Question: Calculate the equilibrium value of vacancies concentration for copper per atom of the order of 10^4 at 1000K the energy for vacancy formation is 0.9eV/atom.
Answer:N=N_0+exp(−Q/RT)is something that I know.However all that I could arrive is to calculate −Q/RT which calculates to give me 82.1.I have taken R to be equal to 0.0821 units and have taken T=1000K. Now I cannot proceed any further.I am not even sure what I am to calculate.
Consult [1], specifically Chapter 10 herein.
[1] D Frenkel, and B Smit, Understanding Molecular Simulations: From Algorithms to Applications, 2nd edition (Academic Press, San Diego, 2002).
Dear Rajorshi Koyal :
I have a PPT presentation in my research uploaded items (Presentations), it is called Semiconductor Physics (Principles) or something similar, you can check it out if you want, I wrote the explanation and description of those concepts in this ppt..
In addition, I don't think the answer is "vacancies concentration for copper per atom", since we don't call vacancies to missing particles in atoms. So it should be... "per unit of volume".
Regarding your logic, I think you are on the right path with this equation: N=N_0+exp(−Q/RT)
N_0 is an experimental value which you can refer to, in tables, so check the ppt presentation I mentioned or other notes on the internet, along with an introductory book on Semiconductor Physics, I would recommend :
Chemming-Hu from UC (University of California) Berkeley
But I think you are on the right path using that equation, of couser if you want to solve the ideal scenario.
Regards !
Dear Rajorshi Koyal,
to calculate the equilibrium vacancy concentration per atom you should use:
C=exp-(Q/kT)
Since you have the enthalpy of formation Q in eV, this value is for 1 vacancy. Consequently you should use the Boltzmann-Constant k instead of R (R is for 1 mol; k is for 1 Atom: R=k x N)
To calculate the vacancy concentration per atom is quite usefull, because one gets a feeling for the relation between atoms and the un-occupied places (vacancies). To calculate the absolute number of vacancies, you shoud multiply your result per atom with the number of atoms per mol (Avogadro-Constant). Knowing the density of copper brings you to the number of vacancies per volume, as was proposed in one answer.
With the best wishes
Bernd Kieback
Sorry, in my answer I vorgot the C0 . The correct formula is:
C=C0 exp-(Q/kT).
C0 is not temperature dependent and is defined by the local change of entropy if 1 vacancy is formed. C0 should be taken from experimental literature, since it is not easy to calculate theoretically.
Just a small addition. If Q is in eV, then it is easier to convert the temperature to eV also. The rule is simple, 1eV = 11600 K. So 1000 K = 1000/11600 eV = 0.0862 eV, and Q/T = 0.9/0.0862 = 10.44.
This calculations is a liitle bit easier because you do not need to think about k and R.
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