Problem 67 - Chemistry: The Central Science

Question

A piece of dry ice (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 sublimed, what is the partial pressure of the resultant \(\mathrm{CO}_{2}\) gas, and the total pressure in the container at \(24^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

The partial pressure of the resultant CO2 gas is 3.07 atm, and the total pressure in the container at \(24^{\circ} \mathrm{C}\) is 3.997 atm.

Step 1: Calculate moles of CO2

First, let's calculate the moles of dry ice (solid CO2) using the molar mass of CO2 (44.01 g/mol). Moles of \(\mathrm{CO_{2}}\) = \(\frac{mass}{molar~mass}\) Moles of \(\mathrm{CO_{2}}\) = \(\frac{5.50~g}{44.01~g/mol}\) = 0.125 moles

Step 2: Convert temperature to Kelvin

The ideal gas law uses temperature in Kelvin(K), so convert the given temperature from Celsius to Kelvin. Temperature in Kelvin = \(24^{\circ} \mathrm{C} + 273.15\) = 297.15 K

Step 3: Calculate the partial pressure of CO2 gas

Use the ideal gas law: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (0.0821 L atm/mol K), and T is the temperature in Kelvin, to find the partial pressure of CO2 gas. P = \(\frac{nRT}{V}\) P_CO2 = \(\frac{0.125~mol \times 0.0821~L~atm/mol~K \times 297.15~K}{10.0~L}\) P_CO2 = 3.07 atm

Step 4: Convert the initial pressure to atm

We need to have the pressures in the same unit to add them together. Convert the initial pressure of the air in the container from torr to atm. Initial pressure of air in atm = \(\frac{705~torr}{760~torr/atm}\) = 0.927 atm

Step 5: Calculate the total pressure

Add the initial pressure of the air and the partial pressure of the CO2 gas to find the total pressure in the container. Total pressure = Initial pressure of air + Partial pressure of CO2 gas = 0.927 atm + 3.07 atm = 3.997 atm The partial pressure of the resultant CO2 gas is 3.07 atm, and the total pressure in the container at \(24^{\circ} \mathrm{C}\) is 3.997 atm.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is central to understanding how gases behave under various conditions of temperature, pressure, volume, and amount. It is expressed as the equation
\[ PV = nRT \]
where

• \( P \) represents the pressure of the gas,
• \( V \) is the volume it occupies,
• \( n \) is the number of moles of the gas,
• \( R \) is the ideal gas constant (0.0821 L atm/mol K), and
• \( T \) is the temperature of the gas in Kelvin.

This relationship is derived from experimental observations and is a good approximation for most gases at standard conditions. However, it tends to deviate at very high pressures or low temperatures, where the behavior of gases becomes non-ideal. Through the Ideal Gas Law, we can find any one of the variables if the others are known. For example, in calculating partial pressures, this equation helps us understand how much pressure a single type of gas (a component of a gas mixture) exerts on the walls of its container. By rearranging the equation, the pressure exerted by any gas can be calculated as long as the temperature, volume, and moles of the gas are known.

Converting Temperature to Kelvin

The Kelvin scale is the absolute temperature scale used by scientists and in the field of physics, including thermodynamics and kinetics. It's critical for gas law calculations because it is essential to use an absolute scale that starts at absolute zero. To convert Celsius to Kelvin, which is crucial for accurately using the Ideal Gas Law, you add 273.15 to the Celsius temperature.
\[ T(K) = T(^\circ C) + 273.15 \]
This conversion ensures that all thermal energy calculations are standardized, as temperature in any other scale would not properly account for the absolute zero point - the temperature at which particles theoretically have minimum thermal motion. For instance, in the exercise given, converting the temperature from 24°C to Kelvin is necessary to plug into the Ideal Gas Law correctly. Neglecting to convert Celsius to Kelvin would lead to incorrect results due to inappropriate temperature units. Remember, even a seemingly small mistake in temperature conversion can cause a significant error in the final calculations of pressure.

Calculating Moles of a Substance

Understanding the concept of moles in chemistry is crucial. A mole is a unit that measures the amount of a substance. One mole of any substance contains Avogadro's number (\(6.022 \times 10^{23}\)) of molecules or atoms, depending on the substance.
For gas calculations, it is often necessary to convert the mass of a gas into moles to utilize the Ideal Gas Law effectively. The number of moles (\(n\)) can be calculated using the formula:
\[ n = \frac{mass}{molar~mass} \]
where the mass is the given mass of the substance (in grams in many cases), and the molar mass is the mass of one mole of the substance (in grams per mole). The molar mass differs for each element and compound and can be found on the periodic table or by summing the atomic masses of constituent elements for a compound. In our exercise, calculating the moles of carbon dioxide is a vital first step. The mass of carbon dioxide is divided by its molar mass (44.01 g/mol) to determine how many moles are present. This value is a key component in calculating the partial pressure of CO2 in the vessel.