# Density solver for ideal mixture of real gases

I am referring the paper on GERG-2008 equation of state. I am interested in calculating the properties of ideal mixtures of real gases. The GERG-2008 EOS is in Helmholtz free energy and hence the independent variables are mixture molar density ($\rho$) and mixture temperature ($T$). But usually the total pressure and temperature of the mixture are available to me. Consequently, I want to write a density solver.

If I suppose that the mixing rules derived for ideal gases are valid for real gases as well, then I can write the total pressure of the mixture as

$p(T,\rho,\bar{x}) = \sum_{k=1}^{N}x_k p_k$

where $\bar{x}$ is the mixture composition, $N$ is the number of pure components in the mixture, $x_k$ is the mole fraction of the $k$-th component and $p_k$ is the partial pressure of the $k$-th component. Following the formula for pressure in the paper mentioned above, I can write

$p = \sum_{k=1}^{N} x_k \rho R T \left( 1 + \frac{\rho}{\rho_{c,k}} \alpha^r_{\delta,k}\ \left(\delta_k, \tau_k \right) \right)$

where the reduced density of the $k$-th component $\delta_k$ and inverse reduced temperature of the $k$-th component $\tau_k$ are given by

$\delta_k = \frac{\rho}{\rho_{c,k}}$

$\tau_k = \frac{T_{c,k}}{T}$

and $\alpha^r_{\delta,k} = \frac{\partial \alpha_{r,k}}{\partial \delta}$ is the residual Helmholtz energy, $\rho_{c,k}$ is the critical molar density and $T_{c,k}$ is the critical temperature of the $k$-th component. Now, I need to solve the above equation with an iterative solver to obtain the molar density of the mixture.

I want to confirm if the above formulation is correct. My main questions are

1. Is the mixing rule that I used valid?
2. Instead of using molar densities for evaluating $\delta_k$, should I be using mass densities i.e. $\delta_k = \frac{\rho_m}{\rho_{m,c,k}}$ ($m$ indicates mass density)? Note that this is different from the above expression. In fact, here $\delta_k = \frac{M_{mix}}{M_k} \frac{\rho}{\rho_{c,k}}$ where $M$ is molar mass.
• Possibly related: physics.stackexchange.com/q/139524/25301 – Kyle Kanos Mar 23 '16 at 0:09
• @KyleKanos In the question that you have linked, the user wants to know how to numerically solve for density given pressure and temperature. I am asking if the equation that I want to solve is correct. – Colorless Photon Mar 23 '16 at 0:11
• That question is also about a similar modified ideal gas EOS, that's why I suggested it was related to yours. – Kyle Kanos Mar 23 '16 at 2:06

• I don't see how they would give nearly the same answers. Suppose I have a mixture of carbon dioxide (CO2) and nitrogen (N2). They are in a ratio of 0.9:0.1 by mole fraction. Then, $M_{mix}$ = 42.4 If I look at the contribution of N2, then $M_{mix}/M_{N2} \approx 1.5$. The values obtained for $\delta_{N2}$ from the two expressions mentioned in the question would be very different considering similar values of $\rho$. – Colorless Photon Mar 22 '16 at 18:20
• If you look at the equation for pressure in the question, it depends on $\alpha^r_{\delta,N2}$ (considering my earlier example). This expression will yield very different values considering the two different expressions for reduced density. I want to know which one is right. Is this clearer? – Colorless Photon Mar 22 '16 at 18:58