Problem 67 - Chemistry The Central Science In Si Units

Question

A piece of dry ice (solid carbon dioxide) with a mass of \(20.0 \mathrm{~g}\) is placed in a 25.0-L vessel that already contains air at \(50.66 \mathrm{kPa}\) and \(25^{\circ} \mathrm{C}\). After the carbon dioxide has totally sublimed, what is the partial pressure of the resultant \(\mathrm{CO}_{2}\) gas, and the total pressure in the container at \(25^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The partial pressure of CO2 is 4.49 kPa, and the total pressure is 55.15 kPa.

Step 1: Calculate Moles of CO2

First, calculate the number of moles of CO2. Use the formula \( n = \frac{m}{M} \), where \( m \) is the mass and \( M \) is the molar mass of CO2. The molar mass of CO2 is \( 44.01 \text{ g/mol} \).\[ n = \frac{20.0 \text{ g}}{44.01 \text{ g/mol}} \approx 0.454 \text{ mol} \]

Step 2: Calculate Partial Pressure of CO2

Using the ideal gas law \( PV = nRT \), where \( R = 8.314 \text{ J/(mol K)} \) and \( T = 298.15 \text{ K} \), calculate the partial pressure \( P_{\mathrm{CO}_2} \). We rearrange the gas law to \( P = \frac{nRT}{V} \).\[ P_{\mathrm{CO}_2} = \frac{0.454 \text{ mol} \times 8.314 \text{ J/(mol K)} \times 298.15 \text{ K}}{25.0 \times 10^{-3} \text{ m}^3} \approx 4494 \text{ Pa} \approx 4.49 \text{ kPa} \]

Step 3: Calculate Total Pressure

Add the partial pressure of CO2 to the initial pressure of the air to find the total pressure in the container. Initial pressure = 50.66 kPa. \[ P_{\text{total}} = P_{ ext{initial}} + P_{ ext{CO}_2} = 50.66 \text{ kPa} + 4.49 \text{ kPa} = 55.15 \text{ kPa} \]

Key Concepts

Partial Pressure

Partial pressure is a fundamental concept in gas chemistry. It refers to the pressure exerted by an individual gas within a mixture of gases. When a gas, such as carbon dioxide, is introduced into a container that already contains other gases, the gas molecules exert a force against the walls of the container.
The pressure due to each gas in such a mixture is its partial pressure. The partial pressure is calculated using the ideal gas law, which considers temperature, volume, and the number of moles of the gas. Partial pressure is crucial for understanding gas mixtures, helping us predict how gases will behave under different conditions.

Moles of Gas

The concept of moles of gas is a cornerstone of chemistry. It quantifies the amount of substance based on Avogadro's number, which is approximately \(6.022 \times 10^{23}\) particles per mole. This quantity helps scientists and students convert between mass and the number of molecules or atoms.
In the exercise, calculating moles is essential before determining the partial pressure of carbon dioxide. By dividing the mass of carbon dioxide by its molar mass, you find the amount in moles. This calculation enables you to accurately determine how much substance will undergo sublimation and contribute to gas pressure inside the container.

Sublimation

Sublimation is a phase transition where a substance goes from solid to gas without passing through the liquid state. This process is seen in substances like dry ice, which is solid carbon dioxide. Under normal atmospheric conditions, dry ice sublimates rapidly, transitioning directly into the gaseous state.
This concept is pivotal in the given exercise because it explains how 20.0 grams of dry ice becomes carbon dioxide gas. Understanding sublimation allows us to predict changes in pressure and volume when a solid is introduced into an environment where it sublimates.

Total Pressure Calculation

Calculating total pressure in a container with a mixture of gases involves summing the partial pressures of all individual gases present. The formula used is the sum of the initial pressure inside the container and the partial pressure contributed by additional gas.
For instance, in our problem, after the carbon dioxide has sublimed, the total pressure is the sum of the initial air pressure and the partial pressure of carbon dioxide. This calculation is fundamental to predicting how the addition of more gas will impact the overall pressure, which is essential in safety assessments and various applications involving gas mixtures.