Problem 68 - 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 -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 \(P_{\mathrm{O}_{2}}=0.208\) 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 (C2H5OC2H5) after it evaporates is 0.206 atm, and the total pressure in the container is 1.165 atm.

Step 1: Calculate the number of moles of diethylether

First, let's find the mass of diethylether: mass = density * volume = 0.7134 g/mL * 5.00 mL = 3.567 g Next, we need to calculate the molar mass of diethylether (C2H5OC2H5). The molar masses of C, H, and O are 12.01 g/mol, 1.01 g/mol and 16.00 g/mol, respectively: molar mass = 4(12.01 g/mol) + 10(1.01 g/mol) + 16.00 g/mol = 74.14 g/mol. Now we can find the number of moles of diethylether: moles = mass / molar mass = 3.567 g / 74.14 g/mol = 0.0481 mol

Step 2: Calculate the partial pressure of diethylether using the ideal gas law

The ideal gas law is given by: PV = nRT Where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant (0.0821 L*atm/mol*K), and T is the temperature in Kelvin. First, convert the given temperature from Celsius to Kelvin: T = 35.0 + 273.15 = 308.15 K Now, rearrange the ideal gas law to solve for the partial pressure of diethylether, P_ETH: P_ETH = (n * R * T) / V Plugging in the values, we get: P_ETH = (0.0481 mol * 0.0821 L*atm/mol*K * 308.15 K) / 6.00 L = 0.206 atm

Step 3: Calculate the total pressure in the container

To find the total pressure, we will sum up the partial pressures of diethylether, nitrogen, and oxygen: P_total = P_ETH + P_N2 + P_O2 = 0.206 atm + 0.751 atm + 0.208 atm = 1.165 atm So, the total pressure in the container is 1.165 atm.

Key Concepts

Ideal Gas Law

One of the fundamental principles used in the problem is the ideal gas law, represented by the equation \( PV = nRT \). This law relates the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas, where R is the universal gas constant. To apply this law effectively, one should understand the conditions under which it is most accurate — ideally low pressures and high temperatures, where gases behave more ideally.

In the given exercise, the ideal gas law is utilized to calculate the partial pressure of diethylether vapor after it has evaporated into the vessel. To use the ideal gas law, the temperature must be in Kelvin (\(T = 35.0 + 273.15 = 308.15 \text{K}\)), and all other inputs should be in their respective units: pressure in atmosphere (atm), volume in liters (L), and the amount of substance in moles (mol). The calculation neatly demonstrates the direct relationship between the amount of substance and pressure for a fixed volume and temperature.

Molar Mass Calculation

Precisely determining the molar mass of a compound is critical in many chemical calculations, especially when using the ideal gas law. Molar mass is defined as the mass of one mole of a substance and is usually expressed in grams per mole (g/mol).

To calculate the molar mass of diethylether, or any compound, sum the molar masses of each individual element present, multiplied by the number of times each element appears within the molecule. In our example, the molecules of carbon (C), hydrogen (H), and oxygen (O) are present in distinct quantities; thus the molar mass is computed as \(4(12.01 \, \text{g/mol}) + 10(1.01 \, \text{g/mol}) + 16.00 \, \text{g/mol} = 74.14 \, \text{g/mol}\). Understanding how to calculate molar mass is essential for converting between the mass of a substance and the number of moles, which then feeds into other calculations, like determining the partial pressure in the ideal gas law.

Diethylether Properties

Diethylether is an organic compound with unique physical and chemical properties that influence how it behaves under different conditions. It is a relatively volatile ether that evaporates quickly at room temperature, a property that's used in the exercise to simulate a scenario in which diethylether becomes a gas.

Some key properties that are relevant to the problem include its density (\(0.7134 \text{g/mL}\)), which helps in determining the mass from a given volume, and its molar mass (\(74.14 \text{g/mol}\)), required for mole calculations. Diethylether's volatility means it will assume a gaseous form in the vessel, affecting the total pressure of the gas mixture inside. A student can see how a combination of a substance's properties, such as density and molar mass, and the conditions to which it is subjected (like temperature and volume), feed into the calculations that dictate behavior under the ideal gas law.