Question

In the Dumas-bulb technique for determining the molar mass of an unknown liquid, you vaporize the sample of a liquid that boils below \(100^{\circ} \mathrm{C}\) in a boiling-water bath and determine the mass of vapor required to fill the bulb. From the following data, calculate the molar mass of the unknown liquid: mass of unknown vapor, \(1.012 \mathrm{~g}\); volume of bulb, \(354 \mathrm{~cm}^{3}\); pressure, 742 torr; temperature, \(99^{\circ} \mathrm{C}\).

Short answer

The molar mass of the unknown liquid is approximately \(27.8\,\mathrm{g/mol}\).

Step-by-step solution

Convert units

Convert the given data to the appropriate SI units: - Volume: \(354 \mathrm{~cm}^{3}\) to \(m^{3}\) - Pressure: \({742 \mathrm{~torr}}\) to \({Pa}\) - Temperature: \(99^{\circ} \mathrm{C}\) to \(K\) Volume: \(1 \mathrm{~cm}^{3} = 1\times10^{-6} \mathrm{~m}^{3}\) So, \(V = 354 \mathrm{~cm}^{3} \times 1\times10^{-6} \mathrm{~m}^{3}/\mathrm{~cm}^{3} = 3.54\times10^{-4} \mathrm{~m}^{3}\) Pressure: \(1 \mathrm{~torr} = 133.322 \mathrm{~Pa}\) So, \(P = 742 \mathrm{~torr} \times 133.322 \mathrm{~Pa}/\mathrm{~torr} = 98884.6 \mathrm{~Pa}\) Temperature: \(T = 99^{\circ}\mathrm{C} + 273.15 = 372.15\,\mathrm{K}\)

Find the number of moles using Ideal Gas Law

Use the Ideal Gas Law equation \(PV = nRT\) to find the number of moles, where: - R = \(8.314 \mathrm{~J/ (mol\cdot K)}\) - Mass of unknown vapor = \(1.012 \mathrm{~g}\) \(n = \frac{PV}{RT}\) Substitute with the converted units: \(n = \frac{98884.6\,\mathrm{Pa} \times 3.54\times10^{-4}\,\mathrm{m}^{3}}{8.314\,\mathrm{J/(mol\cdot K)}\times 372.15\,\mathrm{K}}\) \(n \approx 0.0364 \,\mathrm{mol}\)

Calculate the molar mass of the unknown liquid

Use the formula: Molar mass = \(\frac{\text{Mass of unknown vapor}}{\text{Number of moles}}\) Molar mass = \(\frac{1.012\,\mathrm{g}}{0.0364\,\mathrm{mol}}\) Molar mass \(\approx 27.8\,\mathrm{g/mol}\) The molar mass of the unknown liquid is approximately \(27.8\,\mathrm{g/mol}\).

Key concepts

Dumas-bulb Technique

The Dumas-bulb technique is an established method used in chemistry for determining the molar mass of a liquid with a boiling point below 100°C. During this process, a liquid sample is vaporized in a bulb that's immersed in a boiling water bath. As the liquid turns into gas, it displaces the air in the bulb until it is filled with the vapor of the liquid alone. By measuring the mass of the vapor that fills the bulb and using the Ideal Gas Law, the molar mass can be calculated.

To ensure accuracy, the experimenter must carefully control the temperature and pressure within the bulb and ensure that the entire liquid has vaporized, without any air leaks. This method is particularly useful for substances that decompose at higher temperatures or are sensitive to the presence of air.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure (P), volume (V), temperature (T), and number of moles (n) of a gas through the equation PV=nRT, where R represents the ideal gas constant.

It is based on the assumption that the gas being measured behaves ideally, meaning the particles of the gas do not attract or repel each other and occupy no volume themselves. While no gas is truly ideal, many gases behave closely enough to ideal that they can be measured with the Ideal Gas Law under certain conditions, particularly at high temperatures and low pressures. This law is critical for the Dumas-bulb method as it allows us to determine the number of moles of gas present once we've measured the other variables.

Vaporization of Liquids

Vaporization of liquids is the process by which a liquid substance turns into a gas or vapor. This transition occurs when the molecules in a liquid gain enough energy to overcome intermolecular forces and enter the gaseous phase.

Vaporization can occur in two forms: boiling and evaporation. Boiling happens at a specific boiling point where the vapor pressure of the liquid equals the external pressure, causing the liquid to form bubbles throughout and vaporize rapidly. Evaporation, on the other hand, can occur at any temperature and typically happens at the surface of the liquid. In the context of the Dumas-bulb technique, vaporization is key to converting the liquid into a measurable gas.

Unit Conversion

Unit conversion is a critical element in scientific measurements and calculations, enabling precise communication and understanding of the quantities involved. Since science typically utilizes the International System of Units (SI), it's often necessary to convert given measurements into these standard units before proceeding with any computations.

The conversions from centimeters to meters for volume, torr to pascals for pressure, and degrees Celsius to Kelvin for temperature are essential to apply the Ideal Gas Law correctly because the constants used in the equation are based on these SI units. This step is crucial in the Dumas-bulb technique to calculate the molar mass accurately. In chemistry, even small discrepancies due to improper unit conversion can significantly impact the results of an experiment.