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Calculate the latent heat

  1. Jun 9, 2013 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    I'm stuck in the following problem: In a particular solid-liquid phase transition the point ##T_0##, ##P_0## lies on the coexistence curve. The latent heat of vaporization at this point is ##l_0##. A nearby point on the existence curve has pressure ##P_0+p## and temperature ##T_0+t##; the local slope of the coexistence curve in the P-T plane is p/t. Assuming ##v##, ##c_p##, ##\alpha## and ##\kappa _T## to be known in each phase in the vicinty of the states of interest, find the latent heat at the point ##P_0+p##, ##T_0+t##.


    2. Relevant equations
    ##l=T_{\text{transition}}(s_{\text{liq.}}-s_{\text{sol.}})##. (1)
    ##c_p=\left ( \frac{\partial s}{\partial T} \right ) _P##
    ##\alpha = \frac{1}{v} \left ( \frac{\partial v}{\partial T} \right ) _P##
    ##\kappa _T =-\frac{1}{v} \left ( \frac{\partial v}{\partial P} \right )##
    3. The attempt at a solution
    So I want to use equation (1). My idea is to expression the entropies of the liquid and solid phase as functions of alpha, kappa, etc.
    So I've thought of s as a function of P and T to start with. I then took the total differential of s to reach ##ds=-\alpha v dP+c_p dT##. I've tried to make appear ##\kappa _T## without any success.
    That's basically where I've been stuck for the last days, almost a week now. My friend told me to make a Taylor's expansion for the latent heat, so I've written down ##l(T) \approx l(T_0)+ l'(T_0)(T-T_0)## though I don't think that's correct since the latent heat depends on both the temperature and entropy; not the temperature alone.
    So I'm looking for getting the ##\Delta s##. I've ran out of ideas, any tip is welcome.
     
  2. jcsd
  3. Jun 10, 2013 #2
    Check out the derivation of the Clapeyron equation for vapor-liquid equilibrium. The derivation for going from solid to liquid should parallel this.
     
  4. Jun 11, 2013 #3

    fluidistic

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    I've solved the problem via a Taylor's expansion, the derivation up to the solution is rather lengthy in latex. This differs quite a lot from the 2 derivations given in wikipedia of the Clausius-Clapeyron's relation.
    So I'll write final answer that I got:
    [tex]l(T_0+t)\approx l_0 + \frac{tl_0}{T_0}+t(c_P^{\text{liq}}-c_P^{\text{sol.}})-T_0p(\alpha ^{\text{liq.}}v ^{\text{liq.}}-\alpha ^{\text{sol.}}v ^{\text{sol.}})[/tex]
    . I've found no dependence on ##\kappa _T##.
     
  5. Jun 11, 2013 #4
    You're right. The Clapeyron equation is not the way to go. I've solved this problem starting with general equation for the differential change in enthalpy [itex]dH=C_pdT+V(1-Tα)dP[/itex] in conjunction with the Clapeyron equation, and have confirmed your result. Very nice job.
     
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