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 \(=12.5 \mathrm{mg}\) 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 (d) \( \mathrm{B}_{6} \mathrm{H}_{10} \).

Step 1: Convert Units

First, convert the given units into SI units for calculation. Convert the mass from milligrams to grams: \( 12.5 \, \text{mg} = 0.0125 \, \text{g} \). Convert the volume from \( \text{mL} \) to \( \text{L} \): \( 125 \, \text{mL} = 0.125 \, \text{L} \). Convert the pressure from \( \text{mm Hg} \) to \( \text{atm} \): \( 24.8 \, \text{mm Hg} \times \frac{1 \, \text{atm}}{760 \, \text{mm Hg}} = 0.03263 \, \text{atm} \). Convert the temperature from Celsius to Kelvin: \( 25^\circ \text{C} + 273.15 = 298.15 \, \text{K} \).

Step 2: Use the Ideal Gas Law

Use the ideal gas law \( PV = nRT \) to find the number of moles \( n \) of the gas. Rearrange to solve for \( n \), \( n = \frac{PV}{RT} \).Substitute the values: \( P = 0.03263 \, \text{atm}, V = 0.125 \, \text{L}, R = 0.08206 \, \text{L} \cdot \text{atm} / \text{mol} / \text{K}, T = 298.15 \, \text{K} \).Calculate: \[ n = \frac{0.03263 \, \text{atm} \times 0.125 \, \text{L}}{0.08206 \, \text{L} \cdot \text{atm} / \text{mol} / \text{K} \times 298.15 \, \text{K}} \approx 1.66 \times 10^{-4} \, \text{mol} \].

Step 3: Calculate Molar Mass

To find the molar mass \( M \) of the gas, use the formula \( M = \frac{\text{mass}}{\text{moles}} \).Given mass = 0.0125 g and calculated moles \( n \approx 1.66 \times 10^{-4} \) mol, substitute into the equation:\[ M = \frac{0.0125 \text{ g}}{1.66 \times 10^{-4} \text{ mol}} \approx 75.3 \text{ g/mol} \].

Step 4: Determine Molecular Formula

Compare the calculated molar mass (approximately 75.3 g/mol) to the given options:- (a) \( \mathrm{B}_{2} \mathrm{H}_{6} \): Molar mass = 27.67 g/mol- (b) \( \mathrm{B}_{4} \mathrm{H}_{10} \): Molar mass = 53.32 g/mol- (c) \( \mathrm{B}_{5} \mathrm{H}_{9} \): Molar mass = 63.12 g/mol- (d) \( \mathrm{B}_{6} \mathrm{H}_{10} \): Molar mass = 75.31 g/mol- (e) \( \mathrm{B}_{10} \mathrm{H}_{14} \): Molar mass = 133.83 g/molThe calculated molar mass of 75.3 g/mol matches closest to (d) \( \mathrm{B}_{6} \mathrm{H}_{10} \).

Key Concepts

Boron Hydride

Boron hydrides are compounds composed of boron and hydrogen. They are quite fascinating due to their unique bonding and structures. One of the most intriguing aspects is their hydride bonding which forms clusters of boron atoms surrounded by hydrogen. These compounds are not only interesting to study but also have applications in different fields such as fuel cells and chemical synthesis. Boron hydrides have varying compositions and molecular formulas like \( \text{B}_x\text{H}_y \), where \( x \) and \( y \) denote the number of boron and hydrogen atoms respectively.

In this exercise, we examined a compound whose molar mass was required to determine its formula. Hence, knowing the specific stoichiometry is crucial for understanding its chemical behavior and properties. Boron hydride compounds can have different structures even with the same number of atoms, which further adds to their complexity and the necessity for accurate mass determination.

Molar Mass Calculation

Calculating molar mass is essential for identifying substances in chemical analysis. It represents the mass of one mole of a substance and is typically expressed in grams per mole \(\text{g/mol}\). The process involves comparing the empirical data—such as the measured gas mass—and using it to calculate the molar mass, helping identify the chemical formula of a compound.

In our problem, we determined the molar mass of a boron hydride by using the Ideal Gas Law, which is pivotal in gas-related calculations. First, we solved for the number of moles using the equation \( n = \frac{PV}{RT} \), where \( P \) is pressure, \( V \) is volume, \( R \) is the universal gas constant, and \( T \) is temperature in Kelvin.

Once we had the moles, the molar mass was computed with the equation \( M = \frac{\text{mass}}{\text{moles}} \). This method allows us to precisely identify unknown gases by comparing the calculated molar mass with known values of possible formulas.

Conversion of Units

Converting units accurately is critical in chemistry, as it ensures that the units in calculations are consistent. In the exercise, multiple conversions were necessary to use the Ideal Gas Law properly. Here’s a quick rundown of these conversions:

• Mass was converted from milligrams to grams: \( 12.5 \text{ mg} = 0.0125 \text{ g} \).
• Volume converted from milliliters to liters: \( 125 \text{ mL} = 0.125 \text{ L} \).
• Pressure was converted from mm Hg to atmospheres using \( 24.8 \text{ mm Hg} \times \frac{1 \text{ atm}}{760 \text{ mm Hg}} = 0.03263 \text{ atm} \).
• Temperature was converted from Celsius to Kelvin: \( 25^\circ \text{C} + 273.15 = 298.15 \text{ K} \).

Each of these conversions follows straightforward mathematical operations, yet they are fundamental for the precision of calculations and ensuring outcomes align with scientific standards. Engaging in unit conversions is a common practice in not just chemistry but all fields of science and engineering.