Sackur-Tetrode Equation and Gibbs Paradox

1. Counting States: The Path to the Sackur-Tetrode Equation

Our starting point is the Boltzmann definition of entropy, $S=k_B\ln \Omega (N,V,E)$, where $\Omega$ is the multiplicity, or the number of microstates corresponding to the macrostate defined by particle number $N$, volume $V$, and energy $E$. For a classical ideal gas, a microstate is a point in the $6N$-dimensional phase space. The challenge is to count how many such points correspond to our macrostate.

Classically, position and momentum are continuous, suggesting an infinite number of microstates. The resolution comes from a semi-classical approach: we partition phase space into discrete cells. The size of these fundamental cells is dictated by the uncertainty principle, with each cell having a volume of $h^{3N}$. This introduction of Planck's constant, $h$, is the first crucial ingredient.

The second ingredient is recognizing that for an ideal gas, the particles are indistinguishable. If we swap two identical particles, the microstate remains unchanged. This means we must divide our total count by $N!$, the number of ways to permute $N$ particles.

The following video provides an excellent and detailed walkthrough of the entire derivation. It systematically builds up the calculation, from counting states for a single particle to handling $N$ particles in a high-dimensional phase space, and finally using Stirling's approximation to arrive at the celebrated equation.

Let's summarize the key steps and the final result.

The number of microstates $\Omega (E)$ is found by calculating the volume of a thin shell of constant energy in phase space. For large $N$, it's sufficient to calculate the total phase space volume $\Phi (E)$ for all states with energy up to $E$ and then take its logarithm, as the vast majority of these states lie very close to the surface.

This volume is given by:

• The position space volume for $N$ particles in a box of volume $V$ is simply $V^N$.
• The momentum space volume is that of a $3N$-dimensional hypersphere of radius $p=\sqrt{2mE}$. The volume of a D-sphere is $V_D(R)=\frac{{\pi }^{D/2}}{\Gamma (D/2+1)}{R}^D$.
• The factor $h^{3N}$ correctly normalizes the phase space volume, making the multiplicity dimensionless.
• The factor $N!$ corrects for the overcounting of states due to the indistinguishability of particles.

Putting this together and taking the logarithm $S=k_B\ln \Phi (E)$, we apply Stirling's approximation ($\ln n!\approx n\ln n-n$) to the factorials (including the Gamma function, $\Gamma (z+1)=z!$). After some algebra, we arrive at the Sackur-Tetrode equation:

This equation expresses the absolute entropy of a monatomic ideal gas in terms of its macroscopic state variables. The presence of $h$ shows that even classical statistical mechanics is incomplete without this quantum correction. Now, let's use this result to investigate a famous puzzle.

2. The Gibbs Paradox and Its Resolution

The Gibbs paradox arises when we consider the process of mixing two gases. Let's analyze it with the help of the following text, which concisely lays out the derivation and the paradox.

Let's break down the paradox as described in the notes.

Scenario 1: Mixing Two Identical Gases
Imagine a box of volume $2V$ divided by a partition. On each side, we have the same ideal gas with $N$ particles in volume $V$ at temperature $T$. The total initial entropy is $S_{initial}=S(N,V,T)+S(N,V,T)=2S(N,V,T)$.

Now, we remove the partition. The gas expands to fill the entire volume $2V$. The final state is a gas of $2N$ particles in volume $2V$ at temperature $T$. Since entropy is an extensive property, we expect the final entropy to be $S_{final}=S(2N,2V,T)=2S(N,V,T)$. Therefore, the change in entropy, $\Delta S=S_{final}-S_{initial}$, should be zero. This makes physical sense, as removing a partition between two identical gases is a reversible process (we can just reinsert the partition to restore the original thermodynamic state).

Let's verify this with the Sackur-Tetrode equation. The key feature of the equation is its dependence on the density, $V/N$.

Indeed, $S_{final}=S_{initial}$, so $\Delta S=0$. The Sackur-Tetrode equation gives the correct, physically intuitive result.

The "Paradox"
The paradox emerges if we use an incorrect, non-extensive form for entropy. Suppose we had neglected the $1/N!$ correction. Our entropy formula would then depend on $\ln (V^N)$ instead of $\ln (V^N/N!)$, leading to a term like $N{k}_B\ln V$ instead of $N{k}_B\ln (V/N)$.
Let's call this incorrect entropy $S^{\prime }$.

