Effusion remaining fraction calculator

You supply positive molar masses M1 and M2 in consistent units, typically grams per mole, plus a non-negative time_ratio equal to elapsed time divided by the heavier gas one-over-e lifetime, and the engine returns the remaining fraction of the lighter gas, the companion remaining fraction of the heavier gas, the Graham factor square root of M heavy over M light, and a percent form of the lighter remainder. A familiar textbook check is hydrogen near 2.016 grams per mole against oxygen at 32 grams per mole with time_ratio 0.5, which leaves roughly fourteen percent of the hydrogen while about sixty-one percent of the oxygen remains.

What the gas-leak remaining fraction means and what the response contains

When a closed vessel loses gas through a tiny orifice into vacuum or a much lower-pressure region, ideal kinetic theory says the instantaneous effusion rate of each species is proportional to its partial density divided by the square root of its molar mass. That first-order decay integrates to an exponential remaining fraction n over n zero equals e to the minus kappa t over square root of M for each pure component under a shared orifice factor kappa. This calculator anchors the clock to the heavier gas by defining time_ratio as t divided by tau heavy, where tau heavy is the heavier gas one-over-e lifetime square root of M heavy over kappa, so the heavier remaining fraction is simply e to the minus time_ratio. The lighter remaining fraction is then e to the minus time_ratio times the Graham factor square root of M heavy over M light, which is always smaller than or equal to the heavier remainder when time_ratio is positive. The response echoes molar_mass_1, molar_mass_2, and time_ratio after rounding, names lighter_molar_mass and heavier_molar_mass, reports graham_factor, remaining_fraction_lighter, remaining_fraction_heavier, remaining_percent_lighter as one hundred times the lighter fraction, the method tag graham_effusion_leak_exponential, the precision used, a formula reminder, and a unit note. There is no lookup of molar masses from chemical formulas, no temperature or pressure correction beyond the ideal shared kappa, and no symbolic CAS path: the engine either evaluates finite floating-point exponentials inside the supported magnitude range or rejects the input with invalid_input. Rejected cases include missing required fields, non-finite values, magnitudes above the documented ceiling, any molar mass that is zero or negative, and a negative time_ratio, because elapsed time scaled by a positive lifetime cannot run backward in this model.

How M1, M2, and time_ratio are read and how results stay reproducible

Units for the two molar masses are your responsibility: both values must use the same molar-mass unit, almost always grams per mole in general chemistry tables; mixing grams per mole with kilograms per mole would scale the Graham factor incorrectly without any automatic conversion in this tool. Optional leading plus signs and scientific notation on numeric strings are allowed when they parse as finite decimals; embedded spaces, hex prefixes, and non-numeric tokens are rejected with a clear message so a bad cell does not become a wrong remaining fraction. The preferred fields are molar_mass_1 for M1, molar_mass_2 for M2, and time_ratio for the dimensionless elapsed ratio t over tau heavy, but aliases M1, m1, mm1, mass1, and molarMass1 for the first gas, M2, m2, mm2, mass2, and molarMass2 for the second, and timeRatio, t_ratio, elapsed_ratio, elapsed_time_ratio, and tau for the time ratio are accepted so spreadsheets, query strings, and typed forms can reuse the same contract. Each molar mass must be strictly positive: if either is zero or negative, the request fails with invalid_input because zero molar mass would place mass under a square root in the kinetic lifetime and negative mass has no meaning for a pure substance. The time_ratio must be non-negative; zero means no elapsed leak time so both remaining fractions equal one, and large positive values drive both fractions toward zero with the lighter gas falling faster. An optional precision field, with aliases decimals and dp, chooses how many decimal places to expose for the numeric fields, defaulting to ten and capped at fifteen so JSON goldens stay stable. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the leak exponentials yourself.

Where lighter-gas remaining fractions fit in study, lab work, and software

Contest and homework practice often needs a fast, offline source of remaining_fraction_lighter so solutions can be checked without graphing calculators under time pressure. On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split — free interactive exploration and a priced, identical API — matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with the relative effusion-rate calculator when you only need rate one over rate two, with STP volume or molar concentration tools when you need related gas-law steps, and with other chem endpoints when you are building a small lab of free, deterministic chemistry helpers.