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Homework Help: Calculating equilibrium number of vacancies and other variables (crystal defects)

  1. Mar 24, 2010 #1
    1. The problem statement, all variables and given/known data

    The number of vacancies in a metal increases by a factor of six when the
    temperature is increased from 800K to 1000K. Calculate the energy for vacancy
    formation assuming that the density of the metal remains the same over this
    temperature range.

    2. Relevant equations

    Nv= N exp ( -Qv / kT)

    where Nv= equilibrium number of vacancies
    N = total no. of atomic sites
    N is dependent on density etc, there is a formula but i dont believe it is needed for this problem.
    -Qv= energy required for vacancy formation ( in eV)
    k = Boltzmann’s constant (1.38 x 10-23JK-1 or 8.62 x 10-5eVK-1)
    T = temperature (in kelvin)

    3. The attempt at a solution

    What I attempted to do was a simultaneous equation, taking the fact that Nv at 1000K is 6x more than Nv at 800k

    Nv= N exp ( -Qv / 8.62 x 10-5eVK-1 x 800)


    6Nv= N exp ( -Qv / 8.62 x 10-5eVK-1 x 1000)

    so hence i thought

    6 (N exp ( -Qv / 8.62 x 10-5eVK-1 x 800)) = N exp ( -Qv / 8.62 x 10-5eVK-1 x 1000)

    From here things got a bit crazy, I still don't really understand what exp means (I used wolfram to try and solve and it told me exp just ment e^(whatever is infront of exp). Not sure if I'm doing things right.. but a few more steps and I just kept getting -Qv as zero. Help would be great thanks alot.
    Last edited: Mar 24, 2010
  2. jcsd
  3. Mar 25, 2010 #2
    Hey, I had a similar question just recently - actually... it was the same except 5 instead of 6 :P

    Anyway, I did pretty much the same as you but once I got Nv = Nexp(-Q/kT) and 6Nv = Nexp(-Q/kT) I did some funky stuff with natural log. Firstly, exp is ex (or an exponent x with base e) so you can do natural logs once you manipulate both equations - like this:

    1) Nv = Nexp(-Q/kT) so Nv/N = e(-Q/kT) therefore ln Nv/N = -Q/kT1 (for the initial temp)

    2) 6Nv = Nexp(-Q/kT) so 6Nv/N = e(-Q/kT) therefore ln 6Nv/N = -Q/kT2 (for final temp)
    Here you can use log laws to manipulate this equation to look match the first -
    (ln x/y = ln x - ln y) so ln 6Nv - ln N
    (ln (xy) = ln x + ln y) so ln 6 + ln Nv - ln N = ln 6Nv/N ok?

    From that you can see that ln Nv/N = -Q/kT2 - ln 6
    Then you just put Equation 1 = Equation 2 and solve for Q :) I hope that helps, I got a decent answer but when I sub back in the two values are 1.00 apart so I'm going to check that out... that's my disclaimer lol
  4. Apr 2, 2010 #3
    What is the formula for Vacancy density? I have question regarding the relationship between temperature and vacancy densities.

  5. Apr 2, 2010 #4
    I only have 2 different temperatures, the energy of introducing 1 defect and lattice parameter "a" ....
  6. Apr 3, 2010 #5
    The formula is: Nv= Ne(-Q/kT) (usually written as exp(-Q/kT)

    Nv is the number of vacancies
    N is the total number of atomic sites (which would relate to the crystal structure and lattice constant)
    e is the natural exponential 2.71828 etc
    Q is the energy required for vacancy formation
    k is Boltzmann's constant (1.38x10-23 J OR 8.62x10-5 eV)
    T is the temperature in Kelvin

    What is the question exactly?
  7. Apr 3, 2010 #6
    hey sorry I should have posted that in the first place.

    BCC cubic iron has a lattice constant of 0.287nm. What is the ratio of vacancy densities at T= 25degree C and T=700 degree C given that the energy associated with introducing 1 defect site is 0.92eV.

    so is N= 0.287nm?
  8. Apr 4, 2010 #7
    Ok first of all, the lattice constant (a) is just the length of one side of the unit cell - not the total number of atomic sites (N) BUT you're right to think the two are related.

    Number of atomic sites is given with:

    N = NAp/AFe

    Where AFe is the atomic mass of iron: 55.845g
    NA is Avagadro's number: 6.023x1023
    And p = density of Iron

    The density isn't given but you can calculate using the lattice constant:

    p = nAFe/VcNA

    n is the number of atoms in the unit cell - 2 for BCC
    Vc is the volume of the unit cell - calculated by a3

    Since you know a, you can work out p and then use that to work out N. Once you know N just sub in your temperatures - remembering to convert to Kelvin first - and you'll get the number of vacancies at those temperatures, then just find the ratio. Note that the number of vacancies should never be more than the number of atomic sites, for obvious reasons!

    Good luck and let me know how you go or if you need any more explanation!

    PS: In case you're wondering, A is found from the periodic table and n is found since there is always a set number of atoms associated with each type of unit cell, BCC is 2, FCC is 8 etc.
  9. Apr 4, 2010 #8
    Wow, that's awesome. Thanx!!

    Just wondering, if vacancy density depends on temperature, can't we just compare the temperatures to obtain a ratio? bcz all the other parameters in the equation is the same and just temperature changes. ??

    Is "vacancy density" and "number of vacancies" the same thing? Just wanted to make sure =)
  10. Apr 4, 2010 #9
    I'm assuming the vacancy densities would mean number of vacancies per number of atomic sites... And the ratio then is a comparison between the two densities at those two temperatures? Number of vacancies is usually just a number of how many there are at a given temperature and input energy...

    Most of the parameters are the same, but the number of vacancies will change depending on the temperature (so 25C will have much less than 700C) so you will need to calculate how many vacancies at those temps, but that is just a question of putting 25C (in K) and 700C into the formula to fnd the number of vacancies then probably divide by the total number of sites. So yeah, in a way it is just comparing the temps, but mostly comparing the number of vacancies/total atomic sites at those temps :)
    Hope that helps :D
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