An ideal solution of 5.00 mole benzene and 3.25 mole toluene. At 298 K, the vapor pressure of the pure benzene and toluene were 96. Torr, and 28.9 Torr, respectively.
At what pressure does the vapor phase appear first, if the pressure above this solution is reduced from 760 Torr.
We can use Raoult's Law to determine the vapor pressure of each component in the solution. Raoult's Law states that the partial vapor pressure of a component in a solution is equal to the mole fraction of that component times its vapor pressure in its pure state.
First, let's find the mole fraction of each component in the solution:
Mole fraction of benzene (X_benzene) = moles of benzene / total moles in solution
X_benzene = 5.00 moles / (5.00 moles + 3.25 moles) = 5.00 / 8.25 ≈ 0.606
Mole fraction of toluene (X_toluene) = moles of toluene / total moles in solution
X_toluene = 3.25 moles / (5.00 moles + 3.25 moles) = 3.25 / 8.25 ≈ 0.394
Now we can apply Raoult's Law for each component:
Partial vapor pressure of benzene (P_benzene) = X_benzene * P_pure_benzene
P_benzene = 0.606 * 96.0 Torr ≈ 58.2 Torr
Partial vapor pressure of toluene (P_toluene) = X_toluene * P_pure_toluene
P_toluene = 0.394 * 28.9 Torr ≈ 11.4 Torr
The total vapor pressure of the solution (P_solution) is the sum of the partial vapor pressures of each component:
P_solution = P_benzene + P_toluene ≈ 58.2 Torr + 11.4 Torr ≈ 69.6 Torr
Therefore, the vapor phase will appear first when the pressure above the solution is reduced to approximately 69.6 Torr.
To determine at what pressure the vapor phase appears first, we need to calculate the vapor pressures of the individual components in the solution and compare them to the reduced overall pressure of 760 Torr.
1. Calculate the mole fractions of benzene and toluene in the solution:
The total moles of the solution = 5.00 + 3.25 = 8.25 moles
Mole fraction of benzene = moles of benzene / total moles = 5.00 / 8.25 ≈ 0.6061
Mole fraction of toluene = moles of toluene / total moles = 3.25 / 8.25 ≈ 0.3939
2. Calculate the partial pressures of benzene and toluene in the vapor phase using Raoult's law:
Partial pressure of benzene = mole fraction of benzene * vapor pressure of benzene
≈ 0.6061 * 96.0 Torr ≈ 58.31 Torr
Partial pressure of toluene = mole fraction of toluene * vapor pressure of toluene
≈ 0.3939 * 28.9 Torr ≈ 11.39 Torr
3. Add the partial pressures to determine the total pressure at which the vapor phase appears first:
Total pressure = partial pressure of benzene + partial pressure of toluene
= 58.31 Torr + 11.39 Torr
= 69.70 Torr
Therefore, the vapor phase appears first when the pressure above the solution is reduced to approximately 69.70 Torr.
To determine the pressure at which the vapor phase appears first when the pressure above the solution is reduced from 760 Torr, we need to calculate the partial pressures of benzene and toluene in the solution and compare them to their respective vapor pressures.
First, we need to calculate the mole fraction of benzene and toluene in the solution. The mole fraction (X) is the ratio of moles of a component to the total moles of all components in the solution.
Mole fraction of benzene (Xbenzene) = moles of benzene / total moles of the solution
Xbenzene = 5.00 mol / (5.00 mol + 3.25 mol) = 0.6061
Mole fraction of toluene (Xtoluene) = moles of toluene / total moles of the solution
Xtoluene = 3.25 mol / (5.00 mol + 3.25 mol) = 0.3939
Next, we can calculate the partial pressures of benzene (Pbenzene) and toluene (Ptoluene) using Raoult's law. According to Raoult's law, the partial pressure of a component in the vapor phase above an ideal solution is proportional to its mole fraction in the solution and its vapor pressure:
Pbenzene = Xbenzene * Pbenzene vapor pressure
Pbenzene = 0.6061 * 96 Torr = 58.2416 Torr
Ptoluene = Xtoluene * Ptoluene vapor pressure
Ptoluene = 0.3939 * 28.9 Torr = 11.3801 Torr
Now, we need to determine the pressure at which the vapor phase appears first. When the pressure above the solution is reduced to a certain value, if the partial pressure of either component (benzene or toluene) reaches or exceeds its vapor pressure, the vapor phase will appear.
To find the pressure at which the vapor phase appears first, we compare the partial pressures to their respective vapor pressures:
If Pbenzene <= Pbenzene vapor pressure and Ptoluene <= Ptoluene vapor pressure, the liquid phase is still present.
However, if Pbenzene > Pbenzene vapor pressure or Ptoluene > Ptoluene vapor pressure, the vapor phase appears.
Therefore, we compare the calculated partial pressures to their respective vapor pressures:
If Pbenzene > 96 Torr or Ptoluene > 28.9 Torr, the vapor phase appears.
In this case, from our calculations:
Pbenzene = 58.2416 Torr < 96 Torr
Ptoluene = 11.3801 Torr < 28.9 Torr
Since neither partial pressure exceeds its respective vapor pressure, the vapor phase does not appear yet. The pressure at which the vapor phase first appears is less than the current pressure (which is 760 Torr) but greater than the calculated partial pressures. Unfortunately, without additional information, we cannot determine the exact pressure at which the vapor phase appears first.