Problem 53 - Chemistry The Central Science

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 g; volume of bulb, \(354 \mathrm{cm}^{3} ;\) pressure, 742 torr; temperature, \(99^{\circ} \mathrm{C}\) .

Step-by-Step Solution

Answer

The molar mass of the unknown liquid is approximately 25.18 g/mol, calculated using the given data and the Ideal Gas Law equation.

Step 1: Convert units of given data

Before we proceed to use the Ideal Gas Law equation, we should make sure that all the data is in the correct units. The ideal gas constant (R) is given in units of L atm/mol K. So, we need to convert the volume to liters, temperature to Kelvin, and pressure to atm. 1. Volume conversion: Volume = 354 cm³ = 0.354 L (since there are 1,000 cm³ in a liter) 2. Temperature conversion: Temperature = 99°C = 99 + 273.15 = 372.15 K 3. Pressure conversion: Pressure = 742 torr = (742/760) atm ≈ 0.976 atm (since there are 760 torr in 1 atm) Now we have the required units for the Ideal Gas Law equation: Volume (V) = 0.354 L Temperature (T) = 372.15 K Pressure (P) = 0.976 atm

Step 2: Calculate the number of moles using the Ideal Gas Law equation

Now that we have all the data in the appropriate units, we can use the Ideal Gas Law equation to calculate the number of moles (n). Rearranging the Ideal Gas Law equation for n: \(n = \frac{PV}{RT}\) Using the given data: n = (0.976 atm) * (0.354 L) / (0.0821 L atm/mol K) * (372.15 K) n ≈ 0.0402 moles

Step 3: Calculate the molar mass of the unknown liquid

Now that we have the number of moles of the unknown vapor, we can calculate the molar mass of the unknown liquid using the formula: Molar mass = mass of unknown vapor/number of moles Molar mass = (1.012 g) / (0.0402 moles) Molar mass ≈ 25.18 g/mol The molar mass of the unknown liquid is approximately 25.18 g/mol.

Key Concepts

Molar Mass Calculation

Calculating the molar mass of a substance is a crucial step in many chemistry experiments, including the Dumas-bulb technique. The molar mass is essentially the mass of one mole of a particular substance, usually expressed in grams per mole (g/mol). In this exercise, we determine the molar mass of an unknown liquid vaporized in a bulb.
To find the molar mass, we use the formula:

• Molar mass = \( \frac{\text{mass of vapor}}{\text{number of moles}} \)

First, we need to calculate the number of moles of the gaseous sample using the ideal gas law equation (PV = nRT). Once we determine how many moles of the gas fill the bulb, we divide the given mass of the vapor by the number of moles. This provides us with the molar mass of the unknown liquid.
This step is key, as it helps identify the liquid by comparing the calculated molar mass with known values from reference materials or databases.

Unit Conversion

Unit conversion is a vital aspect of working with scientific data, as consistent units are necessary for accurate calculations. In this particular exercise, we must ensure that all measurements are in the correct units to employ the ideal gas law correctly. Let's go through each necessary conversion:

• Volume: The original volume of the bulb was given in cubic centimeters (cm³), which we converted to liters (L). Since there are 1,000 cm³ in a liter, the conversion is straightforward: \( 354 \text{ cm}^3 = 0.354 \text{ L} \).
• Temperature: Originally provided in degrees Celsius (°C), temperature needs converting to Kelvin (K). This is done using the formula \( T(\text{K}) = T(°\text{C}) + 273.15 \). Thus, \( 99°\text{C} = 372.15 \text{ K} \).
• Pressure: Given in torr, pressure needs to be in atmospheres (atm) for use in the ideal gas law. Since 1 atm = 760 torr, the conversion is \( 742 \text{ torr} \approx 0.976 \text{ atm} \).

Properly converting these units ensures that when we apply the ideal gas law, the calculations will yield correct and reliable results.

Dumas-bulb Technique

The Dumas-bulb technique is an elegant method used to determine the molar mass of an unknown liquid. Named after French chemist Jean-Baptiste Dumas, it involves vaporizing a liquid in a specialized glass bulb, allowing the vapor to occupy a known volume at a given temperature and pressure.
This technique relies on the principles of the ideal gas law, which relates pressure (P), volume (V), and temperature (T) to the amount of gas in moles (n) using the equation \(PV = nRT\). To find the molar mass of the unknown liquid, the experimenter follows these steps:

• The liquid is vaporized, causing it to fill the bulb completely, ensuring no extra air is present inside.
• The mass of the vapor is carefully measured after condensation.
• Using the determined mass and calculated moles of gas, the molar mass is computed.

The Dumas-bulb technique is widely recognized for its simplicity and effectiveness, making it a popular choice in laboratories where sophisticated equipment might not be available.