Problem 39 - General Chemistry: Principles and Modern Applications

Question

A gaseous hydrocarbon weighing \(0.231 \mathrm{g}\) occupies a volume of \(102 \mathrm{mL}\) at \(23^{\circ} \mathrm{C}\) and \(749 \mathrm{mmHg} .\) What is the molar mass of this compound? What conclusion can you draw about its molecular formula?

Step-by-Step Solution

Answer

The molar mass of the given gaseous hydrocarbon is approximately 57.3 g/mol. The molecular formula could be C4H10, C3H8, C2H6 or CH4. However, final judgement cannot be made from molar mass solely.

Step 1: Convert all quantities to appropriate units

First, convert the volume from mL to L by dividing it by 1000. So, \(102 mL = 0.102 L\). Temperature must be converted from Celsius to Kelvin by adding 273.15. So, \(23^{\circ}C = 296.15 K\). Lastly, the pressure is given in mmHg and we need to change it to atm by dividing the given pressure by 760 (since 1 atm = 760 mmHg), so \(749 mmHg = 0.9868 atm\).

Step 2: Calculate the amount of moles

The equation \(PV = nRT\) is used, which is solved for n (amount of gas in moles), \(n = PV / RT\). Plugging in the values \(n = (0.9868 atm)(0.102 L) / ((0.0821 L atm)/(mol K))(296.15 K)\) we get \(n = 0.00403 mol\).

Step 3: Use the number of moles to find molar mass

The molar mass of an element is determined by the formula \(Molar mass = mass / moles\). Plugging in the values, we have \(Molar mass = 0.231 g / 0.00403 mol = 57.3 g/mol\).

Step 4: Draw a conclusion about the molecular formula

Given molar mass 57.3 g/mol, it leads to the conclusion that the molecular formula could be C4H10 (butane) or C3H8 (propane) or C2H6 (ethane) or CH4 (methane) as these have molar masses close to 57.3 g/mol.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation used in chemistry to relate various properties of gases. It is expressed as \(PV = nRT\). Each variable in the equation represents an important characteristic of a gas:

• \(P\) is the pressure of the gas, typically measured in atmospheres (atm).
• \(V\) represents the volume of the gas, usually in liters (L).
• \(n\) is the number of moles of gas present.
• \(R\) is the ideal gas constant, with a value of \(0.0821 \, \text{L atm/mol K}\).
• \(T\) is the temperature of the gas in Kelvin (K).

The Ideal Gas Law allows us to calculate any one of these variables if the other four are known. In the context of the exercise provided, the Ideal Gas Law is used to find the number of moles of the gaseous hydrocarbon. This is a critical step in determining the molar mass of the compound, as establishing the relation between mass, moles, and molar mass is essential in chemistry. The Ideal Gas Law highlights the predictable behavior of gases under different conditions, aligning with principles that gases expand or compress with temperature and pressure changes.

Gas Laws

Gas Laws are collections of relationships that describe the behavior of gases under various conditions. These laws are the foundation for understanding how gases interact and respond to changes in pressure, volume, and temperature. Some key gas laws include:

• Boyle's Law: At constant temperature, the pressure of a gas is inversely proportional to its volume. This is represented as \(P_1V_1 = P_2V_2\).
• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin, or \(V_1/T_1 = V_2/T_2\).
• Avogadro's Law: Equal volumes of gases, at the same temperature and pressure, contain an equal number of moles, or \(V_1/n_1 = V_2/n_2\).

Understanding these laws is crucial when dealing with gas calculations, like those required in the exercise where conversion of units (volume, pressure, and temperature) plays a role in determining the number of moles and ultimately the molar mass of the hydrocarbon. The precise application of these laws allows chemists to accurately predict and quantify the behavior of gases.

Hydrocarbons

Hydrocarbons are organic compounds consisting entirely of hydrogen and carbon atoms. They are the simplest type of organic compound and are the primary constituents of natural gas and petroleum. Hydrocarbons come in various forms, including:

• Alkanes: Saturated hydrocarbons containing single bonds only. Examples are methane (CH4) and butane (C4H10).
• Alkenes: Unsaturated hydrocarbons containing at least one double bond.
• Alkynes: Unsaturated hydrocarbons with at least one triple bond.
• Aromatic Hydrocarbons: Containing rings that adhere to specific structural patterns, such as benzene.

The molecular structure of hydrocarbons affects their physical and chemical properties, like boiling and melting points, and their reactivity. In the context of the original exercise, determining the molar mass of the hydrocarbon helps narrow down its possible identity. For instance, hydrocarbons such as ethane, propane, and butane have differing molar masses and structures, providing clues to the potential composition of the sample. By understanding their properties and applying gas laws, it is possible to deduce important characteristics and identity of the unknown hydrocarbon.