Animated Solution for Physics - Thermodynamics: Calculate the value of mean free path ($\lambda$) for oxygen molecules at temperature $27^{\circ }\text{C}$ and pressure $1.01\times 10^5\text{ Pa}$. Assume the molecular diameter $0.3\text{ nm}$ and the gas is ideal. ($k_B=1.38\times 10^{-23}{\text{ JK}}^{-1}$)

Imagine you are shrunk down to the size of an oxygen molecule, trapped inside a container. It is absolute chaos. You are zipping around at hundreds of meters per second, but you can't go very far before—BAM!—you collide with another molecule. The average distance you manage to travel in a straight line between these violent collisions is what physicists call the mean free path ($\lambda$).

In this problem, we are tasked with finding exactly how far an oxygen molecule travels between collisions at a pleasant room temperature of $27^{\circ }\text{C}$ and standard atmospheric pressure.

To find the mean free path, we rely on a beautiful result from the Kinetic Theory of Gases:

Let's break down why this formula makes perfect physical sense: - Temperature ($T$) is in the numerator: Higher temperature means molecules are moving faster and the gas tends to expand (if not rigidly confined), increasing the space between molecules. - Pressure ($p$) is in the denominator: Higher pressure means the molecules are packed more densely. More crowding equals more frequent collisions, hence a shorter path. - Diameter squared ($d^2$) is in the denominator: The larger the molecule, the bigger its "target area" ($\pi d^2$). A bigger target means it's much harder to avoid hitting others. - The $\sqrt{2}$ factor: This accounts for the fact that all molecules are moving, not just the one we are tracking. It represents the relative velocity between colliding particles.

Before we plug numbers into our master equation, we must ensure absolute harmony in our units. Physics is unforgiving when it comes to mixed units!

- Pressure: $p=1.01\times 10^5\text{ Pa}$ (Already in standard SI units). - Temperature: $T=27^{\circ }\text{C}$. We must convert this to absolute temperature (Kelvin).

- Diameter: $d=0.3\text{ nm}$. We convert this to meters.

- Boltzmann Constant: $k_B=1.38\times 10^{-23}\text{ J/K}$.

Now, we carefully substitute our pristine data into the formula:

Let's tackle the numerator first. It's straightforward:

Next, the denominator. We must square the diameter first:

Now, multiply the constants in the denominator ($\sqrt{2}\approx 1.414$, $\pi \approx 3.14$):

Finally, we divide the numerator by our simplified denominator:

To match our options, we convert the result back into nanometers ($1\text{ nm}=10^{-9}\text{ m}$):

The mean free path is approximately $102\text{ nm}$. This means an oxygen molecule travels about 340 times its own diameter before crashing into a neighbor. The microscopic world is indeed a crowded, chaotic place!