At 20.0°C the vapour pressure of pure methanol is 11.8 kPa and that of pure ethanol is 5.93 kPa. A gaseous mixture of methanol and ethanol is cooled and compressed until the first drops of liquid form at 20.0°C and a total pressure of 10.0 kPa. Assuming the liquid mixture behaves ideally, calculate the mole fractions of methanol in:
a) In liquid phase
b) Gas phase
To calculate the mole fractions of methanol in the liquid and gas phases, we need to use Raoult's law, which states that the partial pressure of a component in a mixture is equal to the product of its mole fraction and its vapor pressure at that temperature.
a) Mole fraction of methanol in the liquid phase:
According to Raoult's law, the partial pressure of methanol in the liquid phase is equal to its mole fraction multiplied by its vapor pressure at that temperature. Therefore, we have the equation:
Partial pressure of methanol in liquid = mole fraction of methanol in liquid x vapor pressure of methanol
Since the liquid phase is at its boiling point, the partial pressure of methanol in the liquid phase is equal to its vapor pressure at that temperature (11.8 kPa). Hence:
11.8 kPa = mole fraction of methanol in liquid x 11.8 kPa
Simplifying the equation, the mole fraction of methanol in the liquid phase is:
Mole fraction of methanol in liquid = 11.8 kPa / 11.8 kPa = 1
Therefore, the mole fraction of methanol in the liquid phase is 1.
b) Mole fraction of methanol in the gas phase:
Since the total pressure of the system is 10.0 kPa and the partial pressure of methanol is 11.8 kPa, we can calculate the mole fraction of methanol in the gas phase using the equation:
Mole fraction of methanol in gas = partial pressure of methanol in the gas / total pressure
Plugging in the values, we get:
Mole fraction of methanol in gas = 11.8 kPa / 10.0 kPa = 1.18
Therefore, the mole fraction of methanol in the gas phase is 1.18.
Note: Since the gas phase mole fraction of methanol is greater than 1, it suggests that the liquid mixture is not at equilibrium and some additional factor may be affecting the behavior of the system.