Question

A new boron hydride, \(\mathrm{B}_{x} \mathrm{H}_{\mathrm{v}}\), 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 \(=25^{\circ} \mathrm{C}\) 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) \(\mathbf{B}_{6} \mathbf{H}_{10}\) (e) \(\mathrm{B}_{10} \mathrm{H}_{14}\)

Short answer

The calculated molar mass is much larger than any provided option, indicating an error or discrepancy.

Step-by-step solution

Convert the Mass of Gas to Grams

The mass is given as 12.5 mg. First, convert this mass into grams by dividing by 1000. \[\text{Mass in grams} = \frac{12.5 \text{ mg}}{1000} = 0.0125 \text{ g}\]

Convert the Pressure to Atmospheres

The pressure is provided in mm Hg. Use the conversion factor of 1 atm = 760 mm Hg to convert the pressure to atmospheres.\[\text{Pressure in atm} = \frac{24.8 \text{ mm Hg}}{760 \text{ mm Hg/atm}} \approx 0.0326 \text{ atm}\]

Convert Temperature to Kelvin

Convert the temperature from Celsius to Kelvin using the formula:\[T(K) = T(^{\circ}C) + 273.15\]For 25°C, this gives:\[T = 25 + 273.15 = 298.15 \text{ K}\]

Convert Volume to Liters

The volume of the flask is given in mL. Convert this volume to liters because the ideal gas law equation uses liters.\[V \text{ in Liters} = \frac{125 \text{ mL}}{1000} = 0.125 \text{ L}\]

Calculate the Number of Moles

Use the ideal gas law, \(PV = nRT\), to solve for the number of moles \(n\). The gas constant \(R\) is 0.0821 L·atm/mol·K.Rearrange to solve for \(n\):\[n = \frac{PV}{RT}\]Plug in the known values:\[n = \frac{(0.0326 \text{ atm})(0.125 \text{ L})}{(0.0821 \text{ L atm/mol K})(298.15 \text{ K})} \approx 0.00001652 \text{ moles}\]

Calculate the Molar Mass

Now calculate the molar mass \(M\) (g/mol) using the formula:\[M = \frac{\text{Mass}}{n} \approx \frac{0.0125 \text{ g}}{0.00001652 \text{ moles}} \approx 756.96 \text{ g/mol}\]

Compare to Given Options

Compare the calculated molar mass to the molar masses of the given options.A) \(\text{B}_2\text{H}_6\), approximately 27.67 g/molB) \(\text{B}_4\text{H}_{10}\), approximately 53.34 g/molC) \(\text{B}_5\text{H}_9\), approximately 63.12 g/molD) \(\text{B}_6\text{H}_{10}\), approximately 73.68 g/molE) \(\text{B}_{10}\text{H}_{14}\), approximately 122.09 g/molThe calculated value of approximately 756.96 g/mol does not closely match any of these options, indicating a possible calculation error or measurement discrepancy.

Key concepts

Boron Hydrides

Boron hydrides are fascinating compounds consisting of boron and hydrogen. They are known as boranes. Boranes are typically characterized by having complex structures due to the strong electron-deficient bonding, which can lead to interesting chemistry. Understanding the formula for a new boron hydride involves determining the ratio of boron to hydrogen atoms.

In our exercise, the challenge is to identify which borane formula from a list of options corresponds to an experimental molar mass. Boranes like B₂H₆, B₄H₁₀, B₅H₉, and others are part of this family. Each has distinct properties and molecular weights, which are crucial when matching them to a calculated molar mass. This requires precise measurement and careful calculations to determine the accurate borane formula.

Molar Mass Calculation

Molar mass is a fundamental concept in chemistry, used to relate the mass of a compound to the number of moles. It is calculated by dividing the mass of the gas by the number of moles. This is expressed in grams per mole (g/mol).

The Ideal Gas Law, given as \( PV = nRT \), allows chemists to calculate the number of moles \( n \) of a gas when the pressure \( P \), volume \( V \), and temperature \( T \) are known. The gas constant \( R \) has a specific value of 0.0821 L·atm/mol·K for these calculations.

In our example, molar mass was calculated by measuring mass, converting units as necessary, and using the ideal gas law to find moles. However, a discrepancy in the result suggested that experimental errors or assumptions could lead to unexpected outcomes. This underscores the importance of careful technique and double-checking each step of the calculation process.

Gas Pressure Conversion

Gas pressure conversion is an essential step in dealing with gases and applying the ideal gas law. Pressure is often measured in different units such as mm Hg, atm, and pascals. To use the ideal gas law correctly, it’s necessary to convert these units into atmospheres (atm).

One atmosphere is equivalent to 760 mm Hg. In our exercise, the pressure was originally given in mm Hg and needed to be converted to atm to work with the ideal gas law. This conversion ensures that all parameters are consistent and formulas work correctly.

Adjusting units before diving into mathematical equations helps avoid errors and provides accurate, reliable results. It's a straightforward but crucial step ensuring that each variable interacts correctly with the others in calculations and ensuring clarity in scientific communication.

Temperature Conversion to Kelvin

Temperature plays a critical role in gas calculations, and in the context of the ideal gas law, temperatures must always be in Kelvin. Kelvin is the absolute temperature scale used in scientific calculations because it begins at absolute zero, where theoretically, all molecular motion ceases.

To convert from Celsius to Kelvin, which is often necessary in experiments, you simply add 273.15 to the Celsius temperature. This conversion aligns temperature readings with the absolute scale needed for accurate gas law calculations.

In our problem, the initial measurement was in Celsius, so converting to Kelvin ensured that the calculations followed the thermodynamic rules that the ideal gas law relies on. This small conversion detail is a critical step to achieving accuracy in any experimental work when dealing with gases.