Question

The vapor pressure of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) at \(19^{\circ} \mathrm{C}\) is \(40.0\) torr. A \(1.00\)-g sample of ethanol is placed in a \(2.00 \mathrm{~L}\) container at \(19^{\circ} \mathrm{C}\). If the container is closed and the ethanol is allowed to reach equilibrium with its vapor, how many grams of liquid ethanol remain?

Short answer

Approximately 0.368 grams of liquid ethanol remains in the container after reaching equilibrium with its vapor.

Step-by-step solution

Calculate the moles of ethanol vapor at equilibrium

To find out the moles of ethanol vapor at equilibrium, we can use the Ideal Gas Law equation: \(PV = nRT\) Here, Pressure (P) = 40.0 torr Volume (V) = 2.00 L R = Gas constant = 62.364 L.torr/mol.K (Using an appropriate value for R, which matches the given units) Temperature (T) = 19°C = 292 K Now, we can solve for 'n', the moles of ethanol vapor at equilibrium: \(n = \frac{PV}{RT}\)

Convert the given mass of ethanol to moles

We are given the initial mass of ethanol, m = 1.00 g. To express this in moles, we need to use the molecular weight of ethanol, M = 46.07 g/mol: Initial moles of ethanol = \(\frac{m}{M}\)

Calculate the moles of liquid ethanol remaining

Since the mass of ethanol is conserved between the vapor and liquid phases, the sum of the moles of vapor and liquid ethanol must equal the initial moles of ethanol: Moles of liquid ethanol remaining = Initial moles of ethanol - Moles of ethanol vapor at equilibrium

Convert moles of liquid ethanol remaining to mass

Now that we have the moles of liquid ethanol remaining, we can convert it back to mass using the molecular weight of ethanol: Mass of liquid ethanol remaining = Moles of liquid ethanol remaining * M Now, let's calculate the values in each step.

Calculate the moles of ethanol vapor at equilibrium

\(n = \frac{40.0 \text{torr} \times 2.00 \text{L}}{62.364 \frac{\text{L.torr}} {\text{mol.K}} \times 292\text{K}}\) \(n \approx 0.0137\) moles

Convert the given mass of ethanol to moles

Initial moles of ethanol = \(\frac{1.00 \text{g}}{46.07 \frac{\text{g}}{\text{mol}}}\) Initial moles of ethanol ≈ 0.0217 moles

Calculate the moles of liquid ethanol remaining

Moles of liquid ethanol remaining = 0.0217 moles - 0.0137 moles Moles of liquid ethanol remaining ≈ 0.0080 moles

Convert moles of liquid ethanol remaining to mass

Mass of liquid ethanol remaining = 0.0080 moles * 46.07 g/mol Mass of liquid ethanol remaining ≈ 0.368 g So, approximately 0.368 grams of liquid ethanol remains in the container after reaching equilibrium with its vapor.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in the study of gases that relates the pressure, volume, and temperature of a gas with the amount of substance present. The equation is given as:

\(PV = nRT\)

where \(P\) stands for the pressure of the gas, \(V\) is the volume it occupies, \(n\) is the number of moles of the gas, \(R\) is the ideal gas constant, and \(T\) is the temperature of the gas measured in Kelvin. Understanding how these variables interact is essential for predicting the behavior of gases under different conditions. In our exercise, we used this law to calculate the moles of ethanol vapor in equilibrium with its liquid at a known temperature and pressure.

Molar Mass of Ethanol

The molar mass of a substance is the weight of one mole of that substance, typically expressed in grams per mole (g/mol). For ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5}\mathrm{OH}\right)\), its molar mass is calculated based on the atomic weights of carbon, hydrogen, and oxygen. The molar mass of ethanol is approximately 46.07 g/mol. Knowing the molar mass is crucial for converting between the mass of a substance and the number of moles, which is a common procedure in many chemical calculations including the one we tackled in the ethanol vapor pressure problem.

Phase Equilibrium

Phase equilibrium refers to a state where multiple phases of a substance, like solid, liquid, and gas, exist together without any net change in the amount of each phase over time. This happens at specific temperature and pressure conditions, where the rates of transfer between phases are equal. In the context of the exercise, equilibrium is achieved between the liquid ethanol and its vapor within a sealed container. At equilibrium, the vapor pressure of a substance becomes constant, as seen with ethanol's vapor pressure of 40.0 torr at \(19^\circ\)C.

Chemical Thermodynamics

Chemical thermodynamics involves the study of energy and work relating to chemical reactions and physical transformations. It's grounded in thermodynamic laws and helps predict the direction of spontaneous processes, energy changes, and the equilibrium positions of chemical systems. In the exercise, chemical thermodynamics underlies the process of ethanol reaching vapor pressure equilibrium, where system energetics are balanced and the total number of moles of ethanol remains constant, irrespective of the phase.