Problem 93 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

A gas container observes Maxwellian distribution law of speed. If the number of molecules between the speed \(5.0\) and \(5.1 \mathrm{~km}\) per sec at \(298 \mathrm{~K}\) is \(N\), what would be number of molecules between this range of speed if the total number of molecules in the vessel are doubled? (a) \(2 N\) (b) \(N\) (c) \(2 N^{2}\) (d) \(N^{2} / 2\)

Step-by-Step Solution

Answer

2N

Step 1: Understand the Maxwellian distribution

The Maxwellian distribution describes the distribution of speeds of particles in a gas. The number of molecules within a specific range of speeds is proportional to the total number of particles in the system assuming the temperature remains constant.

Step 2: Relate the number of molecules to the total number

The number of molecules with a certain speed is a fraction of the total number of molecules. If the total number of molecules is doubled, the number of molecules in any given speed range (assuming the same range and temperature) will also double.

Step 3: Apply the change to the number of molecules

Since only the total number of molecules is doubled while the temperature and the speed range remain the same, the number of molecules in the specific speed range between 5.0 and 5.1 km/s will also double.

Key Concepts

Statistical Mechanics

Statistical mechanics is a branch of physics that uses probability theory to describe and predict the behavior of particles within a system. This approach is particularly powerful because it allows scientists to study the collective behavior of vast numbers of particles, making sense of phenomena at the macroscopic scale based on the laws of motion at the microscopic scale.

For instance, in the context of our exercise, statistical mechanics explains the Maxwellian distribution, which is the probability distribution for the speeds of particles in a gas. This distribution is particularly significant because it encompasses a large number of particles, and despite the individual particles behaving unpredictably, the overall system conforms to a predictable pattern. By understanding statistical mechanics, you can grasp how temperature and other macroscopic properties affect molecular behavior, such as speed, in a systematic way.

Molecular Speed Distribution

Molecular speed distribution gives us a snapshot of how the speeds of molecules in a gas are spread out at a given temperature. The Maxwellian distribution is a specific example of this, showing that at any given moment, some molecules move quickly, others slowly, and many at moderate speeds. It's a bell-shaped curve that skews slightly to the right, reflecting the fact that there are a few very fast particles and no upper limit to the speeds, though extremely high speeds become statistically less likely.

To understand the exercise provided, it's important to realize that the number of molecules with speeds between 5.0 and 5.1 km/s is a tiny segment under this curve. Doubling the total number of molecules effectively doubles the area under the entire curve, but the shape remains the same, hence the proportional segment also doubles, leading to the answer being twice the original number of molecules, or '2N'.

Gas Kinetics

Gas kinetics involves the study of the motion of gas molecules, and it's a topic intimately related to both statistical mechanics and molecular speed distribution principles. It concerns itself with the rates of gas phase reactions, particle speed distributions, and how these elements contribute to observable gas properties, such as pressure and temperature.

One key takeaway from gas kinetics that relates to our exercise is how changes in a gas's physical parameters, like the number of particles, affect the behavior within the system. In our case, even though the total number of molecules in the gas is doubled, gas kinetics tells us that the temperature and the speed range remain the same. Thus, the kinetic energy distribution among the particles remains unchanged, and so the increment in number reflects directly in the number of molecules in any given speed bracket, preserving the Maxwellian distribution shape.