Question

A sample of \(4.00 \mathrm{~mL}\) 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 5.00-L vessel that already contains a mixture of \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\), whose partial pressures are \(P_{N_{2}}=0.751 \mathrm{~atm}\) 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.

Short answer

The partial pressure of the diethylether after evaporating is \(0.689 \mathrm{~atm}\), and the total pressure in the container is \(1.648 \mathrm{~atm}\).

Step-by-step solution

Convert the volume of diethylether to moles

First, we need to determine the number of moles of diethylether in the 4.00 mL sample using its density and molar mass. The molecular formula of diethylether is C2H5OC2H5. Molar mass of diethylether is: Molar mass = (2 × 12.01) + (5 × 1.01) + 16 + (2 × 12.01) + (5 × 1.01) = 74.12 g/mol Now, using the given density, we can calculate the mass of diethylether: Density = Mass/Volume Mass = Density × Volume = 0.7134 g/mL × 4.00 mL = 2.854 g Now we can find the number of moles of diethylether: moles = mass / molar mass = 2.854 g / 74.12 g/mol = 0.0385 mol

Calculate the partial pressure of diethylether

We'll use the Ideal Gas Law equation, PV = nRT, to find the partial pressure of diethylether: P_diethylether = n × R × T / V Here, n = 0.0385 mol (from step 1) R = 0.08206 L atm / K mol (Ideal Gas Constant) T = 35.0°C + 273.15 = 308.15 K V = 5.00 L P_diethylether = 0.0385 mol × 0.08206 L atm / K mol × 308.15 K / 5.00 L P_diethylether = 0.689 atm

Calculate the total pressure in the container

To find the total pressure in the container, we'll add the partial pressures of N2, O2, and diethylether: Total pressure = P_N2 + P_O2 + P_diethylether Total pressure = 0.751 atm + 0.208 atm + 0.689 atm Total pressure = 1.648 atm The partial pressure of the diethylether after evaporating is 0.689 atm, and the total pressure in the container is 1.648 atm.

Key concepts

Ideal Gas Law

Understanding the Ideal Gas Law is essential when dealing with gases in various conditions. It is an equation of state for a hypothetical ideal gas, and is a good approximation for the behavior of many gases under many conditions. The law is usually formulated as:

\(PV = nRT\)

In this formula, \(P\) stands for pressure in atmospheres (atm), \(V\) is the volume in liters (L), \(n\) represents the number of moles of gas, \(R\) is the ideal gas constant (0.0821 L atm / K mol), and \(T\) is the temperature in Kelvin (K).

For the diethylether vapor in the exercise, we used the Ideal Gas Law to calculate its partial pressure after it evaporated in the container. This process assumed that the vapor behaved like an ideal gas, where interactions between molecules are negligible and the gas occupies an amount of volume proportional to the temperature and number of moles.

Molar Mass Calculation

The molar mass of a substance is the mass of one mole of that substance and is expressed in grams per mole (g/mol). It is an intrinsic property of each substance that we often use in various chemical calculations, including converting between mass and moles of a substance.

For calculating molar mass, we sum up the atomic masses of all atoms in the molecule. Taking diethylether (\(C_4H_{10}O\)) as an example, its molar mass is calculated by adding the atomic masses of 4 carbon atoms, 10 hydrogen atoms, and one oxygen atom. By understanding molar mass, students can solve the first part of our exercise by calculating the number of moles from a given mass of the gas.

Gas Mixture

In scenarios involving a gas mixture, like the vessel containing \(N_2\), \(O_2\), and diethylether vapor from the exercise, it's important to know that each gas in a mixture exerts its own pressure. This is known as partial pressure, and the sum of all partial pressures of the gases equals the total pressure of the mixture.

Understanding this concept is critical when working with gas mixtures, as it enables us to determine the contribution of each component to the total pressure. In our exercise, after calculating the partial pressure of diethylether, we simply added it to the partial pressures of \(N_2\) and \(O_2\) to find the total pressure in the container.

Vapor Pressure

Vapor pressure is the pressure exerted by the vapor in equilibrium with its liquid or solid phase at a given temperature. It's a specific kind of partial pressure that becomes relevant when a substance is introduced into a container in which it can evaporate, as we saw with diethylether in our exercise.

The vapor pressure depends on the substance's temperature and its tendency to evaporate; the higher the temperature, the higher the vapor pressure, because more molecules have enough energy to escape the liquid phase. In the given exercise, the vapor pressure of diethylether was calculated after it fully evaporated in the vessel, utilizing the Ideal Gas Law.