The normal boiling point of methanol is 64.7°C. A solution containing a nonvolatile solute dissolved in methanol has a vapor pressure of 642.3 torr at 64.7°C. What is the mole fraction of methanol in this solution?
The vapor pressure of pure methanol is 760 mm at this temperature.
psolution = Xmethanol*Po
642.3 = Xmethanol*760
Then Xmethanol + Xsolute = 1
Solve for Xsolute.
To determine the mole fraction of methanol 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 vapor pressure of the pure component multiplied by its mole fraction.
In this case, since the solute is nonvolatile, the total vapor pressure of the solution is solely due to methanol. Therefore, we can consider the vapor pressure of the solution to be the same as the vapor pressure of pure methanol at its normal boiling point.
Given:
Vapor pressure of the solution (P) = 642.3 torr
Vapor pressure of pure methanol (P°) = 64.7°C (which corresponds to atmospheric pressure)
Using Raoult's law, we can write the equation as:
P = Xmethanol * P°
Rearranging the equation to solve for the mole fraction of methanol (Xmethanol):
Xmethanol = P / P°
Substituting the values:
Xmethanol = 642.3 torr / 64.7 torr
Calculating:
Xmethanol = 9.92
Therefore, the mole fraction of methanol in this solution is approximately 9.92.
To find the mole fraction of methanol in the given solution, we need to use Raoult's law, which states that the vapor pressure of a solution is directly proportional to the mole fraction of the solvent.
Raoult's law equation is:
P = P0solvent * Xsolvent
where P is the vapor pressure, P0solvent is the vapor pressure of the pure solvent, and Xsolvent is the mole fraction of the solvent.
In this case, we are given P = 642.3 torr, which is the vapor pressure of the solution, and P0solvent = 64.7°C, which is the normal boiling point of methanol. We need to convert this to torr.
1 torr = 1 mmHg
1 atm = 760 mmHg
To convert from °C to K, we use the equation:
K = °C + 273.15
Therefore, the normal boiling point of methanol in K is:
T0solvent = 64.7 + 273.15 = 337.85 K
Now, we can rewrite Raoult's law equation as:
P = (P0solvent / 760) * Xsolvent
Let's rearrange the equation to solve for Xsolvent:
Xsolvent = (P * 760) / P0solvent
Plugging in the given values:
Xsolvent = (642.3 * 760) / 337.85
Now we can calculate the mole fraction of methanol in the solution:
Xsolvent = 144807.72 / 337.85
Finally, calculate the mole fraction:
Xsolvent = 428.81
Therefore, the mole fraction of methanol in this solution is 0.42881.