Applying Raoult's Law to a condensed gas mixture

Question

So, here's the question which I'm having trouble solving :

Suppose that we condense a gas mixture. At 25 celsius, what will be the pressure of benzene and of toluene over the obtained liquid.

We know that vapor pressures of pure benzene and pure toluene at 25 celsius are 12,7 kPa and 3,79 kPa respectively. We also previously prepared a solution that has equal molar fraction of benzene and toluene. We had to determine the vapor pression of benzene and toluene. (this is an ideal solution)

This gave us: $$p_{benz}=x_{benz} * p^°_{benz}= 0.500 * 12.7 \text{ kPa} = 6,35 \text{ kPa}$$

$$P_{tol}=x_{tol} * p^°_{tol}= 0.500 * 3.79 = 1.90 \text{ kPa}$$ Total pressure: $P = 6.35 + 1.90 = 8.25 \text{kPa}$

Finally, there was another question which asked us what was the composition, in molar fraction, of the vapor in balance with the solution of benzene and toluene at 25 celsius.

This gave us $$x_{benz} =p_{benz} /P_{total} = 6.35 / 8.25= 0.770$$

$$x_{tol} =p_{tol} /P_{total} = 1.90 / 8.25 = 0.230$$

By the way, here's the answer to my question :

$$p_{tol} = x_{tol} * p^°_{tol} = 0,770 × 3,79 = 2,91 \text{ kPa}$$

$$p_{benz} = x_{benz} * p^°_{tol} = 0,230 × 12,7 = 2,92 \text{ kPa}$$

So, where does the Xtol= 0.770 and Xbenz=0.230 come from ? Thank you

Answer 1

You have confused the vapor-phase mole fraction with the liquid-phase mole fraction. The liquid-phase is mole fraction in usually denoted as $x_i$ for compound $i$. The vapor-phase mole fraction of $i$ is $y_i$.

Raoult's law is that the partial pressure over an ideal solution is equal to the pure-component vapor pressure times the liquid mole fraction, $p_i=x_i*p^°_i$.

Separate from Raoult's law we have the ideal gas law. It says that that says that partial pressures are proportional to mole fraction in the vapor phase, i.e. that $p_i=y_iP$. So when you took the ratios of $p_{benz}/P$ and $p_{tol}/P$, you were solving for the vapor-phase mole fractions, i.e. for $y_{benz}$ and $y_{tol}$.

Your mistake comes in taking those $y_i$ values and treating them as $x_i$ values. That's a mis-formulation of Raoult's law.

You can read more about Raoult's law and the ideal gas law for binary ideal mixtures in my answer to a previous chem.se question.