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Homework Help: Callen: Thermodynamics (general fail)

  1. Oct 28, 2011 #1
    Greetings, I'm a 4th year physics undergrad, hoping to avoid entering a 5th. Using 2nd edition of Callen for Thermo class in upper div. I'm having a real hard time solving any problem in this book. I haven't had nightmares over homeworks since my math methods class which used Arfken. The problem is just ..I dont seem to have the pre-requisite understanding of thermodynamics / statistical physics. Lecture also covers different topics (yes, really.) I am going to post up a couple problems, not really looking for answers but maybe what I should've read up on first before I tried this problem. This is from the 2nd edition of callen. I have a lot of other problems I just think i'd ask about the worst one first.

    1. The problem statement, all variables and given/known data

    19.3-6. Consider a small quantity of matter consisting of a fixed number N moles in a large fluid system. Let ρN be the average density of these N moles: the mass divided by the volume. Show that equation 19.17 implies that the density fluctuations are

    <(ΔρN)^2> / (ρN)^2 = + kb * T * κt / V

    in which V is the average volume of the N moles.

    ok , i can't seem to understand the sigma symbol, so let me explain

    kb is the boltzmann constant
    κt is k-sub-t (subscript), i actually dont know what this is.
    ΔρN has a hat ( a ^ symbol) on the ρ
    ρN is ρ-sub-N (subscript)

    2. Relevant equations

    eq 19.17 is:

    <(ΔV-hat)^2> = -kb T (dv/dp)t,N1,... = kB T V κT

    3. The attempt at a solution

    I met up with 4 other students and we tried to make sense of it. It doesnt . Read the Callen chapter which is bewildering. Other than ch 19, we went over ch 1-7. I tried reading the other chapters before it. Went home trying to work on it. Tried to just replace ρ with M/V throughout the problem, hoping it would be a plug-and-chug answer. Tried to reverse engineer the goal-equation to see what needs doing (it seems like multiply both sides by M/V^3... can't recall off-hand.. it was something that made no sense. After that I began reading about fluctuations and stat mech online... also didnt help. Basically after that I started trying to just look the answer up somewhere, and that wasnt fruitful. I have..an empty sheet, because I dont even know where to start. I considered maybe attacking the (dv/dp) term in 19.17 since we have been doing manipulations like that up until now, but I am not very sure of what I am doing with that anyway.

    So..I'm sure someone knows how to nail this problem. Can I be provided with some maybe useful links and advise on how to solve it? I mean, I feel like I just dont have the tools. Maybe there is a concept of thermodynamics I missed badly along the way somewhere. I love my professor but wow this problem has made me suffer.
    Last edited: Oct 28, 2011
  2. jcsd
  3. Oct 28, 2011 #2


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    Equation 19.17 is
    [tex]\langle (\Delta \hat{V})^2 \rangle = -k_B T\left(\frac{\partial V}{\partial P}\right)_{T, N_1, \dots} = k_B T V \kappa_T[/tex]where
    [tex]\kappa_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T[/tex]is the isothermal compressibility (see page 84).

    The density is equal to [itex]\hat{\rho}_N = N/\hat{V}[/itex], and its average value is [itex]\rho_N = N/V[/itex], where V is the average volume. Therefore, the fluctuation in density is
    [tex]\Delta \hat{\rho}_N = \hat{\rho}_N - \rho_N = \frac{N}{\hat{V}} - \frac{N}{V}[/tex]Try using [itex]\hat V = V + \Delta \hat V[/itex] and expanding the first term to first order in [itex]\Delta \hat V[/itex].
  4. Oct 28, 2011 #3
    An expansion! I should've known.....
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