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 \(24^{\circ} \mathrm{C}\) ?

Short answer

The number of moles of CO2 is \(n = \frac{5.50 g}{44.01 g/mol} = 0.125 mol\). The temperature in Kelvin is \(T = 24°C + 273.15 = 297.15 K\). The partial pressure of CO2 is \(P_{CO2} = \frac{(0.125 mol \times 0.0821 L\cdot atm/mol\cdot K \times 297.15 K)}{10 L} = 3.07 atm\). The initial air pressure in atm is \(P_{air} = 705 torr \times \frac{1 atm}{760 torr} = 0.927 atm\). The total pressure in the container is \(P_{total} = P_{air} + P_{CO2} = 0.927 atm + 3.07 atm = 3.997 atm\). Therefore, the partial pressure of CO2 is 3.07 atm, and the total pressure in the container is 3.997 atm.

Step-by-step solution

Calculate the number of moles of CO2

First, we need to find the number of moles of CO2. The mass (m) of CO2 is given as 5.50 g, and the molar mass (M) of CO2 is 44.01 g/mol (12.01 g/mol for C and 16.00 g/mol for each O). We can use the formula n = m/M to find the number of moles: n = m / M n = 5.50 g / 44.01 g/mol

Convert temperature to Kelvin

Before we use the Ideal Gas Law, we need to convert the temperature given in Celsius (24°C) to Kelvin (K) using the formula K = C + 273.15: T = 24°C + 273.15 T = 297.15 K

Determine the partial pressure of CO2

Now we can use the Ideal Gas Law (PV = nRT) to determine the partial pressure of CO2: P_CO2 × V = n × R × T where P_CO2 is the partial pressure of CO2, V is the volume of the container (10 L), n is the number of moles of CO2, R is the ideal gas constant (0.0821 L⋅atm/mol⋅K), and T is the temperature in Kelvin (297.15 K). Rearrange the formula to solve for P_CO2: P_CO2 = (n × R × T) / V Plug in the values: P_CO2 = (5.50 g / 44.01 g/mol × 0.0821 L⋅atm/mol⋅K × 297.15 K) / 10 L

Convert initial air pressure to atm

The initial pressure of the air in the container is given as 705 torr. To add this to the pressure of CO2, we need to convert it to atm using the conversion factor 1 atm = 760 torr: P_air = 705 torr × (1 atm / 760 torr)

Calculate total pressure in the container

Now we can sum the initial air pressure (in atm) and the partial pressure of CO2 (in atm) to find the total pressure in the container: P_total = P_air + P_CO2 Plug in the values from previous steps to calculate P_total.

Key concepts

Ideal Gas Law

Understanding the Ideal Gas Law is key to solving problems involving gases in containers. It is represented by the equation \(PV = nRT\), where \(P\) stands for the pressure of the gas, \(V\) is the volume it occupies, \(n\) is the number of moles of gas, \(R\) is the ideal gas constant (0.0821 L⋅atm/mol⋅K is commonly used), and \(T\) is the absolute temperature in Kelvin. This law combines several gas laws, including Boyle's, Charles', and Avogadro's, into a single formula.

In the context of the exercise, the Ideal Gas Law allows us to calculate the partial pressure of carbon dioxide in a container. The partial pressure is the pressure the gas would exert if it were the only gas in the container. Since the vessel contains air, the carbon dioxide will have a partial pressure that contributes to the total pressure of the mixture. It's crucial to use Kelvin for the temperature and to make sure the volume is in liters and the pressure in atmospheres to match the units of the gas constant when using this formula.

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 (\(g/mol\)). To calculate it for a compound, you must sum the molar masses of the individual elements that make up the compound, taking into account their respective quantities as seen in the chemical formula.

For example, to find the molar mass of carbon dioxide (\(CO_2\)), you must account for one carbon atom and two oxygen atoms. Carbon has a molar mass of about 12.01 \(g/mol\), and oxygen has a molar mass of about 16.00 \(g/mol\). Therefore, the molar mass of \(CO_2\) is \(12.01 g/mol + 2(16.00 g/mol) = 44.01 g/mol\). The exercise asked us to use this molar mass to calculate the number of moles of carbon dioxide from its given mass of 5.50 grams. Understanding molar mass is essential for stoichiometry in chemistry, which involves calculations based on the mass of reactants and products in a chemical reaction.

Converting Temperature to Kelvin

Kelvin is the SI unit for temperature and is one of the fundamental units in physics. The Kelvin scale is an absolute temperature scale starting at absolute zero, which is the theoretical point where particles have minimum thermal motion. To convert Celsius to Kelvin, which is necessary in gas law calculations including the Ideal Gas Law, you use the formula \(K = ^\circ C + 273.15\).

In our exercise, the temperature given is \(24^\circ C\), so we converted that to Kelvin by adding 273.15, resulting in \(297.15 K\). It's critical that this conversion is done because the Ideal Gas Law requires the absolute temperature in Kelvin for accurate calculations. This ensures that all the components in the equation are consistent, allowing us to accurately assess how temperature influences the behavior of gases.