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Thermodynamics- thermal expansion coefficient

  1. Aug 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove that the volume thermal expansion coefficient of a solid is equal to the sum of its linear expansion coefficients in the three directions. [itex]\beta[/itex]=[itex]\alpha[/itex]x +[itex]\alpha[/itex]y+[itex]\alpha[/itex]z

    For isotopic solid when [itex]\beta[/itex] = 3[itex]\alpha[/itex]

    2. Relevant equations

    [itex]\beta[/itex]=[1/v][dv/dt]p= [itex]\alpha[/itex]x +[itex]\alpha[/itex]y+[itex]\alpha[/itex]z

    3. The attempt at a solution

    I have looked at several websites but can't seem to get it.
    http://www.ami.ac.uk/courses/topics/0197_cte/index.html [Broken]
    http://en.wikipedia.org/wiki/Coefficient_of_thermal_expansion

    I thought about just plugging in the [itex]\beta[/itex] but I don't think that is correct.
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Aug 29, 2011 #2

    Mapes

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    Consider a cube, and calculate β in terms of changing edge lengths, and then in terms of α. What do you get? What approximations can you make if α is very small?
     
  4. Sep 1, 2011 #3
    Thanks for your help.

    I know see what you mean.

    I did of course only understand it until I found this solution online, I wanted to post it here again just in case someone else wants it.
    --------------------------------------------------------------------------------

    Let's start with that cube of side L.

    If we heat it up, the length of each side will change from L to (L+ΔL), where ΔL = αLΔT.

    In other words, the new length is:

    L′ = L + αLΔT = L(1 + αΔT)

    So the new volume is:

    Vo′ = (L′)^3
    = [L(1 + αΔT)]^3
    = (L^3)(1 + 3αΔT + 3(αΔT)² + (αΔT)^3)
    = Vo(1 + 3αΔT + 3(αΔT)² + (αΔT)^3)

    Now we make an approximation. Since αΔT is quite small compared to 1, we can safely say that the "3(αΔT)²" term and the "(αΔT)^3" term are negligible compared to the "3αΔT" term. Therefore, to a good approximation,

    Vo′ = Vo(1 + 3αΔT)

    But we already know, from the definition of β, that:

    Vo′ = Vo(1 + βΔT)

    So this means that β = 3α.
     
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