Question

Consider a tank that contains moist air at \(3 \mathrm{~atm}\) and whose walls are permeable to water vapor. The surrounding air at \(1 \mathrm{~atm}\) pressure also contains some moisture. Is it possible for the water vapor to flow into the tank from surroundings? Explain.

Short answer

Answer: Based on the information provided, we cannot definitively answer whether water vapor would flow into the tank or not. However, qualitatively speaking, it is less likely for water vapor to flow into the tank due to the higher total pressure inside the tank (3 atm) compared to the surrounding air (1 atm).

Step-by-step solution

Understanding the Concept of Partial Pressure

Partial pressure is the pressure exerted by each gas in a mixture like air. In this case, we want to compare the pressures exerted by water vapor in the tank and the surrounding air. If the partial pressure of water vapor in the surrounding air is higher, it would flow into the tank.

Evaluating the Total Pressure

We know the total pressure of the tank and the surroundings: \(P_{tank} = 3 \mathrm{~atm}\) and \(P_{surroundings} = 1 \mathrm{~atm}\). But we need the values of the partial pressure of water vapor in both systems to determine if the vapor could flow into the tank.

Determining the Partial Pressures Using Dalton's Law

We can use Dalton's Law of Partial Pressures to compare the partial pressures of water vapor in the tank and in the surroundings. Dalton's Law states that the total pressure of a gas mixture is the sum of the partial pressures of all the gases in the mixture. Mathematically, it is given by: $$P_{total} = P_1 + P_2 + ... + P_n$$ Where \(P_{total}\) is the total pressure, and \(P_i\) represents the partial pressures of individual gases. If we represent the partial pressure of water vapor in the tank as \(P^{tank}_v\) and in the surroundings as \(P^{surroundings}_v\), we want to determine their values.

Insufficient Data

Given the information in the problem, we cannot determine the partial pressures of water vapor in the tank and surroundings without additional information (e.g., humidity levels or mole fractions of water vapor in both systems). However, we can still analyze the concept qualitatively.

Analyzing the Situation

For water vapor to flow into the tank from the surroundings, the partial pressure of water vapor in the surrounding air should be greater than that in the tank (i.e., \(P^{surroundings}_v > P^{tank}_v\)). Given that the total pressure of the tank is 3 times higher than the surroundings, it's generally less likely for water vapor to flow into the tank without additional driving forces, like a temperature difference or high external humidity.

Conclusion

While we do not have enough information to definitively answer whether water vapor would flow into the tank from the surroundings, we can qualitatively argue that it is less likely due to the higher total pressure inside the tank (3 atm) compared to the surrounding air (1 atm).

Key concepts

Dalton's Law of Partial Pressures

Understanding how gases behave in mixtures is crucial for comprehending the behavior of atmospheric conditions, processes in chemical engineering, and more. One of the fundamental principles governing gas mixtures is Dalton's Law of Partial Pressures. It posits that within a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of each individual gas. In mathematical terms, if we consider a mixture of gases with partial pressures denoted as \(P_1, P_2, ..., P_n\), then the total pressure \(P_{total}\) can be given by the equation:
\[P_{total} = P_1 + P_2 + ... + P_n\]
Dalton's Law is particularly useful when we analyze situations involving the mass transfer of gases, such as determining if water vapor will flow from one area to another, as it heavily depends on the comparative partial pressures. For effective problem solving, we need to identify the gases involved, their respective partial pressures, and apply Dalton's Law to piece together the behavior of the gas mixture.

Vapor Pressure

When discussing the movement of water vapor, it's essential to understand the concept of vapor pressure. Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature. This equilibrium condition means that the rate at which molecules evaporate from the liquid (or solid) to form vapor is equal to the rate at which they condense back from the vapor phase.

Each substance has its characteristic vapor pressure, which varies with temperature. Vapor pressure is relevant in both natural processes, like evaporation from bodies of water, and industrial applications, such as those found in HVAC systems or chemical reactors. The vapor pressure of water plays a crucial role in weather patterns, climate, and our day-to-day weather experiences. It's also a critical factor in determining whether water vapor will transfer into or out of a system, as seen in the exercises related to whether moisture will enter a tank based on external and internal conditions.

Mass Transfer

Mass transfer is a phenomenon that involves the movement of various chemical species from one location to another. It plays a vital role in numerous processes, from simple diffusion of a scent in a room to complex operations in industrial reactors. In the context of gases, mass transfer often refers to the movement of a particular gas' molecules across a boundary due to differences in concentration or partial pressures.

Driving Forces for Mass Transfer

For mass transfer to occur, there must be a driving force, such as a concentration gradient or a pressure differential. In the example of water vapor potentially entering a tank from the surrounding atmosphere, the partial pressure serves as a possible driving force. If the partial pressure of water vapor outside is higher than inside the tank, the vapor will naturally tend to flow into the tank until equilibrium is reached, abiding by the principles of diffusion.

However, other factors can also influence mass transfer, such as temperature gradients, the presence of barriers or membranes, and even the phase of the substances involved. Mass transfer is a complex interplay of these factors, which are crucial for solving real-world problems involving gas exchange, as well as for the proper understanding and application of scientific concepts in exercises and practical scenarios.