Question

If the collision frequency of a gas at 1 atm pressure is \(Z\), then its collision frequency at \(0.5\) atm is (1) \(1.0 \mathrm{Z}\) (2) \(0.25 \mathrm{Z}\) (3) \(2 \mathrm{Z}\) (4) \(0.50 \mathrm{Z}\)

Short answer

0.50Z

Step-by-step solution

Understand the Relationship

The collision frequency of gas molecules is directly proportional to the pressure of the gas. This means that if the pressure changes, the collision frequency will change by the same factor.

Determine the Factor

Calculate the factor by which the pressure changes. The original pressure is 1 atm, and the new pressure is 0.5 atm. Thus, the pressure is reduced to half its original value.

Apply the Factor

Since the collision frequency is directly proportional to the pressure, reducing the pressure by half will reduce the collision frequency by the same factor. Therefore, the new collision frequency is 0.5 times the original collision frequency (0.5Z).

Key concepts

Gas Pressure

Gas pressure is a measure of the force that gas molecules exert on the walls of their container. This force is the result of collisions between gas molecules and the walls. When gas molecules collide, they exert a tiny push, which accumulates to generate pressure. Imagine a balloon filled with air. The air molecules inside are in constant motion, bouncing off the inner walls of the balloon and each other. These collisions are what keep the balloon inflated and give it shape.

The higher the number of collisions, the higher the pressure. If more gas molecules are added to the balloon (or the volume is reduced), there will be more collisions, and the pressure will increase. Conversely, removing gas or increasing the volume will decrease the number of collisions, lowering the pressure.

Direct Proportionality

Direct proportionality means that if one quantity changes, the other changes by the same factor. In the context of this exercise, collision frequency and gas pressure are directly proportional to each other. This means that if the pressure of the gas doubles, the collision frequency will also double.

• If the pressure increases, the collision frequency increases.
• If the pressure decreases, the collision frequency decreases.

For example, if the working pressure of a gas is reduced from 1 atm to 0.5 atm, this halving of pressure results in a halving of the collision frequency. In mathematical terms, if the pressure is divided by 2, the collision frequency is also divided by 2. This relationship helps simplify calculations, as changes in pressure directly translate to proportional changes in collision frequency.

Collision Factor

The collision factor is an important element when discussing the kinetic theory of gases. It refers to the frequency with which gas molecules collide with each other and with the walls of their container. This factor is crucial for understanding phenomena such as diffusion, effusion, and even reaction rates in chemical kinetics.

In our exercise, the collision factor (or frequency) can be denoted as 'Z'. If we decrease the pressure from 1 atm to 0.5 atm, the collision frequency also decreases proportionally. Since the original pressure is halved, the collision frequency is also halved, resulting in a new collision frequency of 0.5Z.

Understanding the collision factor helps in studying gas behaviors under various conditions. It provides insight into how often gas particles interact in a given environment, which is foundational for more advanced topics like gas laws and kinetic molecular theory.