Problem 29 - Chemistry and Chemical Reactivity

Question

A new boron hydride, \(\mathrm{B}_{x} \mathrm{H}_{y}\) has been isolated. To find its molar mass, you measure the pressure of the gas in a known volume at a known temperature. The following experimental data are collected: Mass of gas \(=12.5 \mathrm{mg} \quad\) Pressure of gas \(=24.8 \mathrm{mm} \mathrm{Hg}\) Temperature \(=25^{\circ} \mathrm{C} \quad\) Volume of flask \(=125 \mathrm{mL}\) Which formula corresponds to the calculated molar mass? (a) \(\mathrm{B}_{2} \mathrm{H}_{6}\) (b) \(\mathrm{B}_{4} \mathrm{H}_{10}\) (c) \(\mathrm{B}_{5} \mathrm{H}_{9}\) (d) \(\mathrm{B}_{6} \mathrm{H}_{10}\) (e) \(\mathrm{B}_{10} \mathrm{H}_{14}\)

Step-by-Step Solution

Answer

The formula is \(\mathrm{B}_6\mathrm{H}_{10}\).

Step 1: Convert pressure to atmospheres

Given pressure is 24.8 mm Hg. To convert it to atmospheres, use the conversion: \(1 \, \text{atm} = 760 \, \text{mm Hg}\). Thus, \(P = \frac{24.8}{760} \, \text{atm} \approx 0.0326 \, \text{atm}\).

Step 2: Convert volume to liters

The volume of the flask is given as 125 mL. Convert this to liters by dividing by 1000. Therefore, \(V = 125 \, \text{mL} = 0.125 \, \text{L}\).

Step 3: Convert temperature to Kelvin

Temperature is given as 25°C. Convert Celsius to Kelvin by adding 273.15. Thus, \(T = 25 + 273.15 = 298.15 \, \text{K}\).

Step 4: Calculate the number of moles using the Ideal Gas Law

Apply the Ideal Gas Law, \(PV = nRT\), where \(R = 0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1}\). Substitute the values: \(0.0326 \, \text{atm} \times 0.125 \, \text{L} = n \times 0.0821 \, \text{L atm K}^{-1} \times 298.15 \, \text{K}\). Solve for \(n\): \[n = \frac{0.0326 \times 0.125}{0.0821 \times 298.15} \approx 1.65 \times 10^{-4} \, \text{mol}\]

Step 5: Calculate molar mass of the compound

The mass of the gas is 12.5 mg, which is 0.0125 g. Use the formula for molar mass: \( \text{Molar mass} = \frac{\text{mass}}{\text{moles}} \). Thus, \[\text{Molar mass} = \frac{0.0125 \, \text{g}}{1.65 \times 10^{-4} \, \text{mol}} \approx 75.76 \, \text{g/mol}\]

Step 6: Compare calculated molar mass to given options

The calculated molar mass is approximately 75.76 g/mol. Compare this to the molar masses of the given compounds:- \(\mathrm{B}_2\mathrm{H}_6\) with molar mass \(2(10.81) + 6(1.01) \approx 27.67 \, \text{g/mol}\)- \(\mathrm{B}_4\mathrm{H}_{10}\) with molar mass \(4(10.81) + 10(1.01) \approx 53.34 \, \text{g/mol}\)- \(\mathrm{B}_5\mathrm{H}_9\) with molar mass \(5(10.81) + 9(1.01) \approx 63.12 \, \text{g/mol}\)- \(\mathbf{\mathrm{B}_6\mathrm{H}_{10}}\) with molar mass \(6(10.81) + 10(1.01) \approx 75.50 \, \text{g/mol}\)- \(\mathrm{B}_{10}\mathrm{H}_{14}\) with molar mass \(10(10.81) + 14(1.01) \approx 133.41 \, \text{g/mol}\)The closest match to 75.76 g/mol is \(\mathrm{B}_6\mathrm{H}_{10}\) with molar mass 75.50 g/mol.

Key Concepts

Molar Mass Calculation

Understanding molar mass is key in chemistry, as it relates to the mass of one mole of a particular substance. In simple terms, it's the weight of all atoms in a molecule combined. For this exercise, we focused on boron hydride compounds.

To find the molar mass, we used the formula: \[ \text{Molar mass} = \frac{\text{mass}}{\text{moles}} \]In the experiment, we determined the number of moles using the Ideal Gas Law and then measured the mass of the gas sample. This allowed us to calculate the molar mass of the isolated boron hydride compound.

Having calculated the molar mass as approximately 75.76 g/mol, we compared it to the molar masses of possible compounds to identify the correct formula. This process demonstrates the practical use of molar mass calculations in identifying chemical compounds.

Conversion Units

Unit conversion is a fundamental skill when dealing with chemical equations and reactions. In this exercise, multiple conversions were necessary to use the Ideal Gas Law effectively.

Firstly, pressure was given in mm Hg, which we needed to convert to atmospheres. We used the conversion factor 1 atm = 760 mm Hg, resulting in:\[ P = \frac{24.8}{760} \text{ atm} \approx 0.0326 \text{ atm} \]Secondly, volume was provided in milliliters, so we converted it to liters using 1 L = 1000 mL:\[ V = 125 \text{ mL} = 0.125 \text{ L} \]Lastly, temperature conversion from Celsius to Kelvin is achieved by adding 273.15:\[ T = 25 + 273.15 = 298.15 \text{ K} \]Clearly understanding these conversions is crucial for accurate calculations in chemistry.

Boron Hydrides

Boron hydrides are chemical compounds composed of boron and hydrogen. These are an interesting group of compounds with unique properties, primarily used in research and various industrial applications.

In this exercise, we isolated a new boron hydride and needed to determine its formula. Several boron hydrides were considered as potential candidates based on their molar masses. Understanding the composition and structure of these compounds is highly beneficial in fields like materials science and organic chemistry.

Boron hydrides are often used as reducing agents in chemical reactions, underscoring the importance of identifying their exact chemical formulas correctly through experiments like the one described. This ensures their correct application in scientific and industrial processes.

Thermodynamics

Thermodynamics is the study of heat, work, and the associated energy transfers. In the context of the Ideal Gas Law, which was pivotal in this exercise, thermodynamics provided the theoretical basis to calculate the number of moles from physical data.

Using the Ideal Gas Law: \[ PV = nRT \]we captured the relationship between pressure (P), volume (V), and temperature (T) to find the moles (n) of the boron hydride gas. The constant R represents the ideal gas constant, which in atmospheric calculations is 0.0821 L atm K⁻¹ mol⁻¹.

This relationship exemplifies the practical application of thermodynamic principles in everyday chemistry problems. Understanding these concepts helps clarify the challenges and methodologies used in chemical analysis and synthesis.