= Molar mass measurement model
For a synthetic illustration, choose $m=0.3000\,\mathrm g$, $V=125.0\,\mathrm{mL}$, $p=101325\,\mathrm{Pa}$ and $T=373.15\,\mathrm K$. Using <molar gas constant> $R\approx8.314462618\,\mathrm{J\,mol^{-1}\,K^{-1}}$ gives $n\approx0.00408234\,\mathrm{mol}$ and $M_{\rm inferred}\approx73.4873\,\mathrm{g\,mol^{-1}}$ under the <ideal gas law>. These are chosen model inputs, not observations of a particular <chemical substance>.
\Image[/molar-mass-vapor-measurement-model.png]
{title=Vapor <molar mass> inference: apparatus, independent readouts, and the assumptions behind a synthetic <ideal gas law> calculation}
{id=molar-mass-vapor-measurement-model}
An error's direction follows the model. Extra exterior <water> raises the apparent $m$ and hence $M_{\rm inferred}$. If a fraction $f$ of the hot <gas> particles is residual air, and only the unknown vapor contributes to recovered $m$, an otherwise <ideal gas> calculation gives $M_{\rm inferred}=(1-f)M_{\rm true}$. For a pure nonideal vapor, define $Z=pV/(nRT)$; then $M_{\rm inferred}=M_{\rm true}/Z$, so the direction depends on $Z$. The dye endpoint alone cannot establish gas purity, <thermal equilibrium>, or ideality. These conditional checks explain why the same apparatus cannot provide a universal <molar mass> measurement for arbitrary <mixtures> or substances that decompose on heating.
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