Problem 69 - Chemistry: The Central Science

Question

A rigid vessel containing a \(3: 1 \mathrm{~mol}\) ratio of carbon dioxide and water vapor is held at \(200^{\circ} \mathrm{C}\) where it has a total pressure of \(2.00 \mathrm{~atm}\). If the vessel is cooled to \(10^{\circ} \mathrm{C}\) so that all of the water vapor condenses, what is the pressure of carbon dioxide? Neglect the volume of the liquid water that forms on cooling.

Step-by-Step Solution

Answer

The final pressure of carbon dioxide after cooling and condensation of water vapor is approximately \(0.90 \mathrm{~atm}\).

Step 1: Convert Temperatures to Kelvin

To work with the Ideal Gas Law, we have to convert the temperatures from Celsius to Kelvin. To do so, we add 273.15 to the given Celsius temperature. Initial Temperature: \(200^{\circ}C + 273.15 = 473.15 K\) Final Temperature: \(10^{\circ}C + 273.15 = 283.15 K\)

Step 2: Find the partial pressures of the gases at the initial condition

Using the given mol ratio of 3:1, we can determine the partial pressures of carbon dioxide (CO2) and water vapor (H2O) at the initial condition. Total Pressure (P_total) = 2.00 atm Mole ratio of CO2 to H2O: \[P_{CO_2} / P_{H_2O} = 3 / 1\] We can set the equations as follows: \(P_{CO_2} + P_{H_2O} = P_{total}\) Substitute the mole ratio: \(P_{CO_2} + \frac{P_{CO_2}}{3} = 2.00\,atm\) Now, solve for the partial pressures: \[P_{CO_2} = 1.50\,atm\] \[P_{H_2O} = 0.50\,atm\]

Step 3: Calculate the moles of carbon dioxide

Now that we have the initial pressure of carbon dioxide, we can calculate the number of moles of CO2 present. From the Ideal Gas Law: \[PV = nRT\] Rewriting for moles of CO2 (n_CO2): \[n_{CO_2} = \frac{P_{CO_2}V}{RT_1}\] where \(P_{CO_2}\) is the initial pressure of CO2, V is the volume, R is the gas constant, and \(T_1\) is the initial temperature. As the vessel is rigid, the volume remains constant throughout the process. So, we can use the same volume while calculating the final pressure of carbon dioxide.

Step 4: Calculate final pressure of carbon dioxide

Now that we have the moles of CO2 and know that the volume is conserved, we can calculate the final pressure of carbon dioxide at the decreased temperature. Using the Ideal Gas Law again: \[PV = nRT\] Rearranging for final pressure of CO2 (\(P_{CO_2f}\)): \[P_{CO_2f} = \frac{n_{CO_2}R T_2}{V}\] However, we can rewrite the equation in terms of initial conditions: \[P_{CO_2f} = P_{CO_2i}\frac{T_2}{T_1}\] Plugging in the values: \[P_{CO_2f} = 1.50\,atm\frac{283.15\,K}{473.15\,K}\] Solve for the final pressure of carbon dioxide: \[P_{CO_2f} \approx 0.90\,atm\] The final pressure of carbon dioxide after cooling and condensation of water vapor is approximately 0.90 atm.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that describes the relationship between the pressure (P), volume (V), the number of moles of gas (n), and the temperature (T) of an ideal gas. It is expressed as the formula:
\[ PV = nRT \]
where R is the gas constant with a value of 0.0821 atm·L/(mol·K). This law assumes that the gas molecules do not interact with each other and occupy no volume, which is a close approximation for actual gases at high temperatures and low pressures. The Ideal Gas Law is pivotal when it comes to understanding gas behavior under various conditions, such as in the textbook exercise we're analyzing, where it's used to calculate the final pressure of carbon dioxide in a rigid vessel when the temperature changes.

Partial Pressure

Partial pressure is the pressure that each gas in a mixture of gases would exert if it alone occupied the entire volume. It's a concept critical to gas mixtures, like air or our example with carbon dioxide and water vapor in the exercise.
For mixtures, Dalton's Law states that the total pressure exerted by the mixture is equal to the sum of the partial pressures of individual gases. This can be represented by:
\[ P_{total} = P_{gas1} + P_{gas2} + P_{gas3} + ... \]
In our exercise, by knowing the total pressure and the molar ratio of gases, we were able to calculate the partial pressure of carbon dioxide, which is crucial for determining how the system's pressure changes with temperature.

Moles of Gas

The mole is a unit of measurement for the amount of substance. It is one of the seven base SI units and is defined as containing exactly 6.02214076 x 10^23 (Avogadro's number) particles, be they atoms, molecules, ions, or electrons. In the context of gases, the number of moles represents the amount of gas molecules present.
In the Ideal Gas Law (\(PV = nRT\)), the variable 'n' stands for the number of moles. The ability to calculate moles is essential in chemical reactions and for understanding the physical behavior of gases. For example, knowing the moles of carbon dioxide in our exercise lets us calculate how much the pressure of the gas changes when the vessel is cooled.

Gas Temperature Conversion

Temperature conversion between Celsius and Kelvin is crucial when working with gas laws, as equations like the Ideal Gas Law require absolute temperatures, which are measured in Kelvin. The Kelvin is the SI base unit of thermodynamic temperature and starts at absolute zero, the theoretical point at which molecular motion stops.
To convert Celsius to Kelvin, which is needed in the Ideal Gas Law, you add 273.15 to the Celsius temperature. The conversion formula is simple:
\[ T(K) = T(\degree C) + 273.15 \]
In the exercise, performing this conversion correctly is vital for determining how the temperature change affects the pressure inside the vessel containing the gas mixture.