Problem 68 - Chemistry: The Central Science

Question

A sample of \(5.00 \mathrm{~mL}\) of diethylether \(\left(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{OC}_{2} \mathrm{H}_{5}\right.\) ? density \(=0.7134 \mathrm{~g} / \mathrm{mL}\) ) is introduced into a \(6.00-\mathrm{L}\) vessel that already contains a mixture of \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\), whose partial pressures are \(P_{\mathrm{N}_{2}}=0.751 \mathrm{~atm}\) and \(P_{\mathrm{O}_{1}}=0.208 \mathrm{~atm}\). The temperature is held at \(35.0^{\circ} \mathrm{C}\), and the diethylether totally evaporates. (a) Calculate the partial pressure of the diethylether. (b) Calculate the total pressure in the container.

Step-by-Step Solution

Answer

The partial pressure of diethylether in the container is approximately \(0.206 \mathrm{~atm}\). The total pressure in the container is approximately \(1.165 \mathrm{~atm}\).

Step 1: Calculate the moles of diethylether

First, we need to calculate the moles of diethylether present in the container. To do this, we can use the mass, density, and molar mass of diethylether. First, let's find the mass of diethylether using its density and volume: Mass of diethylether = Density × Volume Mass of diethylether = 0.7134 g/mL × 5.00 mL = 3.567 g Now, we can use the molar mass of diethylether (C₂H₅OC₂H₅): Molar mass of diethylether = 2(12.01 g/mol) + 5(1.01 g/mol) + 16.00 g/mol + 2(12.01 g/mol) + 5(1.01 g/mol) = 74.12 g/mol Now, we can find the moles of diethylether: Moles of diethylether = Mass of diethylether / Molar mass of diethylether Moles of diethylether = 3.567 g / 74.12 g/mol ≈ 0.0481 mol

Step 2: Find the partial pressure of diethylether using the Ideal Gas Law

Now we can use the Ideal Gas Law to find the partial pressure of diethylether in the container: PV = nRT Where: P is the partial pressure of diethylether V is the volume of the container (6.00 L) n is the moles of diethylether (0.0481 mol) R is the Ideal Gas constant (0.0821 L atm / K mol) T is the temperature in Kelvin (35.0 + 273.15 = 308.15 K) We need to solve for P: P = nRT / V P = (0.0481 mol × 0.0821 L atm / K mol × 308.15 K) / 6.00 L P ≈ 0.206 atm So, the partial pressure of diethylether in the container is approximately 0.206 atm.

Step 3: Calculate the total pressure in the container

To calculate the total pressure in the container, we simply add up the partial pressures of all gases: Total pressure = P(N₂) + P(O₂) + P(diethylether) Total pressure = 0.751 atm + 0.208 atm + 0.206 atm ≈ 1.165 atm Hence, the total pressure in the container is approximately 1.165 atm.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a critical tool in understanding the behavior of gases. It is expressed with the equation PV = nRT, where P stands for pressure, V for volume, n for the number of moles of the gas, R for the ideal gas constant, which is 0.0821 L atm/mol K, and T for temperature in Kelvin. It's essential to convert Celsius to Kelvin by adding 273.15 because the Kelvin scale is an absolute scale with its zero point at absolute zero, where theoretically gases have no volume.

For a single gas, the Ideal Gas Law can be rearranged to solve for its partial pressure, as seen in the textbook exercise where the partial pressure for diethylether vapor is calculated. In a mixture of gases, each gas exerts pressure as if it alone occupied the volume, and the total pressure is the sum of each gas's partial pressure. This principle is fundamental when working with gas mixtures and figuring out the overall pressure within a container.

Molar Mass

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is equivalent to the molecular weight, which can be calculated by summing the atomic weights of each element within a compound. In the original exercise, the molar mass of diethylether is found by adding together the molar masses of carbon, hydrogen, and oxygen appropriately.

Understanding molar mass is vital when converting between the mass of a substance and the amount of moles present — a fundamental step in many chemical calculations. The precision in calculating molar mass is crucial because it directly affects the calculation of moles, and any error could alter subsequent steps, such as the determination of partial pressure in a gas law calculation.

Moles of Substance

The 'mole' is a foundational concept in chemistry, representing a specific number of particles, usually atoms or molecules. One mole is equivalent to Avogadro's number, approximately 6.022 x 10²³ particles. The moles of a substance can be calculated by dividing the mass of the substance by its molar mass.

In practical terms, such as in the solution provided, knowing the moles of diethylether is a stepping stone for utilizing the Ideal Gas Law, which integrates the moles of a gas to relate its volume, temperature, and pressure. This quantitative expression allows for the prediction of gas behavior under varying conditions. For the Ideal Gas Law to accurately represent the real-world behavior of gases, it is assumed that the gas particles are small and do not interact with each other, which is a reasonable approximation for many but not all situations.

Gas Mixture

A gas mixture consists of two or more different gases occupying a shared space. The pressure exerted by a single gas in a mixture is its partial pressure, which can be calculated individually for each gas using the Ideal Gas Law. In a gas mixture, such as the one in the container from the textbook exercise, the total pressure is the sum of the individual gases' partial pressures.

When dealing with gas mixtures, it is essential to remember that each gas behaves independently. This property means that in our textbook example, even though diethylether, nitrogen, and oxygen are in the same container, we calculate each one's pressure contribution separately and then add them to find the total. Adjusting for the presence and effects of a gas mixture in calculations is essential in fields ranging from chemistry to engineering and environmental science.