Equation of State of PVT system

1. Feb 19, 2015

KeithPhysics

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. Feb 25, 2015