14-45 Consider a rubber membrane separating carbon dioxide gas that is maintained on one side at \(2 \mathrm{~atm}\) and on the opposite at \(1 \mathrm{~atm}\). If the temperature is constant at \(25^{\circ} \mathrm{C}\), determine (a) the molar densities of carbon dioxide in the rubber membrane on both sides and \((b)\) the molar densities of carbon dioxide outside the rubber membrane on both sides.

The Ideal Gas Law is a fundamental principle in chemistry and physics that describes the behavior of an ideal gas. This law is generally expressed as the equation \( PV = nRT \), where \(P\) represents the pressure of the gas, \(V\) is the volume it occupies, \(n\) is the amount of substance in moles, \(R\) is the universal gas constant, and \(T\) is the absolute temperature in Kelvin.

In real-world applications, the Ideal Gas Law can help us predict how a gas will behave under various conditions. For example, if we know the pressure, temperature, and volume, we can calculate the amount of gas in moles. Conversely, knowing the amount of gas and the temperature can allow us to find the volume it will occupy at a certain pressure.

In the provided exercise, the molar density (moles of gas per unit volume) inside a rubber membrane is determined by rearranging the Ideal Gas Law to \( \rho = \frac{P}{RT} \), where \(\rho\) represents molar density. This equation illustrates that molar density is directly proportional to pressure \(P\) and inversely proportional to absolute temperature \(T\).

By plugging into this equation the known values of pressure and temperature, the student can calculate the molar density of the gas on either side of the membrane. This is an essential step for understanding gas behavior in enclosed systems and is a practical tool for engineers and scientists in fields such as chemical engineering and environmental science.

Membrane separation is a technology used to separate substances based on their physical or chemical properties. This process utilizes a membrane, which is a selective barrier; it allows some particles to pass through while blocking others. The selectivity of the membrane can depend on various factors such as size, charge, or chemical affinity.

In the context of gases, membrane separation is often applied in industries for gas purification or the separation of different gas mixtures. For instance, a rubber membrane might be used to separate carbon dioxide from other gases based on its permeability to CO2.

The effectiveness of membrane separation can be influenced by the pressure and temperature across the membrane, as well as the membrane's material properties. The example in the exercise shows a rubber membrane separating carbon dioxide at two different pressures. It is implied that the membrane allows for a certain degree of CO2 transfer, which may be related to industrial applications such as carbon capture or the production of enriched gas mixtures.

Understanding how membrane separation works and the factors that influence it is crucial for those designing and operating such systems, ensuring efficient and effective separation according to the desired outcomes.

The relationship between gas pressure and temperature is described by Gay-Lussac’s Law, which states that the pressure of a gas is directly proportional to its absolute temperature when the volume is held constant. This is expressed by the equation \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \) when comparing two states of the same amount of gas.

This relationship is an essential component of the Ideal Gas Law. When the temperature increases, the kinetic energy of the gas particles increases, causing more frequent and forceful collisions with the walls of their container, thereby increasing pressure. Conversely, cooling the gas would decrease its pressure.

The exercise presented concerns temperature held constant, which means the pressure difference across the rubber membrane is solely due to the application of different pressures, not temperature change. By maintaining a constant temperature, one can isolate the variable of pressure and explore its direct effect on molar density, as shown in the exercise. This principle is vital in various engineering and scientific scenarios, such as when managing the pressure conditions of chemical reactions or controlling environmental conditions in cultivation industries.