Vapor-density molar mass measurement

An ideal gas law measurement obtains amount of substance without knowing the chemical formula in advance:Here $p$ is vapor pressure, $V$ is its occupied volume, $T$ is absolute temperature, $R$ is the molar gas constant, and $m$ is the mass of vapor occupying that volume at those conditions.

Santa Monica College's teaching experiment uses a $125\mathrm{mL}$ Erlenmeyer flask with a pinhole foil cap, a $600\mathrm{mL}$ water bath on a hot plate, a thermometer, and a barometer. Excess volatile liquid displaces air. A dissolved dye loses its colour and deposits when the liquid evaporates; three more minutes of heating follow. Cooling condenses retained vapor. Subtracting the dry assembly mass from the cooled assembly mass supplies $m$. The flask volume comes from brim-full water mass and mass density at its measured temperature. Controls include a dry exterior, excluding bath water, and three trials.^[ref]

The inference assumes vapor pressure equals atmospheric pressure, vapor and bath reach thermal equilibrium, the retained vapor mass is recovered on cooling, and the vapor obeys the ideal gas law. Residual air, thermal expansion of the flask, and vapor temperature below bath temperature are explicit error questions in the teaching experiment. Repeating trials assesses repeatability; it does not remove a shared bias.^[ref]

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 $R\approx 8.314462618\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ gives $n\approx 0.00408234\mathrm{mol}$ and $M_{\mathrm{inferred}}\approx 73.4873\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$ under the ideal gas law. These are chosen model inputs, not observations of a particular chemical substance.

An error's direction follows the model. Extra exterior water raises the apparent $m$ and hence $M_{\mathrm{inferred}}$. 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 $M_{\mathrm{inferred}}=(1-f)M_{\mathrm{true}}$. For a pure nonideal vapor, define $Z=pV/(nRT)$; then $M_{\mathrm{inferred}}=M_{\mathrm{true}}/Z$, 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.