Problem 70 - Chemistry The Central Science

Question

A sample of \(5.00 \mathrm{~mL}\) of diethylether \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OC}_{2} \mathrm{H}_{5},\right.\) 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 \(P_{\mathrm{O}_{2}}=0.208 \mathrm{~atm} .\) 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

Short answer: (a) To calculate the partial pressure of diethylether, first determine the mass of diethylether using the volume and density (\(m_{diethylether} = V_{diethylether} \times \rho_{diethylether}\)). Convert the mass to moles using the molar mass (\(n_{diethylether} = \frac{m_{diethylether}}{molar\ mass_{diethylether}}\)). Then, use the Ideal Gas Law to calculate the partial pressure (\(P_{diethylether} = \frac{n_{diethylether}RT}{V}\)). (b) To calculate the total pressure in the container, sum up the partial pressures of N2, O2, and diethylether (\(P_{total} = P_{\mathrm{N}_{2}} + P_{\mathrm{O}_{2}} + P_{diethylether}\)).

Step 1: Calculate the mass of diethylether

First, let's calculate the mass of diethylether using its volume and density: \(m_{diethylether} = V_{diethylether} \times \rho_{diethylether}\) where \(m_{diethylether}\) is the mass of diethylether (in grams), \(V_{diethylether}\) is the volume of the diethylether (5.00 mL), and \(\rho_{diethylether}\) is the density of the diethylether (0.7134 g/mL).

Step 2: Convert the mass of diethylether to moles

To convert the mass of diethylether to moles, we need to use the molar mass of diethylether: Molecular formula of diethylether is \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OC}_{2} \mathrm{H}_{5}\), which gives us a molar mass of \(4 * 12.01 + 10 * 1.01 + 1 * 16 = 74.12 \mathrm{g/mol}\). Now we can calculate the moles of diethylether: \(n_{diethylether} = \frac{m_{diethylether}}{molar\ mass_{diethylether}}\)

Step 3: Calculate the partial pressure of diethylether

Now that we have the moles of diethylether, we can use the Ideal Gas Law to calculate the partial pressure of diethylether: \(P_{diethylether}V = n_{diethylether}RT\) where \(P_{diethylether}\) is the partial pressure of diethylether, V is the volume of the container (6.00 L), \(n_{diethylether}\) is the moles of diethylether calculated in Step 2, R is the ideal gas constant (\(0.08206 \mathrm{L~atm/mol~K}\)), and T is the temperature in Kelvin (35.0°C = 308.15 K). Now we can solve for \(P_{diethylether}\): \(P_{diethylether} = \frac{n_{diethylether}RT}{V}\)

Step 4: Calculate the total pressure in the container

Finally, we can calculate the total pressure in the container by summing up the partial pressures of N2, O2, and diethylether: \(P_{total} = P_{\mathrm{N}_{2}} + P_{\mathrm{O}_{2}} + P_{diethylether}\) With this step by step solution, you should now have all the necessary information to solve for the partial pressure of diethylether (a) and the total pressure in the container (b).

Key Concepts

Partial Pressure

In a mixture of gases, each gas has its own pressure, called a partial pressure. Partial pressure is the pressure a gas would exert if it occupied the entire volume by itself without mixing with other gases. When you add the partial pressures of all gases in a container, you obtain the total pressure inside the container. This is expressed by Dalton's Law of Partial Pressures, which states:

• \[P_{total} = P_{1} + P_{2} + \ldots + P_{n} \]

The ideal gas law can also help us find the partial pressure of a gas if we know its amount in moles, the volume of the container, and the temperature. For example, in this exercise, diethylether's partial pressure is found using the ideal gas law: \[P_{diethylether} = \frac{n_{diethylether}RT}{V}\] where:

• \(n_{diethylether}\) is the moles of diethylether
• \(R\) is the ideal gas constant
• \(T\) is the temperature in Kelvin
• \(V\) is the volume of the container

Molar Mass

Molar mass is a fundamental concept in chemistry that tells us the mass of one mole of a substance. It's the sum of the atomic masses of all atoms in the molecule. Molar mass is crucial in converting a substance's mass to its amount in moles.For instance, in this exercise, calculating the molar mass of diethylether involves summing the atomic masses of all atoms present:

• Diethylether is composed of \(\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{OC}_{2}\mathrm{H}_{5}\), which is made up of carbon (C), hydrogen (H), and oxygen (O).
• Its molar mass is calculated as: \(4 \times 12.01 + 10 \times 1.01 + 1 \times 16 = 74.12 \ \mathrm{g/mol} \)

This molar mass is then used to convert the mass of diethylether to moles, which is a critical step for further calculations.

Density

Density is a measure of how much mass is contained in a given volume. It is commonly expressed in grams per milliliter (g/mL) or grams per cubic centimeter (g/cm³). Density is important because it allows us to convert between mass and volume.In this exercise, we used the density of diethylether to determine its mass from its volume as follows:

• The volume of diethylether given is \(5.00 \ \mathrm{mL}\).
• The density is \(0.7134 \ \mathrm{g/mL}\).

The relationship between mass, volume, and density is expressed by the formula:

• \(m = V \cdot \rho \)

where \(m\) is the mass, \(V\) is the volume, and \(\rho\) is the density. This concept allows us to convert volume to mass, which is a necessary step before converting mass to moles.

Chemical Calculations

Chemical calculations are a core part of problem-solving in chemistry, involving the application of ratios, formulas, and constants to find unknowns like mass, moles, and pressure. This exercise employed several key chemical calculation steps:

• First, we calculated the mass of diethylether using its volume and density.
• Then, using the molar mass, we converted this mass to moles.
• With the moles known, we applied the ideal gas law to find the partial pressure of diethylether.
• Finally, we summed partial pressures to find the total pressure in the container.

These steps highlight the importance of understanding and applying basic chemical concepts and calculations to solve complex chemistry problems efficiently and accurately.