Energy of a crystal in thermal equilibrium

• patrykh18
In summary: In a gas, the molecules are constantly moving and exchanging energy so the entropy of any given molecule is not the same as it was the last time you saw it.

Homework Statement

In a monatomic crystalline solid each atom can occupy either a regular lattice site or an interstitial site. The energy of an atom at an interstitial site exceeds the energy of an atom at a lattice site by an amount ε. Assume that the number of interstitial sites equals the number of lattice sites, and also equals the number of atoms N.

b)What is the temperature of the crystal in this state, if the crystal is in thermal equilibrium?

S=kbln(Ω)
1/T=∂S/∂E

The Attempt at a Solution

This is question 2.1 taken from Franz Mandl. Statistical Thermodynamics. Part a) asked me to find the total entropy which was fine. However in part b if you consider the total entropy and total energy of the entire system and use the 2nd formula i posted you will get n/N = exp(-ε/2kbT) (assuming ε >> kbT ) where n is the number of atoms in the interstitial site. Thats the answer given at the back of the book. In my first attempt however I divided my system into two subsystems (subsystem i of atoms in interstitial sites and subsystem r of atoms in regular sites). The entropy of each system is the same because each statistical weight is the same so the entropy of the subsystem i is half the total entropy found part a. The fact that I divided the entropy by two meant that my final answer was nearly the same as the answer at the back of the book but without the 2 in the exponent. I'm just trying to understand why my method of subdividing the system was not correct.

Why would expect that to be right? The lattice positions and the interstitial positions are energy states available to all of the atoms. You are splitting the population up and counting them as two separate isolated populations based on nothing other than the energy state they achieved. In a gas you wouldn’t split the population into those above a certain temperature and those below a certain temperature and then calculate the entropy of the two populations like they were unrelated.

What is the energy of a crystal in thermal equilibrium?

The energy of a crystal in thermal equilibrium is the total amount of energy contained within the crystal at a specific temperature. It includes both the kinetic energy of the atoms or molecules within the crystal and the potential energy of their interactions.

How is the energy of a crystal in thermal equilibrium calculated?

The energy of a crystal in thermal equilibrium is calculated using the Boltzmann distribution, which takes into account the number of energy states available to the atoms or molecules in the crystal and their corresponding probability of being in each state at a given temperature.

What factors affect the energy of a crystal in thermal equilibrium?

The energy of a crystal in thermal equilibrium is affected by the temperature of the crystal, the number of atoms or molecules in the crystal, and the specific properties of the crystal such as its lattice structure and composition.

How does temperature impact the energy of a crystal in thermal equilibrium?

The energy of a crystal in thermal equilibrium increases with temperature, as the atoms or molecules within the crystal gain more kinetic energy and their interactions become more energetic. However, at extremely high temperatures, the crystal may undergo a phase change and release or absorb energy.

What are the practical applications of understanding the energy of a crystal in thermal equilibrium?

Understanding the energy of a crystal in thermal equilibrium is important in fields such as materials science, chemistry, and engineering. It can help in the development of new materials with specific thermal properties, as well as in the design of efficient energy storage and conversion systems.