Problem 108 - Chemistry The Central Science

Question

A 1.42 -g sample of helium and an unknown mass of \(\mathrm{O}_{2}\) are mixed in a flask at room temperature. The partial pressure of the helium is 42.5 torr, and that of the oxygen is 158 torr. What is the mass of the oxygen?

Step-by-Step Solution

Answer

The mass of the oxygen in the flask is approximately 43.36 grams.

Step 1: Find the moles of helium using the Ideal Gas Law

The Ideal Gas Law is given by: \(PV = nRT\) Where: P = Pressure (in atm) V = Volume n = Number of moles R = Ideal Gas Constant (0.0821 L atm/mol K) T = Temperature (in Kelvin) We are given the mass of helium and its partial pressure, but we need to convert the pressure from torr to atm and determine the number of moles of helium. 1 atm = 760 torr Partial pressure of helium in atm = \( \cfrac{42.5\, \text{torr}}{760\, \text{torr/atm}} \) = 0.05592 atm The molar mass of helium is 4.00 g/mol. The mass of helium is 1.42 g. Number of moles of helium = \( \cfrac{1.42\, \text{g}}{4.00\, \text{g/mol}} \) = 0.355 moles

Step 2: Find the volume of the gas mixture

Now, we can use the Ideal Gas Law to find the volume of the gas mixture, using the pressure and moles of helium. We'll assume room temperature to be 298 K. Rearrange the Ideal Gas Law to solve for volume: V = \( \cfrac{nRT}{P} \) V = \( \cfrac{(0.355\, \text{moles})(0.0821\, \text{L atm/mol K})(298\, \text{K})}{0.05592\, \text{atm}} \) = 17.97 L

Step 3: Find the moles of oxygen using the Ideal Gas Law

We can now find the moles of oxygen using the Ideal Gas Law again. First, we need to convert the partial pressure of oxygen to atm. Partial pressure of oxygen in atm = \( \cfrac{158\, \text{torr}}{760\, \text{torr/atm}} \) = 0.2079 atm Rearrange the Ideal Gas Law to solve for moles (n): n = \( \cfrac{PV}{RT} \) Number of moles of oxygen = \( \cfrac{(0.2079\, \text{atm})(17.97\, \text{L})}{(0.0821\, \text{L atm/mol K})(298\, \text{K})} \) = 1.355 moles

Step 4: Calculate the mass of oxygen

Finally, we can calculate the mass of oxygen using its molar mass (32.00 g/mol): Mass of oxygen = (1.355 moles) × (32.00 g/mol) = 43.36 g So, the mass of the oxygen in the flask is approximately 43.36 grams.

Key Concepts

Understanding the Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It is represented by the equation:
\(PV = nRT\)
In this equation, \(P\) stands for pressure, \(V\) represents volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) denotes temperature. The ideal gas constant, \(R\), has a value of 0.0821 L atm/mol K.

To use the Ideal Gas Law for calculations, it's crucial to have all the units match, so often you need to convert measurements, such as pressure from torr to atm or temperature from Celsius to Kelvin.

The law assumes that the gas particles are in random motion and do not interact, which is a good approximation for many gases under normal conditions. It's widely used for stoichiometric calculations and to understand how changes in conditions affect a gas. Whether you're determining moles from given pressure and volume or finding the volume that a gas will occupy under certain conditions, the Ideal Gas Law proves incredibly useful.

Moles Calculation Made Simple

The concept of moles is central to chemistry. A mole represents a specific number of particles, usually atoms or molecules. This number, known as Avogadro's number, is approximately \(6.022 \times 10^{23}\) particles per mole.

The mole allows chemists to count particles by weighing them. Calculating moles involves either relating the mass of a substance to its molar mass or using the Ideal Gas Law in the context of gaseous substances.

For solids and liquids, the number of moles (\(n\)) can be calculated from a given mass (\(m\)) and the substance's molar mass (\(M\)), with the formula:
\(n = \frac{m}{M}\)
This equation is invaluable when dealing with chemical reactions, as it helps to determine the amounts of reactants needed or products formed.

Determining Molar Mass

Molar mass is the weight of one mole of a substance and is expressed in grams per mole (g/mol). It's equivalent to the atomic or molecular weight of a substance but scaled up from atomic mass units (amu) to grams.

Every element's molar mass can be found on the periodic table as it's roughly equal to the element's atomic mass. For compounds, the molar mass is calculated by summing the molar masses of all the individual elements in the molecule, according to their stoichiometric proportions. For example, to find the molar mass of water (\(H_2O\)), you would sum twice the molar mass of hydrogen with the molar mass of oxygen.

Knowing the molar mass allows us to convert between mass and moles of a substance, which is crucial in chemical stoichiometry for balancing equations and thus for determining the proportions in which chemicals react.