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Chapter 13: Problem 53

The vapor pressure of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) at \(20^{\circ} \mathrm{C}\) is \(44 \mathrm{mmHg},\) and the vapor pressure of methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) at the same temperature is \(94 \mathrm{mmHg}\). A mixture of \(30.0 \mathrm{~g}\) of methanol and \(45.0 \mathrm{~g}\) of ethanol is prepared (and may be assumed to behave as an ideal solution). (a) Calculate the vapor pressure of methanol and ethanol above this solution at \(20^{\circ} \mathrm{C}\). (b) Calculate the mole fraction of methanol and ethanol in the vapor above this solution at \(20^{\circ} \mathrm{C}\).

Short Answer

Expert verified
The vapor pressure of methanol above the solution at 20°C is calculated with Raoult's law as 31.3 mmHg, while the vapor pressure of ethanol above the solution at 20°C is calculated as 12.6 mmHg. The mole fraction of methanol in the vapor phase is 0.713, and the mole fraction of ethanol is 0.287. Hence, methanol is more volatile in nature when compared to ethanol in the given ideal solution.

Step by step solution

01

Convert Mass to Moles

Starting with the mass of methanol and ethanol, the amounts in moles will need to be determined. Using the formula: \[\text{{Moles}} = \frac{{\text{{Mass}}}}{{\text{{Molar mass}}}}\]The molar mass of Methanol (\(\mathrm{CH}_{3} \mathrm{OH}\)) is 32.04 g/mol and for ethanol (\(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\)) it's 46.07 g/mol. To determine the number of moles for each, the mass of each will be divided by their molar mass.
02

Calculate Mole Fractions

Next, calculate the mole fraction of methanol (\( x_{\text{{MeOH}}} \)) and ethanol (\( x_{\text{{EtOH}}} \)) in the solution. The mole fraction is calculated by dividing the moles of a component by the total moles in the solution. The total moles is the sum of the moles of methanol and ethanol.
03

Apply Raoult's Law

The vapor pressure of each component can then be calculated by substituting the mole fraction and pure vapor pressure into Raoult's law which is:\[P = x \times P^{0}\]Where:P is the partial pressure of the component,x is mole fraction of the component in the solution,and \( P^{0} \) is the vapor pressure of the pure component.
04

Calculate the Total Vapor Pressure

The total vapor pressure above the solution at 20°C can then be calculated by adding the partial pressures of methanol and ethanol.
05

Determine Mole Fractions in the Vapor Phase

For part (b), the mole fractions of methanol and ethanol in the vapor phase can be determined by dividing the partial pressure of each component by the total vapor pressure.

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