Gas in a box

The gas in a box is a central model in Classical Statistical Mechanics. It serves as a bridge between microscopic mechanics and macroscopic thermodynamic behavior.
The system consists of many particles (atoms or molecules), confined in a box. It can be modeled at different levels:

Microscopically: Each state $i$ corresponds to a full specification of all particles’ positions and momenta, with energy $E_{i}$ .
Macroscopically: We describe bulk properties like temperature, pressure, and average energy.

When the box is placed in contact with a large heat bath at fixed temperature $T$ , the gas eventually reaches thermal equilibrium. In this situation:

The system's energy is not fixed.
Instead, if we assume the principle of maximum entropy, it follows a Boltzmann distribution:

p_{i} = \frac{e^{- β E_{i}}}{Z} .

Each $E_{i}$ corresponds to one or several possible microstates of the gas.
The system fluctuates between these states, exchanging energy with the environment.

Coolness

The parameter $β$ is called coolness. Justification: when $β$ is larger, the probability of the more energetic states it tiny.

Temperature

The parameter that is physically more intuitive than coolness is temperature, $T$ . Temperature is defined as the inverse of coolness, scaled by a fundamental constant.

T = \frac{1}{k β} or β = \frac{1}{k T} $ $ H e r e, $ k $ i s [[01 C O N C E P T S / B o l t z m a n n^{'} s c o n s t a n t ∥ B o l t z m a n n^{'} s c o n s t a n t]] ($ 1.380649 \cdot 10^{- 23} $ j o u l e s / k e l v i n) . I t i s a c o n v e r s i o n f a c t o r t h a t c o n n e c t s o u r m a c r o s c o p i c t e m p e r a t u r e s c a l e (K e l v i n) t o t h e m i c r o s c o p i c e n e r g y s c a l e . W i t h t h i s d e f i n i t i o n, t h e B o l t z m a n n d i s t r i b u t i o n c a n b e w r i t t e n i n i t s m o r e c o m m o n f o r m : $ $ p_{i} = \frac{e^{- E_{i} / k T}}{Z}

A low temperature (large $β$ ) makes the exponential term decay very quickly, meaning only the lowest energy states are probable. A high temperature (small $β$ ) "flattens" the distribution, making higher energy states more accessible.

The Physical Meaning of Temperature

While temperature is fundamentally a statistical parameter, it gains a clear physical meaning when we apply it to the gas in a box. In this model, the energy of the states ( $E_{i}$ ) is primarily the total kinetic energy of the particles.

If we use the Boltzmann distribution to calculate the average or expected energy of a single particle in the gas, the result of the calculation is remarkably simple:

⟨ E_{kinetic} ⟩ = \frac{3}{2} k T

This is a crucial insight, derived from the equipartition theorem. It shows that for a gas, the abstract parameter $T$ is directly proportional to the average kinetic energy of its constituent particles. The "hotness" we feel is a macroscopic manifestation of the average speed of microscopic particles.

This holds only for a classical ideal gas whose energy is purely quadratic. For other systems, temperature must be understood via its thermodynamic definition, via derivatives of entropy or partition functions

The Significance of Boltzmann's Constant

The fact that $k$ is an incredibly small number is a profound discovery. It reveals the vast difference in scale between our world and the atomic world.

Since $k$ is tiny (on the order of $10^{- 23}$ ), the energy of a single particle ( $\frac{3}{2} k T$ ) is also minuscule. However, the total thermal energy of the gas in the box is a tangible, macroscopic quantity. For the total energy to be large when the energy per particle is so small, the gas must be composed of an enormous number of particles.

This logic allows us to estimate the number of atoms in a given amount of gas. The result is on the order of $10^{23}$ , a number known as Avogadro's number. Therefore, the statistical model of a gas in a box, through the tiny value of Boltzmann's constant, provides powerful evidence for the atomic nature of matter.

Pressure

Imagine a cubic box of side length $L$ containing $N$ particles (molecules) of mass $m$ , moving in random directions.

Number density: $ρ = N / V$ , with $V = L^{3}$ .
The pressure $P$ is force per unit area, which comes from particles colliding elastically with the walls.

Take one molecule moving with velocity $v_{x}$ along the $x$ -axis.

When it hits a wall perpendicular to $x$ , it bounces elastically: its $x$ -momentum changes from $+ m v_{x}$ to $- m v_{x}$ .
Change in momentum:

Δ p_{x} = - m v_{x} - (m v_{x}) = - 2 m v_{x} .

So, each collision transfers $2 m v_{x}$ of momentum to the wall.

Now, how often does this particle hit the wall?

Time between successive hits on the same wall is $Δ t = \frac{2 L}{| v_{x} |}$ .
Collision frequency = $| v_{x} | / (2 L)$ .

Thus, average force from this one particle on the wall is

F = \frac{Δ p_{x}}{Δ t} = \frac{2 m v_{x}}{2 L / | v_{x} |} = \frac{m v_{x}^{2}}{L} .

Now we can compute the force per unit area (area of wall = $L^{2}$ ):

P_{one particle} = \frac{F}{L^{2}} = \frac{m v_{x}^{2}}{L^{3}} = \frac{m v_{x}^{2}}{V} .

For $N$ particles, the total pressure is, since $ρ = N / V$ ,

P = \frac{m}{V} \sum_{i = 1}^{N} v_{x, i}^{2} = ρ m ⟨ v_{x}^{2} ⟩,

where $⟨ v_{x}^{2} ⟩$ is the average over all particles.

By symmetry (since velocity distribution is isotropic):

⟨ v_{x}^{2} ⟩ = ⟨ v_{y}^{2} ⟩ = ⟨ v_{z}^{2} ⟩ = \frac{1}{3} ⟨ v^{2} ⟩,

where $⟨ v^{2} ⟩ = ⟨ v_{x}^{2} + v_{y}^{2} + v_{z}^{2} ⟩$ . Thus,

P = ρ m ⟨ v_{x}^{2} ⟩ = \frac{1}{3} ρ m ⟨ v^{2} ⟩ .