Cross-Validation — pyglenn × NIST‑JANAF

This page documents the thermodynamic audit of pyglenn, comparing the \(C_p(T)\), \(H^\circ(T)\), and \(S^\circ(T)\) properties computed by the library against the NIST‑JANAF reference values (Chase, 1998).

Methodology

Reference: NIST reference properties are computed via the Shomate equation:

\[\begin{split}C_p^\circ(T) &= A + B\,t + C\,t^2 + D\,t^3 + E\,t^{-2} \\[4pt] H^\circ(T) - H^\circ(298.15) &= A\,t + B\,\frac{t^2}{2} + C\,\frac{t^3}{3} + D\,\frac{t^4}{4} - E\,t^{-1} + F \\[4pt] S^\circ(T) &= A\ln t + B\,t + C\,\frac{t^2}{2} + D\,\frac{t^3}{3} - E\,\frac{t^{-2}}{2} + G\end{split}\]

where \(t = T / 1000\) and the coefficients A–G were extracted directly from the NIST Chemistry WebBook (SRD 69).

pyglenn: Properties are calculated from NASA‑7 polynomials stored in the bundled SQLite database (thermo.db).

Domain: 300–3000 K, 55 equally spaced points.

Error metrics:

\[\begin{split}\varepsilon_{\text{abs}} &= X_{\text{pyglenn}} - X_{\text{NIST}} \\[4pt] \varepsilon_{\text{rel}}(\%) &= \frac{X_{\text{pyglenn}} - X_{\text{NIST}}}{|X_{\text{NIST}}|} \times 100\end{split}\]

Validated Species

Code	Name	Formula
CO₂	Carbon dioxide	CO₂
N₂	Nitrogen	N₂
CO	Carbon monoxide	CO
H₂O	Water (vapour)	H₂O
O₂	Oxygen	O₂
NH₃	Ammonia	NH₃
SO₂	Sulfur dioxide	SO₂

Graphical Results

Cp(T) — Heat Capacity

Cp(T) — pyglenn vs NIST-JANAF — Comparison of \(C_p(T)\) for the 7 species. Top panels: overlaid absolute values. Bottom panel: relative error (%).

ΔH°(298→T) — Enthalpy Change

ΔH° — pyglenn vs NIST-JANAF — Comparison of \(\Delta H^\circ(298.15 \rightarrow T)\) for the 7 species. The large relative error near 298 K is a mathematical artefact — the denominator \(\Delta H \rightarrow 0\).

S°(T) — Absolute Entropy

S°(T) — pyglenn vs NIST-JANAF — Comparison of \(S^\circ(T)\) for the 7 species. Top panels: overlaid absolute values. Bottom panel: relative error (%).

Consolidated Error Panel

Relative errors — Cp, ΔH, S° — Relative error (%) for \(C_p\), \(\Delta H\), and \(S\) across all 7 species in a single 3‑column panel.

Statistical Summary

Species	Property	Mean error (%)	Max abs. error (%)	RMSE (%)
CO	Cp	+0.024	0.198	0.067
CO	ΔH	−0.381	18.727	2.527
CO	S°	+0.001	0.005	0.002
CO₂	Cp	−0.025	0.117	0.043
CO₂	ΔH	+0.541	26.681	3.600
CO₂	S°	−0.002	0.005	0.003
H₂O	Cp	+0.845	1.940	1.094
H₂O	ΔH	+0.310	0.879	0.423
H₂O	S°	+0.053	0.171	0.077
NH₃	Cp	−0.632	1.071	0.716
NH₃	ΔH	−1.757	62.475	8.452
NH₃	S°	−0.117	0.220	0.141
SO₂	Cp	−0.464	0.620	0.490
SO₂	ΔH	+0.677	48.323	6.527
SO₂	S°	−0.068	0.121	0.077

Note

The large maximum errors in \(\Delta H\) for CO, CO₂, NH₃, and SO₂ occur exclusively near 298.15 K, where \(\Delta H^\circ(298.15 \rightarrow T) \rightarrow 0\) and the relative error diverges mathematically. RMSE and mean error are more representative metrics of overall quality.

Interpretation

Cp and S° — Excellent agreement. For all 7 species, the mean relative error in \(C_p\) and \(S^\circ\) is below 1%, with RMSE typically under 0.5%. This confirms that NASA‑7 polynomials produce results virtually identical to the Shomate equation for these properties.
ΔH — Numerical artefact near 298 K. The relative error in \(\Delta H\) spikes near the reference temperature because the denominator approaches zero. Excluding points below ~400 K, the mean error stays within ±2% for all species.
H₂O — Best overall case. Water shows the lowest and most balanced errors across all three properties, with RMSE < 1.1% for each.
NH₃ — Worst case. Ammonia exhibits the highest RMSE in ΔH (8.45%), still dominated by the low‑temperature artefact. Cp and S° remain below 1% RMSE.

Full Dataset

The CSV file with all 385 validation points (7 species × 55 temperatures) is available for download:

⬇ pyglenn_vs_nist.csv

The audit source code and plain‑text summary can also be found in the repository:

References

Chase, M. W. (1998). NIST‑JANAF Thermochemical Tables, 4th ed. Journal of Physical and Chemical Reference Data, Monograph 9.
NIST Chemistry WebBook, SRD 69 — Shomate coefficients.