Question

A piece of dry ice (solid carbon dioxide) with a mass of \(5.50 \mathrm{~g}\) is placed in a 10.0 - \(\mathrm{L}\) vessel that already contains air at 705 torr and \(24^{\circ} \mathrm{C}\). After the carbon dioxide has totally vaporized, what is the partial pressure of carbon dioxide and the total pressure in the container at \(24{ }^{\circ} \mathrm{C} ?\)

Short answer

The partial pressure of CO2 in the container is approximately 0.306 atm, and the total pressure in the container after the dry ice vaporizes is approximately 1.233 atm.

Step-by-step solution

Find the moles of CO2 using mass and molar mass

We are given the mass of dry ice, which is 5.50 g. To find the moles of CO2, we will use the molar mass of carbon dioxide which is 44.01 g/mol. Number of moles (n) = mass / molar mass n = 5.50 g / 44.01 g/mol = 0.125 mol

Convert temperature to Kelvin

The given temperature of the gas is 24°C. We will convert it into Kelvin by adding 273.15. T (in K) = 24°C + 273.15 = 297.15 K

Convert pressure to atmospheres

The given pressure is 705 torr. We will convert it into atmospheres by dividing by 760. P (in atm) = 705 torr / 760 torr/atm = 0.927 atm

Calculate partial pressure of CO2 using the Ideal Gas Law

Now, we will use the Ideal Gas Law (PV = nRT) to find the partial pressure of CO2, where R is the gas constant 0.0821 L atm/mol K. Let P_CO2 be the partial pressure of CO2, then: P_CO2 * V = n * R * T We know the volume V = 10.0 L, n = 0.125 mol, R = 0.0821 L atm/mol K, and T = 297.15 K. Solve for P_CO2: P_CO2 = (n * R * T) / V P_CO2 = (0.125 mol * 0.0821 L atm/mol K * 297.15 K) / 10.0 L P_CO2 ≈ 0.306 atm So, the partial pressure of CO2 is approximately 0.306 atm.

Calculate the total pressure in the container

To find the total pressure in the container, we will add the initial pressure of air (0.927 atm) to the partial pressure of CO2 (0.306 atm). Total pressure = Initial pressure + Partial pressure of CO2 Total pressure = 0.927 atm + 0.306 atm ≈ 1.233 atm Thus, the total pressure in the container after the dry ice has vaporized is approximately 1.233 atm.

Key concepts

Ideal Gas Law

The Ideal Gas Law is an equation that relates the pressure, volume, temperature, and amount (in moles) of an ideal gas. It's represented by the formula: \( PV = nRT \). In this context, \(P\) stands for the pressure of the gas, \(V\) is the volume it occupies, \(n\) is the number of moles of the gas, and \(T\) represents the temperature in Kelvin. The constant \(R\) is known as the ideal gas constant, and its value depends on the units used for pressure and volume.

In practice, the Ideal Gas Law allows us to calculate any one of the variables if we know the other three. In the exercise, we used this law to find out the partial pressure of carbon dioxide in a container with a known volume and temperature, which is crucial for understanding how gases behave under different conditions.

Molar Mass of Compounds

Molar mass is the mass of one mole of a substance. It's usually expressed in grams per mole (g/mol) and is calculated by adding the atomic masses of all the atoms in a molecule. The molar mass of compounds is fundamental in converting between the mass of a substance and the amount in moles. For example, for carbon dioxide (CO2), with one carbon atom (12.01 g/mol) and two oxygen atoms (16.00 g/mol each), the molar mass is \(12.01 g/mol + 2 \times 16.00 g/mol = 44.01 g/mol\). Knowing the molar mass allowed us to calculate the moles of CO2 produced from dry ice in our problem, which is a stepping stone towards using the Ideal Gas Law.

Gas Constant (R)

The gas constant, denoted as \(R\), is a proportionality constant that appears in the Ideal Gas Law. Its value can vary depending on the units used but for most calculations involving atmospheres and liters, its value is \(0.0821 L \cdot atm / mol \cdot K\). This constant is key to connecting the otherwise disparate units of pressure, volume, and temperature to moles of gas in calculations. It serves as a bridge allowing us to use the Ideal Gas Law, and a correct understanding of \(R\) is necessary to ensure accurate results in gas-related computations. In our exercise, it was used to calculate the partial pressure of carbon dioxide.

Converting Temperature to Kelvin

Temperature conversion to Kelvin is critical in gas law calculations because the Ideal Gas Law requires temperature to be in an absolute scale, which Kelvin is. The Kelvin scale starts at absolute zero, the point where particles theoretically stop moving. To convert Celsius to Kelvin, which we did in our exercise, one must add 273.15 to the Celsius temperature. So, for the dry ice scenario at \(24^\circ C\), the Kelvin temperature is \(24^\circ C + 273.15 = 297.15 K\). Use of the Kelvin scale ensures that all temperature-dependent gas calculations are consistent and accurate.

Pressure Conversion

Pressure conversion is important because different regions and scientific disciplines use various units to measure pressure. In this exercise, we started with pressure in torr, a common unit in chemistry. However, the Ideal Gas Law usually requires pressure in atmospheres (atm). The conversion factor between torr and atm is 760 torr per 1 atm. Therefore, converting \(705\) torr to atmospheres gave us \(705 torr / 760 torr/atm = 0.927 atm\). It’s essential to use the correct conversion to ensure the accuracy of the final calculation of gas pressure.