• $S_{initial}^{\prime }=2\times N{k}_B[\ln V+C^{\prime }(T)]$
• $S_{final}^{\prime }=(2N)k_B[\ln (2V)+C^{\prime }(T)]=2N{k}_B[\ln V+\ln 2+C^{\prime }(T)]$
• $\Delta S^{\prime }=S_{final}^{\prime }-S_{initial}^{\prime }=2N{k}_B\ln 2>0$.

This incorrect result suggests that simply removing a partition between identical gases creates entropy, which is absurd. This non-zero change is called the "entropy of mixing".

Resolution
The resolution is clear: the paradox was never real, but a result of using an incorrect formula for entropy. The crucial factor is the $1/N!$ term, which accounts for particle indistinguishability. This correction ensures that entropy is properly extensive, depending on intensive quantities like density ($V/N$) rather than extensive ones like volume ($V$) alone.

Scenario 2: Mixing Two Distinguishable Gases
What if the two gases are different (e.g., Argon and Neon)? Now, when we remove the partition, an irreversible process occurs. Each gas freely expands to fill the entire volume $2V$. The total entropy change is the sum of the entropy changes for each gas expanding:

For our symmetric case ($N_A=N_B=N$, $V_A=V_B=V$, $V_{total}=2V$), this gives:

This positive change is physically correct. It reflects the fact that the mixed state is more disordered than the separated state, and work would be required (e.g., using semi-permeable membranes) to separate them again.

3. Deeper into the Paradox: A Question of Definition

The resolution via the $1/N!$ factor is standard, but the physicist E.T. Jaynes offered a more profound interpretation. He argued that the paradox arises from a subtle confusion about the very definition of a thermodynamic state.

Jaynes's point is that entropy is a property of the macrostate (defined by macroscopic variables like P, V, T), not the microstate. The paradox appears to create a discontinuity: the entropy of mixing is $2N{k}_B\ln 2$ as long as the gases are even infinitesimally different, but drops to zero the moment they become identical. Jaynes argues the discontinuity is not in the physics, but in what we mean by "restoring the original state".

Jaynes's perspective shifts the focus from a purely mathematical fix ($1/N!$) to a conceptual one. The crucial insight is that thermodynamics deals with what is macroscopically observable and controllable. When two identical gases mix, no macroscopic property changes. When distinguishable gases mix, properties like partial pressure and chemical composition change fundamentally. The entropy change reflects this macroscopic, operational difference.

The two key ingredients from our derivation play distinct but complementary roles in this resolution:

1. Indistinguishability ($1/N!$): This makes the statistical entropy formula extensive, matching the macroscopic requirement highlighted by Jaynes and Pauli. It ensures that $S$ scales correctly with the system size, depending on density $V/N$.
2. Phase Space Discretization ($h^{3N}$): This makes the argument of the logarithm dimensionless and provides a way to calculate an absolute entropy value. Without $h$, entropy would only be defined up to an arbitrary constant, making a definitive statement about the "entropy of mixing" ambiguous.

Conclusion

In this lesson, we have derived one of the foundational results of statistical mechanics and used it to resolve a classic puzzle.

Key Takeaways:

• The Sackur-Tetrode equation gives the absolute entropy of a classical monatomic ideal gas. Its derivation requires two semi-classical inputs: the quantization of phase space into cells of volume $h^{3N}$ and the indistinguishability of particles, which introduces a factor of $1/N!$.
• The Gibbs paradox is the apparent creation of entropy when mixing two identical gases, which arises from using an incorrect, non-extensive formula for entropy.
• The paradox is resolved by the $1/N!$ factor, which makes the entropy an extensive function of state. The entropy correctly depends on the density $V/N$, ensuring that no entropy change occurs when mixing identical gases.
• The mixing of distinguishable gases is an irreversible process that does lead to a positive entropy change, known as the entropy of mixing.
• Conceptually, the paradox highlights that entropy is a property of the macroscopic state. For identical gases, mixing does not change the macrostate, whereas for distinguishable gases, it does.

We have now seen how the statistical framework can be used to calculate a fundamental thermodynamic quantity for a specific system. In our next lesson, we will step back to a more general question: given the different ensembles we have constructed (microcanonical, canonical, grand canonical), how do we justify the choice of a specific ensemble for a given physical problem? We will explore this based on the physical constraints of a system and the computational convenience each ensemble offers.