Statistical mechanics, structure of matter Foundations of equilibrium statistical mechanics Proposition

Sackur–Tetrode equation

Builds on Microcanonical ensemble Thermodynamic limit Stirling's formula

Statement

Consider a classical monatomic ideal gas of $N$ identical particles of mass $m$ in a container of volume $V$ in $\R^3$ , with Hamiltonian $H = \sum_{i=1}^N \abs{p_i}^2 / (2m)$ . Let $h > 0$ be the Planck constant and $k_B$ the Boltzmann constant. Define the microcanonical entropy at energy $U > 0$ by $S(U, V, N) = k_B \ln \frac{\abs{\set{H \le U}}}{N! \, h^{3N}},$ where $\abs{\cdot}$ is the phase-space volume. In the thermodynamic limit $N \to \infty$ with $V/N$ and $U/N$ fixed, $\frac{S}{N} = k_B \left[ \ln \left( \frac{V}{N} \left( \frac{4 \pi m U}{3 N h^2} \right)^{3/2} \right) + \frac{5}{2} \right] + o(1).$

Remarks

Proof idea: the set $\set{H \le U}$ is $V^N$ times a ball of radius $\sqrt{2 m U}$ in $\R^{3N}$ . Its volume is $V^N (2 \pi m U)^{3N/2} / \Gamma(3N/2 + 1)$ , and Stirling's formula applied to $\Gamma(3N/2 + 1)$ and $N!$ gives the result. With $U = \frac{3}{2} N k_B T$ the formula becomes $S = N k_B [\ln (V / (N \lambda^3)) + \frac{5}{2}]$ , where $\lambda = h / \sqrt{2 \pi m k_B T}$ is the thermal wavelength. The factor $1/N!$ is necessary: without it the entropy is not extensive, which is the Gibbs paradox. The formula is reliable only when $V/N \gg \lambda^3$ ; for low temperatures quantum statistics takes over, and it would give negative entropy.

Going deeper

For a student

The formula shows that $S$ is extensive: doubling $U$ , $V$ and $N$ doubles $S$ . Differentiating gives back the ideal gas laws: $1/T = \partial S / \partial U$ gives $U = \frac{3}{2} N k_B T$ , and $p/T = \partial S / \partial V$ gives $pV = N k_B T$ . The appearance of $h$ in a classical formula shows that absolute entropies need quantum mechanics.

Reading path

To understand this page from the ground up, read these 82 pages in order. The foundations come first.

Big O notation definition
Limit point of a set definition
Local maximum definition
Microcanonical ensemble concept
Partition of an interval definition
Sequence definition
Thermodynamic limit concept
Little o notation definition
Limit of a function definition
Local minimum definition
Mesh of a partition definition
Riemann sum definition
Real sequence definition
Subsequence definition
Asymptotic equivalence definition
Continuity at a point definition
Derivative definition
Riemann integrable function definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Continuous function definition
Differentiable function definition
Fermat's theorem on stationary points statement
Limit superior definition
Cauchy sequences are bounded statement
Sum of a finite geometric series statement
Monotone convergence theorem for sequences statement
Convergent sequence definition
Every sequence has a monotone subsequence statement
Pointwise convergence definition
Squeeze theorem statement
Partial sum definition
Power series definition
Intermediate value theorem statement
Differentiable functions are continuous statement
Derivative function definition
Bolzano–Weierstrass theorem statement
The sequence r^n tends to zero for |r| < 1 example
Radius of convergence definition
Convergent series definition
Uniform convergence definition
Sine and cosine definition
Boundedness theorem statement
Product rule statement
Higher derivative definition
Cauchy criterion for sequences statement
Uniform convergence on a set definition
Absolute convergence definition
A series of nonnegative terms converges iff its partial sums are bounded statement
Series of functions definition
Sum of a series definition
Extreme value theorem statement
Continuously differentiable function definition
Uniformly Cauchy sequence of functions definition
Geometric series example
Comparison test statement
Rolle's theorem statement
Root test statement
Ratio test statement
Weierstrass M-test statement
Mean value theorem statement
Cauchy–Hadamard theorem statement
Ratio formula for the radius of convergence statement
Power series converge uniformly on smaller discs inside the radius of convergence statement
Second fundamental theorem of calculus statement
Uniform convergence of derivatives allows term-by-term differentiation statement
The exponential series has infinite radius of convergence statement
Integration by parts statement
Term-by-term differentiation statement
Exponential function definition
Reduction formula concept
e object
Power series are differentiable inside the radius of convergence statement
Derivative of cosine statement
Wallis product statement
pi object
Stirling's formula statement
Sackur–Tetrode equation statement

Tag	`H2HGR6T5V0`
URI	`m2://H2HGR6T5V0`
Kind	statement, proposition
Writing	W0 stub
MSC	82B05, 82D05, 82B30
Status	This copy of the site has no build data. Run `m2 status H2HGR6T5V0`.
About	Microcanonical ensemble
Uses	Microcanonical ensemble, Thermodynamic limit, Stirling's formula
Same as	Wikidata Q1426583 Wikipedia Sackur–Tetrode equation
Changed	3 November 2026