Problem 63 - Chemistry: The Central Science

Question

A piece of solid carbon dioxide with a mass of \(5.50 \mathrm{~g}\) is placed in a 10.0-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 \(24{ }^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

After the carbon dioxide has totally vaporized in the container at \(24^{\circ}C\), the partial pressure of carbon dioxide is \(0.306 \mathrm{~atm}\), and the total pressure in the container is \(1.233 \mathrm{~atm}\).

Step 1: Calculate the moles of carbon dioxide

To find the moles of carbon dioxide, we need to use its molar mass: Molar mass of CO2 = 12.01 (C) + 2*16.00 (O) = 44.01 g/mol Given mass of CO2 = 5.50 g Now, we can calculate the moles of CO2: moles of CO₂ = (mass of CO₂)/(molar mass of CO₂) moles of CO₂ = (5.50 g)/(44.01 g/mol) = 0.125 mol

Step 2: Convert the temperature to Kelvin

The temperature is given in Celsius; we need to convert it to Kelvin using the formula: Temperature (K) = Temperature (°C) + 273.15 Temperature = 24 °C + 273.15 = 297.15 K

Step 3: Find the initial pressure of air

We are given the initial total pressure of air inside the container, but we need to calculate the pressure after the CO₂ gas is added. To do this, we first need to calculate the initial pressure of air. The total pressure is given in torr. Let's convert it to atm. 1 atm = 760 torr Initial pressure of air = (705 torr)/(760 torr/atm) = 0.927 atm

Step 4: Calculate the partial pressure of CO₂

Now, we can use the Ideal Gas Law (PV=nRT) to find the partial pressure of CO₂ gas: P_CO₂ (Partial pressure of CO₂) = n_CO₂ * R * T / V where: n_CO₂ = 0.125 mol R (gas constant) = 0.0821 atm L/mol K T = 297.15 K V = 10.0 L P_CO₂ = (0.125 mol) * (0.0821 atm L/mol K) * (297.15 K) / (10.0 L) = 0.306 atm

Step 5: Calculate the total pressure in the container

Finally, we can calculate the total pressure in the container by adding the initial pressure of air and the partial pressure of CO₂: Total pressure = Pressure of air + Pressure of CO₂ Total pressure = 0.927 atm + 0.306 atm = 1.233 atm After the carbon dioxide has totally vaporized in the container at 24 °C, the partial pressure of carbon dioxide is 0.306 atm, and the total pressure in the container is 1.233 atm.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry which relates the pressure, volume, temperature, and amount of gas in a closed system. The law is expressed by the formula: \( PV = nRT \). Each variable represents a property of the gas: \( P \) is the pressure, \( V \) the volume, \( n \) the number of moles, \( R \) the universal gas constant, and \( T \) the temperature in Kelvin. This law assumes that gas particles are in constant, random motion, that they have negligible volume, and that there are no intermolecular forces between them. As long as the conditions approximate those of an 'ideal' gas, this law allows us to predict and calculate the behavior of gases in reactions and in contained environments, such as our CO2 in the vessel.

For solving problems involving partial pressure like the one provided, we apply the Ideal Gas Law to find the pressure that a single gas, in this case CO2, exerts within a mixture of gases. We do this by considering the moles of just our gas of interest, CO2, and assuming it occupies the entire volume of the container on its own.

Molar Mass

Molar mass is the mass of one mole of a substance and is expressed in grams per mole (g/mol). It is an essential concept when working with the Ideal Gas Law because it allows us to convert between mass and moles. Every chemical element has a specific molar mass, which can be found on the periodic table. For compounds, the molar mass is calculated by summing the masses of all the atoms in the molecule. For instance, the molar mass of CO2 is found by adding the molar mass of one carbon atom (approximately 12.01 g/mol) to the molar masses of two oxygen atoms (each approximately 16.00 g/mol), leading to a total of 44.01 g/mol.

This calculation enables us to find out how many moles of a substance we have, which is critical when using the Ideal Gas Law. Knowing the molar mass, as we did with our 5.50 g of solid CO2, allows us to calculate that we have 0.125 moles of CO2 – a stepping stone towards finding out its partial pressure in the vessel.

Gas Constant

The gas constant (R) is a proportionality factor that appears in the Ideal Gas Law equation and is crucial for solving problems related to gas behavior. Its value appears to link the other variables (pressure, volume, temperature, and number of moles) in the equation. The value of R is constant and depends on the units used for the other terms in the Ideal Gas Law. For pressure in atmospheres (atm), volume in liters (L), and temperature in Kelvin (K), as in our exercise, the value of R is 0.0821 atm L/mol K.

When using the Ideal Gas Law to calculate the partial pressure of a gas, the value of R provides the necessary scale to balance the units across the equation, ensuring that we can derive the correct pressure in atmospheres (as we needed to for the partial pressure of CO2 in our container).

Converting Temperatures to Kelvin

Temperature conversions are important in gas laws since all gas law formulas require temperature to be in Kelvin (K). The Kelvin scale is an absolute temperature scale, which means it starts at absolute zero, the theoretical point where particles have minimal thermal motion. To convert Celsius (°C) to Kelvin, one needs to add 273.15 to the Celsius temperature. The formula is: \( T(K) = T(°C) + 273.15 \).

In the context of our problem, the conversion from 24°C to Kelvin is critical because the Ideal Gas Law operates strictly with absolute temperatures. Using other temperature scales would not accurately represent the kinetic energy of the gas particles. By converting and using Kelvin, we get a consistent and direct measure of this energy, which we then used to solve for the partial pressure of CO2 in the vessel.