A solution of methanol and water has a vapour pressure of 213 Torr. What would you predict as the mole fractions of each component, assuming ideal behaviour? The vapour pressures of methanol and water are 256 Torr and 55.3 Torr, respectively
By the way those are estimates. You should recalculate to get more exact numbers.
To predict the mole fractions of methanol and water in the solution, we can use Raoult's Law, which states that the partial pressure of a component in a solution is directly proportional to its mole fraction.
Raoult's Law:
P_total = P_methanol * X_methanol + P_water * X_water
Given:
P_total = 213 Torr
P_methanol = 256 Torr
P_water = 55.3 Torr
Let's assume X_methanol as the mole fraction of methanol and X_water as the mole fraction of water.
We can rearrange Raoult's Law for each component:
213 Torr = 256 Torr * X_methanol + 55.3 Torr * X_water
Now, we have two unknowns and only one equation. To solve this, we need another equation. The mole fractions must add up to 1, so we have a second equation:
X_methanol + X_water = 1
We can solve this system of equations to determine the mole fractions.
Let's substitute X_water in the first equation:
213 Torr = 256 Torr * X_methanol + 55.3 Torr * (1 - X_methanol)
Simplifying, we get:
213 Torr = 256 Torr * X_methanol + 55.3 Torr - 55.3 Torr * X_methanol
Rearranging the equation:
256 Torr * X_methanol - 55.3 Torr * X_methanol = 55.3 Torr - 213 Torr
Combining like terms:
200.7 Torr * X_methanol = -157.7 Torr
Finally, solving for X_methanol:
X_methanol = (-157.7 Torr) / (200.7 Torr)
X_methanol ≈ -0.785
Considering mole fractions cannot be negative, this implies an error in the calculation or setup. Please review the given values and equations to verify the information provided.
To predict the mole fractions of methanol and water in the solution, we can use Raoult's law, which states that the partial pressure of a component in a solution is equal to the product of its mole fraction and its vapor pressure.
Let's assume the mole fraction of methanol in the solution is xM, and the mole fraction of water is xW.
According to Raoult's law, the vapor pressure of methanol in the solution (PtotalMethanol) would be given by:
PtotalMethanol = xM * Pmethanol
Similarly, the vapor pressure of water in the solution (PtotalWater) would be given by:
PtotalWater = xW * Pwater
Given that the total vapor pressure of the solution is 213 Torr, we can write the following equation:
PtotalMethanol + PtotalWater = 213 Torr
Substituting the values of the known vapor pressures, we have:
(xM * Pmethanol) + (xW * Pwater) = 213 Torr
Pmethanol = 256 Torr is the vapor pressure of pure methanol
Pwater = 55.3 Torr is the vapor pressure of pure water
Now, let's solve the equation to find the mole fractions:
xM * 256 Torr + xW * 55.3 Torr = 213 Torr
Simplifying the equation, we get:
256xM + 55.3xW = 213
This equation represents a constraint on the mole fractions of the components.
However, we have one more constraint: the sum of the mole fractions should be equal to 1:
xM + xW = 1
Now we have a system of two equations, and we can solve it to find the values of xM and xW.
Using these two equations together, we can solve for xM and xW simultaneously.
Difference in p of solvents = 256-55.3 = 200.7
(256-213)/200.7 = about 0.2
(213-55.3)/200.7 = about 0.8
So the 0.2 is closer to the Xmethanol.
The 0.8 must be XH2O