Question

A closed vessel contains equal number of nitrogen and oxygen molecules at a pressure of \(P \mathrm{~mm}\). If nitrogen is removed from the system, then the pressure will be (a) \(P\) (b) \(2 P\) (c) \(P / 2\) (d) \(P^{2}\)

Short answer

The pressure will be \(P / 2\).

Step-by-step solution

Understanding the relationship between gas molecules and pressure

According to the ideal gas law, pressure (P) is directly proportional to the number of moles (n) of the gas when volume and temperature are held constant. The equation can be written as PV = nRT, where R is the gas constant and T is the temperature. Since the number of molecules is directly proportional to the number of moles, if we remove half of the molecules (all nitrogen molecules, in this case), the number of moles will also reduce to half.

Applying Dalton's Law of Partial Pressures

Dalton's Law states that the total pressure exerted by the mixture of non-reacting gases is equal to the sum of the partial pressures of individual gases. Here, initially, the pressure is due to the equal contribution of nitrogen and oxygen. When the nitrogen is removed, only the pressure due to oxygen remains.

Calculating the new pressure

Considering the vessel initially contained equal numbers of nitrogen and oxygen molecules, they would have exerted equal pressures, thus each contributing to half of the total pressure. So, if nitrogen is removed, the pressure will drop to the contribution of oxygen alone, which is half of the total initial pressure.

Key concepts

Partial Pressures

In the realm of gases, understanding partial pressures is crucial to comprehending how multi-component gas systems behave. Partial pressure refers to the pressure that one component of a mixture of gases would exert if it occupied the entire volume of the mixture alone.
Picturing a party balloon filled with both helium and air provides a real-world analogy. If you could magically remove all the air, leaving only the helium, the balloon would still remain inflated, but the pressure inside it would decrease. This is because only the helium would now be contributing to the total pressure inside the balloon.

Relation to Moles and Volume

In quantifiable terms, the partial pressure of a gas is directly associated with the number of moles of that gas in a given volume, assuming the temperature remains constant—a fundamental aspect of the ideal gas law formula:
\( P_{gas} = n_{gas} \times \frac{RT}{V} \)
where \( P_{gas} \) is the partial pressure of the gas, \( n_{gas} \) is the number of moles of the gas, \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( V \) is the volume. The beauty of this equation lies in its simplicity—showing how the pressure of a gas in a mixture links to the amount of that gas present.

Dalton's Law of Partial Pressures

Building on the concept of partial pressures, we encounter Dalton's Law of Partial Pressures, an essential principle in understanding gas mixtures. Stated by John Dalton, the law posits that the total pressure exerted by a mixture of non-reacting gases is the sum of their individual partial pressures.
Imagine our gas mixture as a team of runners, each runner's speed being akin to the partial pressure of a gas. Dalton's Law tells us that the speed of the team (total pressure) is the sum of each runner's speed (partial pressures). It doesn't matter how fast one runner is, the total speed is the collective effort of all.

Mathematical Expression

Mathematically, Dalton's Law can be expressed as:
\( P_{total} = P_{1} + P_{2} + … + P_{n} \)
where \( P_{total} \) is the total pressure of the gas mixture, and \( P_{1}, P_{2}, …, P_{n} \) are the partial pressures of each component gas. This formula becomes particularly relevant when investigating the behavior of a gas mixture when the composition changes, such as removing one of the gases, which leads to a direct change in the total pressure.

Gas Molecules and Pressure

Delving into the molecular level provides a more foundational perspective on how pressure is generated by gas molecules. Pressure, in the context of gases, is essentially the force exerted by the gas molecules as they collide with the walls of their container.
The more molecules you have in a given space, the more frequent these collisions become, and this increase in collision frequency leads to a higher pressure. It is as if you have a crowd of people in a room—more people moving around means more bumping into the walls and each other.

Direct Proportionality with Number of Molecules

This concept is underpinned by the ideal gas law, wherein the pressure is directly proportional to the number of molecules, which also translates to the number of moles of a gas. When the question of removing gas molecules arises, as in the exercise given, this relationship helps us predict the resultant change in pressure.
To use a basic analogy, if you have a box half-filled with tennis balls (gas molecules) shaking the box (the container) results in a certain noise level (pressure). If you remove half of the balls, the noise level decreases accordingly. Likewise, removing half of the gas molecules from a container reduces the pressure by half, assuming the temperature and volume remain constant, which aligns with the finding in the exercise where the pressure dropped to \( P / 2 \) after removing the nitrogen molecules.