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Equation of State of PVT system

  1. Feb 19, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the Equation of State of a fluid with Volume expansivity ##\alpha_P## and an isothermal compresibility ##\kappa_T## are given by

    $$\alpha_P=\alpha_0 \Big(1-\frac{P}{P_0}\Big) \\ \: \: \: \: \kappa_T=\kappa_0[1+\beta_0(T-T_0)]$$

    ¿ What conditions should the constants ##\alpha_b,P_0,\kappa_0## y ##\beta_0## have for the problem to have a solution?

    2. Relevant equations

    I used this relation given in my book for ##PVT## systems

    $$\Big(\frac{\partial P}{\partial T}\Big)_V=\frac{\alpha_P}{\kappa_T}$$

    definition of isothermal compresibility ##\kappa_T##:
    $$\kappa_T=-\frac{1}{V} \Big(\frac{\partial V}{\partial P}\Big)_{T}=\kappa_0[1+\beta_0(T-T_0)]$$


    3. The attempt at a solution

    I solve the equation with constant ##V##

    $$\Big(\frac{\partial P}{\partial T}\Big)_V=\frac{\alpha_0 (1-\frac{P}{P_0})}{\kappa_0(1+\beta_0(T-T_0))}$$
    $$=\Big(\frac{\alpha_0}{\kappa_0 P_0}\Big)\frac{P_0-P}{(1+\beta_0(T-T_0))}$$

    $$\frac{d P}{P_0-P}=\Big(\frac{\alpha_0}{\kappa_0 P_0}\Big)\frac{dT}{(1+\beta_0(T-T_0))}$$
    $$-\ln(P_0-P)=\Big(\frac{\alpha_0}{\kappa_0 P_0 \beta_0}\Big) \ln(1+\beta_0(T-T_0))+f(V) \tag{5}$$

    The problem is that I find this ##f(V)## to be a function of ##T## and ##P## with the problem conditions any Help ?

    > Derivation of the problem **(Not Necesary)**
    Using the definition of isothermal compresibility ##\kappa_T##:
    $$\kappa_T=-\frac{1}{V} \Big(\frac{\partial V}{\partial P}\Big)_{T}=\kappa_0[1+\beta_0(T-T_0)]$$
    Now taking ##T=cte##
    $$\Big(\frac{1}{P_0-P}\Big) \Big(\frac{\partial P}{\partial V}\Big)_{T}=f'(V)$$
    $$\Big(\frac{1}{P_0-P}\Big)\Big(\frac{1}{f'(V)}\Big)=\Big(\frac{\partial V}{\partial P}\Big)_{T}$$
    $$\Big(\frac{1}{P_0-P}\Big)\Big(\frac{1}{f'(V)}\Big)\Big(\frac{-1}{V}\Big)=-\frac{1}{V} \Big(\frac{\partial V}{\partial P}\Big)_{T}=\kappa_0[1+\beta_0(T-T_0)]=\kappa_T$$
    $$\Big(\frac{1}{P_0-P}\Big)\Big(\frac{1}{f'(V)}\Big)\Big(\frac{-1}{V}\Big)=\kappa_0[1+\beta_0(T-T_0)]$$
    Now we obtain ##f(V)##
    \begin{align*}
    \Big(\frac{1}{P_0-P}\Big)\Big(\frac{1}{f'(V)}\Big)\Big(\frac{-1}{V}\Big)&=\kappa_0[1+\beta_0(T-T_0)]\\
    \Big(\frac{1}{P_0-P}\Big)\Big(\frac{1}{\kappa_0[1+\beta_0(T-T_0)]}\Big)\Big(\frac{-1}{V}\Big)&=f'(V)\\
    \Big(\frac{1}{P_0-P}\Big)\Big(\frac{1}{\kappa_0[1+\beta_0(T-T_0)]}\Big)-\ln(V)&=f(V)\\
    \frac{-\ln(V)}{\kappa_0(P_0-P)[1+\beta_0(T-T_0)]}&=f(V)
    \end{align*}​

    That finished, any other idea of finding the equation of state or the constants conditions ?
     
  2. jcsd
  3. Feb 25, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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