Heat capacity of a solid using the Einstein model

In summary, the Einstein model for heat capacity assumes that the frequencies of all the harmonic oscillators in a material are the same, but the energy varies with temperature. To plot the heat capacity as a function of temperature for aluminium, we can use the dimensionless plot C(1/x) / 3Nk vs. x, which is independent of specific values and focuses on the shape of the plot. This model raises questions about the frequencies being the same for all temperatures or just a given temperature and whether a function for \epsilon (T) is needed to avoid having two free variables in the Cv equation.
  • #1
Antti
27
0

Homework Statement



Plot the heat capacity as a function of temperature for aluminium, using the Einstein model.

Homework Equations



The Einstein model for heat capacity:

b6ae86f1d6f5881e5c3582f77dfd8e04.png


be968b7166a250b07b92e04c2cb0702e.png


The Attempt at a Solution



The model assumes that the frequencies of all the harmonic oscillators in the material are the same. But does it mean "same for all temperatures" or "same for a given temperature"? And if the latter is true, will I need to find a function [tex]\epsilon (T)[/tex]? Because otherwise the right hand side of the Cv equation wil have two free variables. Thankful for help, my textbook is very unclear about this.
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
In this model, the frequency of each oscillator is the same at all temperatures, but the energy varies with temperature. Define

[tex]\frac{1}{x}=\frac{\epsilon}{kT}[/tex]

Plot C(1/x) / 3Nk vs. x which is a dimensionless plot, good for any choice of specific values. Here, it's the shape that counts.
 

FAQ: Heat capacity of a solid using the Einstein model

1. What is the Einstein model?

The Einstein model is a theoretical model used to describe the heat capacity of a solid. It was proposed by physicist Albert Einstein in 1907 and is based on the assumption that atoms in a solid vibrate at a single frequency.

2. How does the Einstein model explain heat capacity?

The Einstein model explains heat capacity by considering the energy levels of the atoms in a solid. According to this model, the atoms can only absorb or release energy in discrete packets, or quanta. As the temperature of the solid increases, more energy is available for the atoms to absorb, leading to an increase in heat capacity.

3. What is the formula for calculating heat capacity using the Einstein model?

The formula for calculating heat capacity using the Einstein model is C = 3NkB(θE/T)2 eθE/T/(eθE/T-1)2, where C is the heat capacity, N is the number of atoms, kB is the Boltzmann constant, θE is the Einstein temperature, and T is the temperature.

4. How is the Einstein temperature determined?

The Einstein temperature is determined by fitting experimental heat capacity data to the Einstein model equation. It is the temperature at which the energy levels of the atoms are equal to the thermal energy of the solid, and all modes of vibration are excited.

5. Are there any limitations to the Einstein model?

Yes, there are limitations to the Einstein model. It assumes that all atoms in a solid vibrate at the same frequency, which is not entirely accurate. It also does not take into account anharmonic effects and the possibility of multiple energy levels. Other models, such as the Debye model, have been developed to address these limitations.

Similar threads

Einstein's heat capacity model .and law of corresponding
Replies
5
Views
2K
How Do You Determine Heat Capacity for Liquids?
Replies
2
Views
1K
Heat Capacity of a Van Der Waals' gas in an Open system
Replies
6
Views
4K
I How to understand experiment to measure heat capacity of solid?
Replies
1
Views
991
Which Graph Matches Each Energy Structure in a 3-Level System?
Replies
1
Views
2K
Solid State Heat capacity optical and acoustic
Replies
1
Views
1K
What is the relationship between heat capacity and temperature?
Replies
5
Views
2K
What factors affect the mean free path of phonons in Germanium at 300K?
Replies
1
Views
3K
Heat Capacity of a classical ideal gas and SHO
Replies
1
Views
2K
I Why does the heat capacity of solids drop exponentially at low temperatures?
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top