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 (see drawing). 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 \(80.84 \, \frac{\textit{g}}{\textit{mol}}\).

Step-by-step solution

Convert given values to appropriate units

First, we need to convert the given values to appropriate units to be able to use them in the Ideal Gas Law equation. We will convert temperature to Kelvin, pressure to atm and volume to liters. Temperature: \(T = 99^{\circ} C + 273.15 = 372.15 K\) Pressure: \(P = 742 \, \textit{torr} \times \frac{1 \, \textit{atm}}{760 \, \textit{torr}} = 0.976 \, \textit{atm}\) Volume: \(V = 354 \, \textit{cm}^3 \times \frac{1 \, \textit{L}}{1000 \, \textit{cm}^3} = 0.354 \, \textit{L}\)

Apply the Ideal Gas Law equation

Now, we will apply the Ideal Gas Law equation to calculate the amount of moles in the vapor sample. Ideal Gas Law Equation: \(PV = nRT\) Where: P is pressure in atm, V is volume in liters, n is the number of moles, R is the Ideal Gas Constant, which has a value of \(0.0821 \frac{\textit{L}\cdot\textit{atm}}{\textit{mol}\cdot\textit{K}}\), T is the temperature in Kelvin. So, we have: \(0.976 \, \textit{atm} \times 0.354 \, \textit{L} = n \times 0.0821 \frac{\textit{L}\cdot\textit{atm}}{\textit{mol}\cdot\textit{K}} \times 372.15 \, \textit{K}\)

Solve for the number of moles

From step 2, we can now solve for the number of moles (n) in the vapor sample. \(0.976 \, \textit{atm} \times 0.354 \, \textit{L} = n \times 0.0821 \frac{\textit{L}\cdot\textit{atm}}{\textit{mol}\cdot\textit{K}} \times 372.15 \, \textit{K}\) \(n = \frac{0.976 \, \textit{atm} \times 0.354 \, \textit{L}}{0.0821 \frac{\textit{L}\cdot\textit{atm}}{\textit{mol}\cdot\textit{K}} \times 372.15 \, \textit{K}} = 0.01252 \, \textit{mol}\)

Calculate the molar mass of the unknown liquid

Now that we have the number of moles and the mass of the vapor sample, we can calculate the molar mass of the unknown liquid. Molar mass = \(\frac{\textit{mass of unknown vapor}}{\textit{number of moles}}\) Molar mass = \(\frac{1.012 \, \textit{g}}{0.01252 \, \textit{mol}} = 80.84 \, \frac{\textit{g}}{\textit{mol}}\) The molar mass of the unknown liquid is approximately \(80.84 \, \frac{\textit{g}}{\textit{mol}}\).

Key concepts

Molar Mass Calculation

The calculation of molar mass is crucial when it comes to identifying an unknown substance, such as a liquid in the Dumas-bulb technique. It's the mass in grams for one mole of a substance. To find it, you need the mass of the vapor and the number of moles present in that vapor. This is a vital part of determining the substance's identity. Let's break it down:

• First, measure the mass of the unknown vapor—this is the mass of the sample in the gas form, filling the Dumas bulb.
• Next, determine the number of moles using the Ideal Gas Law and convert necessary units to be compatible for calculations.
• Finally, use the molar mass formula: Molar Mass = \( \frac{\text{mass of gas}}{\text{number of moles}} \).

In our solution, the mass of the vapor was 1.012 grams, and the amount of moles calculated was 0.01252. The resulting molar mass of the mystery liquid was around 80.84 g/mol, providing insight into its possible chemical makeup.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that describes the behavior of gases. It's given by the formula: \[ PV = nRT \] where:

• \( P \) = Pressure, measured in atmospheres (atm),
• \( V \) = Volume, measured in liters (L),
• \( n \) = Number of moles of the gas,
• \( R \) = Universal gas constant (0.0821 \( \frac{\text{L atm}}{\text{mol K}} \)),
• \( T \) = Temperature, measured in Kelvin (K).

This equation is essential for calculating the number of moles a gas occupies under specific conditions of temperature, volume, and pressure. In the exercise, by substituting the measured pressure, volume, and temperature into the Ideal Gas Law, we calculated the moles of vapor necessary to determine the liquid's molar mass. Remember, all values must be in their respective SI units for the equation to work correctly.

Unit Conversion

Correct unit conversion is indispensable when working with the Ideal Gas Law. Having units in alignment ensures accurate calculations. Here's what you need to do in this context:

• Temperature should be in Kelvin. Add 273.15 to the degrees Celsius to convert: \( T[K] = T[°C] + 273.15 \).
• Pressure must be in atmospheres (atm). Convert from torr to atm using: \( P[atm] = P[torr] \times \frac{1 \, ext{atm}}{760 \, ext{torr}} \).
• Volume should be in liters. Convert from cubic centimeters (cm³) to liters (L) using: \( V[L] = V[cm³] \times \frac{1 \, ext{L}}{1000 \, ext{cm³}} \).

These conversions ensure all values are compatible with the Ideal Gas Law. Proper unit conversion is an essential mathematical skill in chemistry and many other scientific fields.

Vaporization Process

The vaporization process converts a liquid into vapor by applying heat. In the Dumas-bulb technique, the unknown liquid is heated in a boiling water bath until it turns entirely into vapor. This process allows us to capture the vapor inside the bulb and measure its mass under controlled conditions. Some key points about vaporization include:

• It's an endothermic process—heat is absorbed by the liquid to change into gas.
• Occurs when the liquid molecules gain enough energy to overcome intermolecular attractions and enter the gas phase.
• The boiling point must be known, to ensure complete vaporization without decomposition.

By understanding the vaporization process, you can accurately determine the mass of the vapor, crucial for calculating the molar mass of the liquid using the Dumas-bulb technique. When the liquid is fully vaporized, the gas occupies the bulb, making it possible to measure the necessary parameters for further calculations.