Kinetic Theory and Ideal Gas Pressure

1. The Microscopic Model of an Ideal Gas

Before we can derive anything, we need to define our model. The kinetic theory simplifies the complex reality of a gas into a manageable model called an "ideal gas." This model is built on a few key assumptions.

To summarize, the core assumptions are:

1. Large Number of Molecules: The gas consists of a very large number of identical molecules in continuous, random motion.
2. Negligible Molecular Volume: The volume of the molecules themselves is negligible compared to the volume of the container they occupy.
3. No Intermolecular Forces: Molecules do not exert any long-range forces on each other. They interact only during collisions.
4. Elastic Collisions: All collisions, both between molecules and with the walls of the container, are perfectly elastic. This means kinetic energy and momentum are conserved.

These assumptions allow us to use mechanics and statistics to build our theory.

2. From a Single Collision to Total Pressure

Our goal is to calculate the pressure, which is force per unit area ($P=F/A$). The force exerted by the gas on a container wall comes from the continuous bombardment of countless molecules colliding with it. We will derive the expression for this force by following these steps:

1. Calculate the momentum transferred to the wall by a single molecule in one collision.
2. Calculate the rate at which molecules collide with the wall.
3. Combine these to find the total force, and thus the pressure.

The following reading from the Feynman Lectures provides an elegant and insightful derivation.

Let's break down the key steps from that reading to ensure the logic is crystal clear.

Step 1: Momentum Change per Collision
Consider a molecule with mass $m$ and x-component of velocity $v_x$ moving towards a wall parallel to the y-z plane. In an elastic collision, its x-velocity reverses to $-v_x$.

• Initial x-momentum: $p_{ix}=m{v}_x$
• Final x-momentum: $p_{fx}=-m{v}_x$
• Change in molecule's momentum: $\Delta p_{molecule}=p_{fx}-p_{ix}=-2m{v}_x$
• By Newton's third law, the momentum transferred to the wall is the opposite of this: $\Delta p_{wall}=2m{v}_x$.

Step 2: Collision Rate
How many molecules hit an area $A$ of the wall per second?

• In a time interval $\Delta t$, only molecules within a distance $v_x\Delta t$ from the wall can reach it.
• The volume of this "collision cylinder" is $V_{cyl}=A\cdot v_x\Delta t$.
• If the number of molecules per unit volume (number density) is $n$, there are $n\cdot V_{cyl}=nA{v}_x\Delta t$ molecules in this volume.
• Since the motion is random, we assume on average half of these molecules are moving towards the wall and half are moving away.
• Number of collisions in time $\Delta t$: $N_{coll}=\frac{1}{2}(nA{v}_x\Delta t)$.
• The collision rate (collisions per second) is $\frac{N_{coll}}{\Delta t}=\frac{1}{2}nA{v}_x$.

Step 3: Force and Pressure
Force is the total momentum transferred per unit time.
$F=(\text{momentum per collision})\times (\text{collision rate})$
$F=(2m{v}_x)\times \left(\frac{1}{2}nA{v}_x\right)=nAm{v}_x^2$
Pressure is force per area, so we divide by $A$:
$P=\frac{F}{A}=nm{v}_x^2$

3. Averaging and the Final Expression

The equation $P=nm{v}_x^2$ is for a simplified case where all molecules have the same x-velocity. In reality, molecules have a wide distribution of velocities. We must therefore use the average value of $v_x^2$, denoted as $⟨v_x^2⟩$.

So, the pressure is correctly written as:
$P=nm⟨v_x^2⟩$

The final step is to relate the average of the x-component of velocity to the overall average speed of the molecules. Because the molecular motion is random, there is no preferred direction. This principle is called isotropy.
This implies that the average motion along each axis is the same:
$⟨v_x^2⟩=⟨v_y^2⟩=⟨v_z^2⟩$
The total squared speed for any molecule is given by the Pythagorean theorem in 3D: $v^2=v_x^2+v_y^2+v_z^2$. Averaging this over all molecules gives:
$⟨v^2⟩=⟨v_x^2⟩+⟨v_y^2⟩+⟨v_z^2⟩=3⟨v_x^2⟩$
From this, we find the crucial relationship:
$⟨v_x^2⟩=\frac{1}{3}⟨v^2⟩$
Here, $⟨v^2⟩$ is the mean square speed of the molecules.

Substituting this back into our pressure equation gives the final result:
$P=nm\left(\frac{1}{3}⟨v^2⟩\right)⟹P=\frac{1}{3}nm⟨v^2⟩$

The following video provides a complete walkthrough of a similar derivation, which can help reinforce the concepts. Note that it calculates the force from a single particle by considering the time between its successive collisions with the same wall, a slightly different but equally valid logical path to the same result.

4. Summary of the Derivation

The entire derivation can be visualized with the following steps.

Let's restate our final equation in a few useful forms:

1. In terms of number density ($n$) and mean square speed ($⟨v^2⟩$):
   $P=\frac{1}{3}nm⟨v^2⟩$
2. In terms of total number of molecules ($N$) and Volume ($V$): Since $n=N/V$,
   $PV=\frac{1}{3}Nm⟨v^2⟩$
3. In terms of gas density ($\rho$): Since $\rho =nm$,
   $P=\frac{1}{3}\rho ⟨v^2⟩$
4. In terms of average kinetic energy ($⟨K.E.⟩$): Noting that $\frac{1}{2}m⟨v^2⟩$ is the average translational kinetic energy of a molecule,
   $PV=\frac{2}{3}N\left(\frac{1}{2}m⟨v^2⟩\right)=\frac{2}{3}N⟨K.E.{⟩}_{avg}$
   This last form is powerful because it directly links the macroscopic quantities $P$ and $V$ to the average microscopic kinetic energy of the gas molecules.

Conclusion

In this lesson, you have seen how the abstract concept of pressure can be derived from the concrete mechanics of molecular collisions. This is a cornerstone of statistical mechanics.

Key Takeaways:

• Gas pressure is the result of momentum transfer from countless molecular collisions with the container walls.
• The derivation hinges on three main steps: calculating momentum change per collision, finding the collision rate, and averaging over all molecules.
• The assumption of isotropy (random motion with no preferred direction) is critical for relating the one-dimensional motion ($⟨v_x^2⟩$) to the overall three-dimensional motion ($⟨v^2⟩$), giving us $⟨v_x^2⟩=\frac{1}{3}⟨v^2⟩$.
• The final expression, $P=\frac{1}{3}nm⟨v^2⟩$, provides a direct link between the macroscopic property of pressure and the microscopic average motion of molecules.

Preview of the Next Lesson:
We ended with the equation $PV=\frac{2}{3}N⟨K.E.{⟩}_{avg}$. You also know the empirical Ideal Gas Law, $PV=nRT=N{k}_{B}T$. In our next lesson, we will combine these two powerful equations. This will reveal one of the most profound results in kinetic theory: the direct relationship between temperature and the average kinetic energy of gas molecules. We will then define and calculate the different types of molecular speeds: mean, root-mean-square, and most probable.