Problem 1 - Chemical Principles in the Laboratory

Question

A student weighs an empty flask and stopper and finds the mass to be \(53.256 \mathrm{g}\). She then adds about \(5 \mathrm{mL}\) of an unknown liquid and heats the flask in a boiling water bath at \(98.8^{\circ} \mathrm{C}\). After all the liquid is vaporized, she removes the flask from the bath, stoppers it, and lets it cool. After it is cool, she momentarily removes the stopper, then replaces it and weighs the flask and condensed vapor, obtaining a mass of \(53.780 \mathrm{g}\). The volume of the flask is known to be \(231.1 \mathrm{mL}\). The absolute atmospheric pressure in the laboratory that day is \(728 \mathrm{mm}\) Hg. a. What was the pressure of the vapor in the flask in atm? b. What was the temperature of the vapor in \(\mathrm{K}\) ? the volume of the flask in liters? c. What was the mass of condensed vapor that was present in the flask? d. How many moles of condensed vapor were present? e. What is the mass of one mole of vapor (Eq. 2 )?

Step-by-Step Solution

Answer

Pressure: 0.958 atm. Mass of vapor: 0.524 g. Moles of vapor: 0.00726 mol. Molar mass: 72.16 g/mol.

Step 1: Convert Pressure to atm

The atmospheric pressure is given as \(728\) mm Hg. To convert this to atmospheres, use the conversion factor: \(1 \text{ atm} = 760 \text{ mm Hg}\).\[\text{Pressure in atm} = \frac{728}{760} \approx 0.958 \text{ atm}\]

Step 2: Convert Temperature to Kelvin

The boiling point is given as \(98.8^{\circ} \mathrm{C}\). Convert this temperature to Kelvin using the formula:\[T(K) = T(^{\circ}C) + 273.15\]\[T(K) = 98.8 + 273.15 = 371.95 \text{ K}\]

Step 3: Convert Volume to Liters

The volume of the flask is given as \(231.1\) mL. Convert this volume to liters by dividing by \(1000\):\[V(L) = \frac{231.1}{1000} = 0.2311 \text{ L}\]

Step 4: Find Mass of Condensed Vapor

To find the mass of the condensed vapor, subtract the mass of the empty flask from the mass of the flask with the condensed vapor:\[\text{Mass of condensed vapor} = 53.780 \text{ g} - 53.256 \text{ g} = 0.524 \text{ g}\]

Step 5: Calculate Moles of Condensed Vapor

Use the Ideal Gas Law formula, \( PV = nRT \), to solve for \(n\) (moles of gas), where \(R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \):\[n = \frac{PV}{RT} = \frac{0.958 \times 0.2311}{0.0821 \times 371.95}\approx 0.00726 \text{ moles}\]

Step 6: Calculate Molar Mass of the Vapor

The molar mass can be found by dividing the mass of the vapor by the moles:\[\text{Molar mass} = \frac{0.524 \text{ g}}{0.00726 \text{ mol}} \approx 72.16 \text{ g/mol}\]

Key Concepts

Pressure Conversion

When dealing with gases, it's important to understand how to properly convert pressure between units. The Ideal Gas Law often requires pressure input in atmospheres (atm), but lab measurements may give pressure in other units like millimeters of mercury (mmHg). To convert from mmHg to atm, use the conversion factor:

• 1 atm = 760 mmHg

For example, if the atmospheric pressure in your exercise problem is 728 mmHg, then the pressure in atmospheres can be calculated by dividing the mmHg value by 760. \[ P = \frac{728}{760} \approx 0.958 \, \text{atm} \] This conversion is crucial for using the Ideal Gas Law correctly, as it relies on consistent units.

Temperature Conversion

In scientific calculations, temperature should always be in Kelvin for use in the Ideal Gas Law. Celsius is common in everyday life, but Kelvin is a starting point for scientific computations. The formula to convert Celsius to Kelvin is:

• \( T(K) = T(^{\circ}C) + 273.15 \)

For instance, if the boiling point in the experiment is given as 98.8°C, convert it to Kelvin:\[ 98.8 + 273.15 = 371.95 \, \mathrm{K} \]This conversion ensures that the temperature scale starts from absolute zero, which is essential for calculations involving gases.

Molar Mass Calculation

Calculating the molar mass of a substance is often necessary to understand its properties and behavior in chemical reactions. In gas calculations, once you know the mass of the condensed vapor and the number of moles, the molar mass can be determined using:

• \( \text{Molar Mass} = \frac{\text{Mass of Vapor}}{\text{Moles of Vapor}} \)

For the problem at hand, if the mass of the condensed vapor is 0.524 g and the moles of vapor are 0.00726 mol, the molar mass can be calculated as:\[ \text{Molar Mass} = \frac{0.524 \, \text{g}}{0.00726 \, \text{mol}} \approx 72.16 \, \text{g/mol} \] Having a good grasp of molar mass calculation aids in determining the identity and quantity of substances present.

Volume Conversion

Volume conversion is often a necessary step in gas-related calculations. The most common conversion needed is from milliliters (mL) to liters (L) because the Ideal Gas Law uses liters. To convert mL to L, remember:

• There are 1000 milliliters in a liter.
• So, divide the volume in mL by 1000 to get the volume in L.

For example, if a flask's volume is given as 231.1 mL, convert this to liters by calculating:\[ V = \frac{231.1}{1000} = 0.2311 \, \text{L} \]Keeping track of units and correctly converting volume ensures accurate and meaningful results in computations.