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 \(R_{\mathrm{N}_{2}}=21.08 \mathrm{kPa}\) and \(P_{\mathrm{O}_{2}}=76.1 \mathrm{kPa}\). 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 diethylether is \(20.87 \text{kPa}\) and the total pressure in the container is \(118.05 \text{kPa}\).

Step-by-step solution

Find moles of diethylether

Given that the density of diethylether is 0.7134 g/mL, and we have 5 mL of diethylether: Mass of diethylether = Density × Volume Mass of diethylether = 0.7134 g/mL × 5 mL = 3.567 g The molecular formula for diethylether is \(C_2H_5OC_2H_5\). To find its molar mass, we can sum up the atomic masses of the constituent elements: Molar mass of diethylether = 2×(Molar mass of C) + 10×(Molar mass of H) + 1×(Molar mass of O) Molar mass of diethylether = 2×(12.01 g/mol) + 10×(1.01 g/mol) + 1×(16.00 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

Calculate partial pressure of diethylether

Temperature = 35.0 °C = 308.15 K Volume of vessel = 6.00 L We will use the Ideal Gas Law to calculate the partial pressure of diethylether: \(PV = nRT\) \(P_{diethylether} = \frac{n_{diethylether} × R × T}{V}\) We will use the gas constant R = 8.314 J/(mol·K), hence we need to convert volume from liters to cubic meters: V = 6.00 L × (1 m³ / 1000 L) = 0.00600 m³ Then, we have: \(P_{diethylether} = \frac{0.0481 \text{mol} \times 8.314 \text{J/(mol·K)} \times 308.15 \text{K}}{0.00600 \text{m³}} = 20868.5 \text{Pa}\) \(P_{diethylether} = 20.87 \text{kPa}\)

Calculate total pressure

Now we have the partial pressures of N2, O2, and diethylether: \(P_{N_2} = 21.08 \text{kPa}\) \(P_{O_2} = 76.1 \text{kPa}\) \(P_{diethylether} = 20.87 \text{kPa}\) Total pressure in the container = Sum of partial pressures Total pressure = \(P_{N_2} + P_{O_2} + P_{diethylether}\) Total pressure = 21.08 kPa + 76.1 kPa + 20.87 kPa = 118.05 kPa So, the partial pressure of diethylether is 20.87 kPa and the total pressure in the container is 118.05 kPa.

Key concepts

Partial Pressure

When dealing with gases in a mixture, partial pressure is a crucial concept. Each gas in a mixture exerts its own pressure, known as the partial pressure. The partial pressure of a gas is the pressure it would exert if it were the only gas in the container, provided that the temperature and volume remain constant. In the context of the Ideal Gas Law, we can determine the partial pressure of a gas using the equation:
\[P_{gas} = \frac{nRT}{V}\]Here,

• \(P_{gas}\) is the partial pressure of the gas
• \(n\) is the number of moles of the gas
• \(R\) is the ideal gas constant, often used as 8.314 J/(mol·K)
• \(T\) is the temperature in Kelvin
• \(V\) is the volume of the container

The sum of all partial pressures equals the total pressure in the container, giving us a complete picture of the pressure dynamics within mixed gases.

Molar Mass

Molar mass is the mass of one mole of a substance and is typically expressed in grams per mole (g/mol). It can be found by summing the atomic masses of all atoms in a molecule. For example, in diethylether (\(C_2H_5OC_2H_5\)), you sum the masses of all the carbon, hydrogen, and oxygen atoms it contains. Here's a simplified breakdown:

• Molar mass of Carbon (\(C\)) = 12.01 g/mol
• Molar mass of Hydrogen (\(H\)) = 1.01 g/mol
• Molar mass of Oxygen (\(O\)) = 16.00 g/mol

For diethylether, the molar mass can be calculated as:
\[\text{Molar mass of diethylether} = 2 \times (12.01) + 10 \times (1.01) + 16.00 = 74.12 \text{ g/mol}\]This efficient calculation aids in determining the number of moles for the mass of diethylether given.

Density

Density is a physical property that describes how much mass is contained in a given volume. It is commonly expressed in grams per milliliter (g/mL) for liquids and can tell us how "heavy" a substance is without needing to weigh it. In practical terms, density is calculated using the formula:
\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]For diethylether, with a provided density of 0.7134 g/mL and a volume of 5.00 mL, the mass can be calculated as:
\[\text{Mass of diethylether} = 0.7134 \text{ g/mL} \times 5.00 \text{ mL} = 3.567 \text{ g}\]Understanding density facilitates conversions between mass and volume, which is critical in chemical analyses and calculations.

Chemical Formula

A chemical formula represents the elements present in a compound and the ratio of these atoms. The formula provides a snapshot of the molecule's composition. For instance, the chemical formula for diethylether is \(C_2H_5OC_2H_5\). This formula reveals:

• Two C2H5 groups, which implies two carbon chains each with two carbon atoms and five hydrogen atoms
• One oxygen atom connecting the two carbon chains

To break it down:

• C (Carbon): essential for organic structures
• H (Hydrogen): common element in organic compounds that bonds with carbon
• O (Oxygen): known for forming single and double bonds, particularly important for the structure of ethers

Understanding a chemical formula helps in visualizing molecular structures and determining the molecular mass, providing insight into the characteristics and reactivity of the compound. This understanding is also essential for furthering studies in stoichiometry and related chemical calculations.