In crystalline solids, atoms are arranged in periodic arrays. The atoms themselves are fixed in place but can vibrate. You can imagine that the atoms in a crystal behave as if they are connected to their neighbors by springs. These vibrations travel as waves called “phonons” (lattice vibrations). The heat capacity of a crystalline solid arises from these vibrational degrees of freedom and is described mathematically by the Debye law: Where N = Avogadro’s number, k = Boltzmann’s constant, and θ is the Debye temperature a constant that is a metric of the stiffness of the “springs” and is correlated with mechanical properties of solids (i.e., mechanically softer solids tend to exhibit lower θ). a. Calculate the change in entropy associated with changing the temperature of diamond from 15 K to 100 K. (θ = 2230 K) b. The Debye temperature of gold (Au) is θ = 170 K. Calculate the entropy change associated with changing the temperature of solid gold from 15 K to 100 K. c. Compare your values from a) and b). Provide an explanation for the differences in entropy of this process for diamond vs. gold.
Thermochemistry
Thermochemistry can be considered as a branch of thermodynamics that deals with the connections between warmth, work, and various types of energy, formed because of different synthetic and actual cycles. Thermochemistry describes the energy changes that occur as a result of reactions or chemical changes in a substance.
Exergonic Reaction
The term exergonic is derived from the Greek word in which ‘ergon’ means work and exergonic means ‘work outside’. Exergonic reactions releases work energy. Exergonic reactions are different from exothermic reactions, the one that releases only heat energy during the course of the reaction. So, exothermic reaction is one type of exergonic reaction. Exergonic reaction releases work energy in different forms like heat, light or sound. For example, a glow stick releases light making that an exergonic reaction and not an exothermic reaction since no heat is released. Even endothermic reactions at very high temperature are exergonic.
In crystalline solids, atoms are arranged in periodic arrays. The
atoms themselves are fixed in place but can vibrate. You can
imagine that the atoms in a crystal behave as if they are
connected to their neighbors by springs.
These vibrations travel as waves called “phonons” (lattice
vibrations). The heat capacity of a crystalline solid arises from
these vibrational degrees of freedom and is described
mathematically by the Debye law:
Where N =
constant that is a metric of the stiffness of the “springs” and is correlated with mechanical
properties of solids (i.e., mechanically softer solids tend to exhibit lower θ).
a. Calculate the change in entropy associated with changing the temperature of diamond
from 15 K to 100 K. (θ = 2230 K)
b. The Debye temperature of gold (Au) is θ = 170 K. Calculate the entropy change
associated with changing the temperature of solid gold from 15 K to 100 K.
c. Compare your values from a) and b). Provide an explanation for the differences in
entropy of this process for diamond vs. gold.
![### Text Transcription:
The equation displayed is:
\[ C_{P,m} = \frac{12\pi^4 Nk}{5\theta^3} T^3 = \frac{1944}{\theta^3} \frac{J}{mol} T^3 \]
### Diagram Explanation:
The diagram appears to represent a three-dimensional lattice structure, commonly used to illustrate the arrangement of atoms or molecules in a solid.
- **Atoms/Molecules:** Orange spheres represent atoms or molecules arranged at the lattice points.
- **Connections:** Green spring-like lines connect these spheres, symbolizing interactions or bonds within the lattice.
This type of structure is typical in crystallography and materials science, representing how particles are organized in a solid state.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf84dde2-f77b-4203-b744-f3414691451c%2F501483c9-63e5-40f7-b9d6-b4e060fefed2%2Fnt7ctov_processed.png&w=3840&q=75)
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