open HCV - 2 chapter - Kinetic Theory of Gases .........

Now in the assumptions of KTG .. the last assumption is explained by taking examples........ i have a problem there...

the first example is clear...

in the second example, it says that, for a gas, if there are 400 molecules in 1cm³ having velocity along x-direction , then there will be 400 molecules in some other 1cm³ having velocity along x-direction. (upto here, its clear)

Then it says, that also there will be 400 molecules in 1cm³ having velocity along y-direction.

This the centre of confusion.

Does existence of 400 molecules having velocity along x-direction in 1cm³ of a gas imply that there will be exactly 400 molecules having velocity along y-direction in 1cm³ of the gas ?????

If yes,then how can we conclude ???... there is no mention about this in the assumptions of KTG.

If no, then why does the book say so ??...

Let's dive into the assumptions of the Kinetic Theory of Gases (KTG) and clarify the point you're grappling with regarding the distribution of gas molecules and their velocities. The confusion seems to stem from the interpretation of how gas molecules behave in three-dimensional space, particularly concerning their velocities in different directions.

Understanding Molecular Distribution

The Kinetic Theory of Gases is built on several key assumptions about the behavior of gas molecules. One of the fundamental ideas is that gas molecules are in constant random motion, and this motion can be described in terms of their velocities in three dimensions: x, y, and z directions.

Velocity Distribution in Gases

When the text states that there are 400 molecules in a 1 cm³ volume moving along the x-direction, it is important to understand that this is just a snapshot of the gas's behavior at a given moment. The assumption of random motion implies that the velocities of gas molecules are not confined to just one direction.

• Random Motion: Gas molecules move in all directions. Therefore, if you have a certain number of molecules moving in the x-direction, it is reasonable to expect that there will be a similar number moving in the y-direction and z-direction, assuming the gas is isotropic (the same in all directions).
• Statistical Nature: The KTG relies on statistical mechanics. While you might have 400 molecules moving in the x-direction at a specific moment, the average distribution of velocities in a large sample of gas will tend to be equal across all directions over time.

Example to Illustrate

Imagine a room filled with people (representing gas molecules) moving randomly. If you observe a group of 400 people moving towards the east (x-direction), it’s likely that, if you look at another group of the same size, you will find a similar number moving north (y-direction) and upwards (z-direction). This is because, in a large enough space and with random motion, the distribution tends to balance out.

Conclusion on the Assumption

So, to address your question directly: the statement in your book is based on the assumption of isotropy in the gas. While it does not explicitly state that there will be exactly 400 molecules moving in the y-direction, it implies that, statistically, in a large volume of gas, the number of molecules moving in each direction will be roughly equal. This is a fundamental characteristic of gases under the assumptions of KTG, which is why the text presents it this way.

In summary, the Kinetic Theory of Gases assumes that gas molecules are uniformly distributed in their velocities across all directions, leading to the conclusion that if you have a certain number of molecules moving in one direction, you can expect a similar number moving in the other directions as well, especially in a large sample. This statistical approach is what underpins the theory and helps explain the behavior of gases in various conditions.