Cases when Raoult's law is unsolvable

Question

Setup

Consider a closed binary mixture of known total molar composition $n_i$, held at volume $V$ and temperature $T$. The equilibrium phase composition is determined by $\mu_i^{vap} = \mu_i^{liq}$. Assume the vapor is an ideal gas mixture, the liquid is an ideal mixture, Poynting's correction is negligible, and $V_{liq} \ll V$. Then Raoult's law holds:

$$y_i P = x_i P_i^{sat}(T)\quad \tag{1}\label{1}$$

Simplify

For convenience, denote vapor quantity = $v_i$, liquid quantity = $l_i$, total quantity = $n_i=v_i+l_i$, and "saturation quantity" = $s_i=\frac{P_i^{sat}(T)\,V}{R T}$. A closed system means $n_i$ are constant, and $s_i$ are constant at constant $T$. For simplicity, choose units such that $R T = V = \sum_i n_i=1$, so that Raoult's law becomes

$$ n_i - l_i = s_i \frac {l_i}{\sum_i l_i} \tag{2}\label{2}$$

where $l_i$ are the only independent variables. (The solution/result is independent of these choices. It holds for any Raoult's law system at constant $N, V, T$. It generalizes to infinite systems with fixed $\sum n_i/V$.)

Question

Eq. ($\ref{2}$) can be algebraically solved. But it has no solution when $s_1>1$ and $s_1 - n_1 ≥ n_2 \frac{s_1}{s_2}$. For example, if $s_1=2$ and $s_2=1/2$, then no solution exists when $n_2≤1/3$ ($x_2=y_1$ and $y_2=x_1$ at $\lim n_2\to \frac{1}{3}^+$).

This seems to predict that any generic species can form an uncondensable mixture if one is uncondensable (as a pure component), and there are few enough of the condensable species. Is this correct?

Answer 1

As you have not provided a source for your definition of $s_i$, I shall redefine it in terms of Raoult's Law and the Ideal Gas Law. To do this, we shall first take Raoult's Law and solve for $P$.

$$\begin{align}y_iP&=x_iP_i^{sat}(T)\\P&=\frac{x_iP_i^{sat}(T)}{y_i}\end{align}$$

we shall then take this form, and plug it into the ideal gas law, considering only the gas phase of the equilibrium system. Note that because the system is at equilibrium, $P_{bulk}=P_{liq}=P_{vap}$ and $T_{bulk}=T_{liq}=T_{vap}$. If the volume of the liquid phase is assumed to be significantly lower than that of the vapor, then $V_{bluk}=V_{vap}$. After plugging in this, we will then work towards the form of $s_i$ as shown.

$$\begin{align}P_{vap}V_{vap}&=n_{vap}RT_{vap}\\PV&=n_{vap}RT\\n_{vap}&=\frac{PV}{RT}\\n_{vap}&=\frac{x_iP_i^{sat}(T)}{y_i}\frac{V}{RT}\\\frac{n_{vap}y_i}{x_i}&=\frac{P_i^{sat}(T)V}{RT}\end{align}$$

From this, and the definition of $s_i$ as you stated, we can show that

$$s_i=\frac{P_i^{sat}(T)V}{RT}=\frac{n_{vap}y_i}{x_i}$$

which is subject only to the reasonable assumptions of ideality as stated above, and the bounds per the definition of each term. Namely, $\left\{P_i^{sat}(T),V,T\right\}$ must be positive, real numbers and that $\left\{x_i,y_i\right\}$ must be positive, real numbers with bounds of $\left\lbrack0,1\right\rbrack$.

No further inequalities are required to describe the system.

To solve such a system, you would then set up a system of equations and numerically backsolve for an answer, in terms $n_{vap},x_{i},y_{i}$. For example you might feed a set of equations the seed values of $n_{vap}=1$ and $x_i=.5=y_i$ such as seen below, then optimize your values to minimize the error between the calculated and target values.

The above seed values could then optimize to a particular solution that looks like. Note that this does not represent the whole of the solution space.

If you wish to further control where your particular solution ends up, you can restrict the solution space by hold one of the values to be a known constant. Below is one such example as described per your comments.

In this case, it is not possible to find a reasonable solution, as $n_{vap}=0.25~\text{mol}$ lies outside the solution space for this system of equations. Without going too heavily into the derivation, it can generally be stated that this value is a bit too low to garuntee that you will be able to find a solution. This more says that if you had this system, it would likely evolve off more of its liquid into the gas phase at in order to reach equilibrium than it says the system could not exist, though yours is not an unreasonable reading of the result.

A good rule of thumb for this would be that you can expect to find a reasonable solution for $\left\{x_i,y_i\right\}$ for any system in which $s_1 \leq n_{vap} \leq s_2$ holds, however this is by far the most complicated way to solve out this system of equations. A simpler approach to this would be to hold either $x_i$ or $y_i$ constant to restrict the solution space, as any constant value on the bounds of $\left\lbrack0,1\right\rbrack$, as already assumed will garuntee a unique solution to the system of equations.