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 \(202.7 \mathrm{kPa}\). 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.

Short answer

The final pressure of carbon dioxide after cooling the rigid vessel to \(10^{\circ}C\) and condensing all the water vapor is approximately \(91.5 kPa\).

Step-by-step solution

Determine the initial partial pressures of carbon dioxide and water vapor

Given the total pressure of the mixture is \(202.7 kPa\), and the mol ratio of carbon dioxide to water vapor is \(3:1\), we can find the partial pressures of each component. Let \(P_{CO_2}\) and \(P_{H_2O}\) be the partial pressures of carbon dioxide and water vapor, respectively. Then, we can write: \[P_{CO_2} + P_{H_2O} = 202.7\] Since the mol ratio of carbon dioxide to water vapor is \(3:1\), we can write: \[P_{CO_2} = 3P_{H_2O}\] Now, we can substitute this expression for \(P_{CO_2}\) back into the first equation: \[3P_{H_2O} + P_{H_2O} = 202.7\] \[4P_{H_2O} = 202.7\] Solving for \(P_{H_2O}\), we get: \[P_{H_2O} = 50.675 kPa\] Now, we can find the partial pressure of carbon dioxide: \[P_{CO_2} = 3(50.675) = 152.025 kPa\]

Find the initial number of moles of carbon dioxide

Now that we have the initial partial pressure of carbon dioxide, we can use the ideal gas law to find the initial number of moles of carbon dioxide in the mixture. The ideal gas law is given by: \[PV = nRT\] Where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature. We can rearrange for the number of moles: \[n_{CO_2} = \frac{P_{CO_2}V}{RT}\] Since the vessel is rigid and the volume of liquid water is negligible, we can assume that the volume remains constant throughout the process. Since we're only interested in the pressure of carbon dioxide, we don't need to directly compute the volume. We'll only need the ratio of the initial and final temperatures. The initial and final temperatures in Kelvin are: \[T_{1} = 200 + 273.15 = 473.15 K\] \[T_{2} = 10 + 273.15 = 283.15 K\]

Calculate the final pressure of carbon dioxide

Now that we have the initial number of moles of carbon dioxide and the initial and final temperatures, we can apply the ideal gas law to determine the final pressure of carbon dioxide. Since the volume and the number of moles of carbon dioxide (water vapor has condensed) remain constant: \[\frac{P_{1}}{T_{1}} = \frac{P_{2}}{T_{2}}\] Where \(P_{1}\) is the initial partial pressure of carbon dioxide, and \(P_{2}\) is the final pressure of carbon dioxide. We can solve for the final pressure, \(P_{2}\): \[P_{2} = \frac{P_{1}T_{2}}{T_{1}}\] Substituting the values we found earlier: \[P_{2} = \frac{152.025 \times 283.15}{473.15}\] Evaluating the expression, we find the final pressure of carbon dioxide: \[P_{2} = 91.5 kPa\] Therefore, the pressure of carbon dioxide after cooling the rigid vessel to \(10^{\circ}C\) and condensing all the water vapor is approximately \(91.5 kPa\).

Key concepts

Partial Pressure

Partial pressure is the pressure exerted by a specific gas in a mixture of gases. In any mixture, each gas contributes a portion of the total pressure of the system. This contribution is called the partial pressure. To understand it better, imagine you have a sealed container holding two different gases. Even though you can't see them individually, each gas applies pressure as if it were the only gas present in the container.
In the example of our rigid vessel, the total pressure is given as 202.7 kPa. To know how much each gas in the mixture—carbon dioxide and water vapor—contributes to this total pressure, we calculate their partial pressures. The given mol ratio helps us deduce that three moles of carbon dioxide exist for every one mole of water vapor. Therefore, using the mol ratio, we can determine that carbon dioxide exerts three times the pressure of water vapor.
Mathematically, this can be expressed as:

• Total pressure: 202.7 kPa
• Carbon dioxide pressure (\(P_{CO_2} = 3P_{H_2O}\))
• Water vapor pressure (\(P_{H_2O}\)

Thus, dividing the total pressure by four yields the partial pressure of the water vapor, which is then multiplied by three to find the partial pressure of carbon dioxide (152.025 kPa).

Mol Ratio

The mol ratio is a way to express the proportions between different substances in a chemical reaction or mixture. It represents the relationship between the amounts in moles of various components, vital for chemical calculations and understanding reactions.
In our exercise, the mol ratio of carbon dioxide to water vapor is given as 3:1. This means for every one mole of water vapor, there are three moles of carbon dioxide present. Knowing this ratio helps us to determine how the partial pressures of each gas relate to one another.
When dealing with problems like these, the mol ratio can guide us in:

• Understanding the relationship between different gases in a mixture.
• Calculating partial pressures based on the known total pressure, as seen with carbon dioxide and water vapor.
• Relating the proportions of different substances to the overall change or process at hand.

For example, by utilizing the initial mol ratio, we deduced that \(P_{CO_2} = 3P_{H_2O}\). This important relationship enabled the breakdown of how much space each gas fills within the total pressure, contributing to the overall pressure based on its volumetric presence in the mixture.

Rigid Vessel

A rigid vessel is a container with a fixed volume. Unlike flexible containers, a rigid vessel does not change its shape or size, regardless of the pressure or temperature changes inside. This characteristic implies that, when the temperature shifts within the vessel, the volume remains constant. Therefore, any changes in pressure are a direct result of changes in temperature, assuming the number of moles of gas remains unchanged.
In our exercise, the vessel prevents the volume from altering. This means we can focus on the other variables of the ideal gas law: pressure, temperature, and number of moles. At a constant number of moles and unchanging volume, any change in temperature directly affects the pressure of the contained gas.
To determine the final pressure of carbon dioxide after cooling, we use the ideal gas law in form of a proportionality:

• Initial condition: \(P_1, T_1\)
• Final condition: \(P_2, T_2\)
• Equation: \[\frac{P_1}{T_1} = \frac{P_2}{T_2}\]

By solving this equation, considering the rigid nature of the vessel, we substitute the known pressures and temperatures (converted to Kelvin) to find that the final pressure of carbon dioxide is 91.5 kPa. This showcases how a rigid vessel can simplify calculations in thermodynamics and help understand the behavior of gases in fixed-volume systems.