helpful idea when analyzing systems like gases, crystals, and even mechanical contraptions.
For a two-dimensional crystal, the degrees of freedom are linked to how atoms in the layer can move:
  • Plane Movement: Atoms can only move within their plane, leading to two degrees of freedom: horizontal and vertical motion.
  • Heat Capacity Relevance: These degrees of freedom contribute to calculating properties like heat capacity. At room temperature, the molar heat capacity is linked to this concept through formulas like \(C = \frac{f}{2} \times R\), where \(f\) is the degrees of freedom.
  • Temperature Impact: At lower temperatures, however, the available energy might not excite these degrees of freedom as much, hence, lowering the heat capacity.
This understanding allows students and scientists alike to predict material behavior in thermal conditions.

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Most popular questions from this chapter

Consider an ideal gas at \(27^{\circ} \mathrm{C}\) and 1.00 atm. To get some idea how close these molecules are to each other, on the average, imagine them to be uniformly spaced, with each molecule at the center of a small cube. (a) What is the length of an edge of each cube if adjacent cubes touch but do not overlap? (b) How does this distance compare with the diameter of a typical molecule? (c) How does their separation compare with the spacing of atoms in solids, which typically are about \(0.3 \mathrm{nm}\) apart?

\(\mathrm{A}\) welder using a tank of volume \(0.0750 \mathrm{~m}^{3}\) fills it with oxygen (molar mass \(32.0 \mathrm{~g} / \mathrm{mol}\) ) at a gauge pressure of \(3.00 \times 10^{5} \mathrm{~Pa}\) and temperature of \(37.0^{\circ} \mathrm{C}\). The tank has a small leak, and in time some of the oxygen leaks out. On a day when the temperature is \(22.0^{\circ} \mathrm{C},\) the gauge pressure of the oxygen in the tank is \(1.80 \times 10^{5} \mathrm{~Pa}\). Find (a) the initial mass of oxygen and (b) the mass of oxygen that has leaked out.

Estimate the number of atoms in the body of a \(50 \mathrm{~kg}\) physics student. Note that the human body is mostly water, which has molar mass \(18.0 \mathrm{~g} / \mathrm{mol}\), and that each water molecule contains three atoms.

The gas inside a balloon will always have a pressure nearly equal to atmospheric pressure, since that is the pressure applied to the outside of the balloon. You fill a balloon with helium (a nearly ideal gas) to a volume of \(0.600 \mathrm{~L}\) at \(19.0^{\circ} \mathrm{C}\). What is the volume of the balloon if you cool it to the boiling point of liquid nitrogen \((77.3 \mathrm{~K}) ?\)

Three moles of an ideal gas are in a rigid cubical box with sides of length \(0.300 \mathrm{~m}\). (a) What is the force that the gas exerts on each of the six sides of the box when the gas temperature is \(20.0^{\circ} \mathrm{C} ?\) (b) What is the force when the temperature of the gas is increased to \(100.0^{\circ} \mathrm{C} ?\)
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