Question

In the \(19^{\text {th }}\) century, \(\mathrm{J}\). \(\mathrm{B}\). A. Dumas devised a method for finding the molar mass of a volatile liquid from the volume, temperature, pressure, and mass of its vapor. He placed a sample of such a liquid in a flask that was closed with a stopper fitted with a narrow tube, immersed the flask in a hot water bath to vaporize the liquid, and then cooled the flask. Find the molar mass of a volatile liquid from the following: Mass of empty flask \(=65.347 \mathrm{~g}\) Mass of flask filled with water at \(25^{\circ} \mathrm{C}=327.4 \mathrm{~g}\) Density of water at \(25^{\circ} \mathrm{C}=\) \(0.997 \mathrm{~g} / \mathrm{mL}\) Mass of flask plus condensed unknown liquid \(=65.739 \mathrm{~g}\) Barometric pressure \(=101.2 \mathrm{kPa}\) Temperature of water bath \(=99.8^{\circ} \mathrm{C}\)

Short answer

The molar mass of the volatile liquid is approximately 45.85 g/mol.

Step-by-step solution

Calculate the volume of the flask

First, use the mass of the flask filled with water and the density of water to find the volume of the flask. The mass of the water alone can be found by subtracting the mass of the empty flask from the mass of the flask filled with water.Mass of the water = 327.4 g - 65.347 g = 262.053 gUsing the density of water (0.997 g/mL), find the volume:Volume = Mass / DensityVolume = 262.053 g / 0.997 g/mL = 262.83 mL

Determine the mass of the vapor

Next, find the mass of the unknown liquid by subtracting the mass of the empty flask from the mass of the flask plus the condensed unknown liquid.Mass of the unknown liquid = 65.739 g - 65.347 g = 0.392 g

Convert temperature to Kelvin

Convert the temperature of the water bath to Kelvin since ideal gas calculations require temperature in Kelvin.Temperature (K) = 99.8°C + 273.15 = 372.95 K

Apply the Ideal Gas Law to find moles of vapor

Use the ideal gas law to find the number of moles of the vapor. Calculate the volume in liters by converting from mL to L (262.83 mL = 0.26283 L). The pressure must also be in Pascals (1 kPa = 1000 Pa), so convert 101.2 kPa to Pa.Pressure (Pa) = 101.2 kPa * 1000 = 101200 PaIdeal Gas Law: PV = nRTRearrange to solve for moles (n): n = PV / RTUse R = 8.314 J/(mol·K)n = (101200 Pa) * (0.26283 L) / (8.314 J/(mol·K) * 372.95 K)n ≈ 0.00855 moles

Calculate the molar mass

Finally, calculate the molar mass by dividing the mass of the vapor by the number of moles.Molar mass = Mass / MolesMolar mass = 0.392 g / 0.00855 moles ≈ 45.85 g/mol

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics. It describes the behavior of an ideal gas under various conditions of pressure, volume, and temperature. The formula for the Ideal Gas Law is \(PV = nRT\), where:

• \(P\) stands for pressure measured in Pascals (Pa).
• \(V\) is the volume measured in liters (L).
• \(n\) represents the number of moles of the gas.
• \(R\) is the universal gas constant, approximately \(8.314 J/(mol·K)\).
• \(T\) is the temperature in Kelvin (K).

In the given exercise, we use the Ideal Gas Law to determine the number of moles of vaporized volatile liquid. By rearranging the equation to \(n = \frac{PV}{RT}\), we can solve for \(n\) using the provided pressure, volume, and temperature.

Dumas Method

The Dumas Method, developed in the 19th century by J. B. A. Dumas, is used to determine the molar mass of a volatile liquid. This method involves the following steps:

• A known mass of the volatile liquid is placed in a flask.
• The flask is heated in a water bath until the liquid vaporizes completely.
• The flask is then cooled to condense the vapor back into liquid form.
• By measuring the mass of the flask before and after condensation, we can determine the mass of the vapor.

Using the Ideal Gas Law, we then calculate the number of moles of vapor and subsequently find the molar mass by dividing the mass of the vapor by the number of moles.

Volatile Liquid

A volatile liquid easily vaporizes at relatively low temperatures. This characteristic is crucial for the Dumas Method as it allows the liquid to fully vaporize when heated in a water bath. Examples include acetone, ethanol, and ether. For this experiment:

• The liquid must be placed in a sealed flask with a narrow tube to ensure complete vaporization.
• The vaporized liquid's properties are then measured at known temperature and pressure conditions.

Understanding the nature of volatile liquids helps in accurately conducting and interpreting the results of molar mass determination experiments.

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. It helps us determine the amount of substances required or produced. In the context of the Dumas Method:

• The molar relationship between the amount of vapor and its corresponding mass is used.
• The Ideal Gas Law aids in calculating the number of moles (\(n\)) of the volatile liquid vaporized in the experiment.
• We use stoichiometry to relate these moles to the mass, thereby finding the molar mass of the volatile liquid.

Accurate stoichiometric calculations ensure precise and reliable outcomes in chemical experiments and computations.

Thermodynamics

Thermodynamics studies the relationships between heat, work, temperature, and energy. In this exercise, it is particularly relevant in the following ways:

• Heating the flask in a water bath increases the kinetic energy of the liquid molecules, causing vaporization.
• The system reaches equilibrium when the flask is maintained at the set temperature of the water bath.
• The laws of thermodynamics guide the understanding of how temperature affects the state of matter (liquid to vapor in this case).

Insight into thermodynamic principles helps elucidate the behavior of the volatile liquid during the Dumas Method experiment and ensures accurate molar mass measurements.