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Statistical physics

  1. Mar 12, 2008 #1
    [SOLVED] statistical physics

    1. The problem statement, all variables and given/known data
    I am working on number 3 part a.
    I am trying to calculate C_P.
    From the first law of thermodynamics: [itex]dQ = dU -dW = dU +PdV[/itex] (does anyone know how to write the inexact differential d in latex?).
    And we know that [itex]C_p \equiv \frac{dQ}{dT}_P[/itex]. But I don't see how to get an explicit expression for dQ. Should I expand dU and dV in terms of the other independent variables or what? What variables should I choose to be independent?

    EDIT: I actually need help with Problem 4 also. I can integrate (dS/dA)_T w.r.t A and get that
    [tex]S(A,T) = -\frac{NkT}{A-b}+\frac{aN^2}{A^2} +f(T)[/tex] but then I have no idea how to find f(T)!
    2. Relevant equations

    3. The attempt at a solution
    Last edited: Mar 12, 2008
  2. jcsd
  3. Mar 12, 2008 #2


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    A good definition for [itex]c_P[/itex] is

    [tex]c_P=\frac{1}{N}\left(\frac{\partial H}{\partial T}\right)_P=\frac{1}{N}\left[\frac{\partial (U+PV)}{\partial T}\right]_P[/tex]

    Recall that for an ideal gas [itex]dU=Nc_V\,dT[/itex].

    Once you find [itex]c_P[/itex] you should be able to integrate your equation for [itex]\delta Q[/itex] as

    [tex]Q=\int Nc_P\,dT[/tex]
  4. Mar 12, 2008 #3


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    Regarding your second question: try inverting [itex]\left(\frac{\partial T}{\partial \mathcal{S}}\right)_A[/itex] and using

    [tex]d\mathcal{S}=\left(\frac{\partial \mathcal{S}}{dT}\right)_A dT+\left(\frac{\partial \mathcal{S}}{dA}\right)_T dA[/tex]
  5. Mar 12, 2008 #4
    Is it in general true that

    [tex]1/\left(\frac{\partial T}{\partial \mathcal{S}}\right)_A = \left(\frac{\partial \mathcal{S}}{\partial T} \right)_A[/tex]

  6. Mar 12, 2008 #5


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    In my experience, it always works in thermodynamics. Outside engineering it may be risky. Mathematicians, want to weigh in?
  7. Mar 12, 2008 #6
    Yes, it would really help me if a mathematician posted exactly when that is true.
  8. Mar 13, 2008 #7
  9. Apr 5, 2008 #8


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    I assume that notation means you're "Treating S (resp. T) as a function of A and T (resp. S), and differentiating, holding A as constant"?

    Or more precisely, S, T, and A are functions of your state [itex]\xi[/itex], and you have a relationship

    [tex]S(\xi) = f( T(\xi), A(\xi) )[/tex]

    and you're interested in [itex]f_1(T(\xi), A(\xi))[/itex], the partial derivative of this function with respect to the first place, evaluated at [itex](T(\xi), A(\xi)[/itex]?

    Well, for any particular value of A, this is just ordinary, one variable calculus -- let [itex]f_a[/itex] denote the function defined by [itex]f_a(x) = f(x, a)[/itex]. If [itex]f_a[/itex] is invertible, then it's easy to find a relationship: just differentiate the identity [itex]x = f_a( f_a^{-1}(x))[/itex].

    For a more geometric flavor, if restricting to a subspace where A is constant means that the differentials dS and dT are proportional (i.e. dS = f dT for some f), then it's just a matter of algebra to express dT in terms of dS where possible.
    Last edited: Apr 5, 2008
  10. Apr 6, 2008 #9
    So it is in general true (as long as we assume differentiability of the function and its inverse and do not divide by 0)! Yay!
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