# Kinetic theory for Charles' Law: does frequency and time interval of collisions change?

So I was given this table in class about Charles' Law: The table I got from class says that the time interval and frequency of collisions increases.

However, if the velocity of molecules and volume of the container increase, I'm under the impression that the two (velocity and volume) should cancel each other out to maintain a constant pressure, and since velocity and volume increased, then should the time interval and frequency of collisions not also stay the same?

KEY:
Red font: What my school says
Green font: What I think

We can show with some elaborate mathematics that the average speed of particles in a gas increases propotionally to the square root of its temperature: $$v_{avg}\propto \sqrt T$$ Now, the average time interval $\tau_{coll}$ between collisions of particles is just the average distance between them (the so-called "mean free path" $\lambda$) divided by their average speed, $$\tau_{coll} = \lambda / v_{avg}\propto \lambda/\sqrt T$$ where we've made use of the previous fact that $v_{avg}\propto\sqrt T$.
The mean free path is inversely proportional to the number density $n$ of the gas, i.e., the number of particles $N$ divided by the volume $V$: $$\lambda\propto 1/n = 1/(N/V) = V/N\propto V$$ so we see that $\lambda\propto V$.
Charles's Law gives us that $V\propto T$, so also $\lambda\propto T$. We substitute this proportionality into our definition of $\tau_{coll}$: $$\tau_{coll}\propto\lambda/\sqrt T\propto T/\sqrt T\propto\sqrt T$$ which shows that, indeed, the average time between collisions increases with temperature, and naturally the collision frequency decreases.