Problem 67 - Chemistry: The Central Science in SI Units, Global Edition

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

\(n_{CO_2} = 0.454 \mathrm{~mol}\) Step 2: Find the partial pressure of carbon dioxide #tag_title#Determine the partial pressure of the carbon dioxide gas #tag_content# Use the Ideal Gas Law equation, \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the Ideal Gas Constant (\(8.314 \mathrm{J/mol\cdot K}\)), and \(T\) is the temperature in Kelvin. First, convert the temperature from Celsius to Kelvin: \(T = 25^{\circ} \mathrm{C} + 273.15 = 298.15 \mathrm{K}\) Now, rearrange the Ideal Gas Law equation to find the pressure: \(P_{CO_2} = \frac{n_{CO_2}RT}{V}\) Calculate the partial pressure of carbon dioxide: \(P_{CO_2} = \frac{(0.454 \mathrm{~mol})(8.314 \mathrm{J/mol\cdot K})(298.15 \mathrm{K})}{25.0 \mathrm{~L}}\) \(P_{CO_2} = 45.75 \mathrm{kPa}\) Step 3: Calculate the total pressure in the container #tag_title#Determine the total pressure in the container #tag_content# Now, simply add the partial pressures of the air and carbon dioxide to find the total pressure. The air pressure is given as \(50.66\ \mathrm{kPa}\): \(P_{total} = P_{air} + P_{CO_2}\) \(P_{total} = 50.66 \mathrm{kPa} + 45.75 \mathrm{kPa}\) \(P_{total} = 96.41 \mathrm{kPa}\) So, the partial pressure of the resultant carbon dioxide gas is \(45.75 \mathrm{kPa}\), and the total pressure in the container at \(25^{\circ} \mathrm{C}\) is \(96.41 \mathrm{kPa}\).

Step 1: Determine the moles of dry ice present in the container

We are given that mass of solid carbon dioxide (dry ice) is \(20.0\ g\). To calculate the moles, we must use the molar mass of carbon dioxide, which is approximately \(44.01\ g/mol\). The formula for the number of moles is given by: Number of moles, \(n = \frac{mass}{molar\ mass}\) Calculate the number of moles of dry ice using this formula: \(n_{CO_2} = \frac{20.0 \mathrm{~g}}{44.01 \mathrm{~g/mol}}\)

Key Concepts

Sublimation of CO2

Sublimation is an interesting process where a solid changes directly into a gas without passing through the liquid phase. This can happen when a substance absorbs enough energy to break the bonds that keep its molecules together in solid form. Carbon dioxide (CO2), commonly known as dry ice when solid, undergoes sublimation. When you place dry ice in an open environment or a container, it absorbs heat from its surroundings and transforms into carbon dioxide gas.
This process is particularly useful because it maintains a constantly cold temperature without getting wet, making it ideal for transportation of perishable goods, among other applications. For our exercise, the dry ice sublime entirely into gaseous form inside the container, increasing the volume of gas and impacting the pressure within the vessel. Understanding sublimation is crucial as it directly affects the pressure calculations in the gas laws used to solve such exercises.

Ideal Gas Law

The Ideal Gas Law is a useful equation that relates the pressure, volume, temperature, and number of moles of a gas in a closed system. The formula is expressed as:\[ PV = nRT \]Where:

• \(P\) is the pressure of the gas
• \(V\) is the volume of the container
• \(n\) is the number of moles of the gas
• \(R\) is the ideal gas constant
• \(T\) is the temperature in Kelvin

Understanding this equation is essential in solving gas problems because it allows us to calculate one property of the gas if the others are known.
In our exercise, once the CO2 is completely sublimed, the Ideal Gas Law helps determine the partial pressure of CO2 in the container. Each gas in a mixture contributes to the total pressure, a concept known as partial pressure, which is essential in solving these types of problems.

Molar Mass of CO2

The molar mass of CO2 is an important constant when dealing with calculations involving this gas. It allows for conversion between the mass and the number of moles, which is vital in chemical reactions and gas laws. The calculation for molar mass involves adding the atomic masses of all the atoms in the chemical formula:

• One carbon atom with an atomic mass of approximately \(12.01\ g/mol\)
• Two oxygen atoms with an atomic mass of approximately \(16.00\ g/mol\) each

Adding these values gives a total molar mass of roughly \(44.01\ g/mol\) for CO2. This precise value helps in calculating the amount of substance (moles) from a given mass and is critical in translating laboratory or theoretical data into practical results.
In the given exercise, the molar mass of CO2 is used to determine how many moles of solid CO2 are initially present, a crucial step needed for further calculations involving gas pressure using the Ideal Gas Law.