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Partial Derivatives Problem

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  1. Sep 24, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the equation of state given that k = aT^(3) / P^2 (compressibility) and B = bT^(2) / P (expansivity) and the ratio, a/b?

    2. Relevant equations
    B = 1/v (DV /DT)Pressure constant ; k = -1/v (DV /DP)Temperature constant D= Partial derivative
    dV = BVdT -kVdP (1)
    ANSWER is V = V0exp(aT^(3)/P)

    3. The attempt at a solution
    a. Integrate (1) and obtain v = voexp (bT^(3)/3P) + 2aT^(3)/P^3) WRONG
    b. Hint by Professor: rewrite as: (Let "D" = partial derivative): D/DP (lnV) = 1/V (DV/DP) then D/DP (lnV) + aT^(3)/P = 0. Write as D/DP(lnv + g(P,T)) = 0 where g is a function only of V. ==> lnV + g(P,T) = f(V) where V is an arbitrary function.

    I don't understand "b" but following attempt "a" gave me the proper result for a very similar problem?
     
  2. jcsd
  3. Sep 25, 2015 #2
    Your starting equations are:

    $$\frac{\partial \ln V}{\partial P}=-a\frac{T^3}{P^2}$$
    $$\frac{\partial \ln V}{\partial T}=b\frac{T^2}{P}$$
    Together with $$d\ln V=\frac{\partial \ln V}{\partial P}dP+\frac{\partial \ln V}{\partial T}dT$$
    The last equation implies that d lnV is an exact differential. What does that imply about the relationship between ##\frac{\partial}{\partial T}\left(\frac{\partial \ln V}{\partial P}\right)## and ##\frac{\partial}{\partial P}\left(\frac{\partial \ln V}{\partial T}\right)##?

    Chet
     
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