Graham's law calculator
Hydrogen at 2.016 g/mol effuses 3.9840953644 times as fast as oxygen at 32 g/mol, about four times, because the effusion rate falls with the square root of molar mass. Enter two molar masses to compare any pair of gases. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Molar mass 1
- 2.016
- Molar mass 2
- 32
- Rate ratio
- 3.9840953644
- Rate2 over rate1
- 0.250998008
- Mass ratio
- 15.873015873
- Precision
- 10
This Graham's law calculator returns the relative rate of effusion of two gases from their molar masses. Graham's law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass, so the ratio of rates is rate one over rate two equals the square root of M two over M one. You supply positive molar masses M1 and M2 in consistent units, typically grams per mole, and the engine returns the dimensionless rate ratio, the inverse ratio, and the mass ratio M2 over M1. A familiar textbook check is hydrogen near 2.016 grams per mole against oxygen at 32 grams per mole, which yields a rate ratio of about four: hydrogen effuses roughly four times as fast as oxygen under the same conditions. Zero or negative molar mass is rejected because the square-root law is undefined for non-positive masses. The path is fully deterministic arithmetic with no network calls, no Monte Carlo sampling, and no date-dependent shortcuts.
Where relative effusion rates fit in study, lab work, and software
Students meet Graham's law when a kinetic-molecular chapter connects molar mass tables to which gas escapes a porous container faster, when a lab compares helium and nitrogen leak rates, or when an exam supplies two molar masses and demands the relative effusion rate without a programmable calculator. Teachers can pre-generate answer keys for modest mass pairs without trusting opaque apps that silently invert which gas is in the numerator or drop the square root.
Process and materials engineers use the same contract as a quick sanity check before full transport models, confirming that a light process gas will out-effuse a heavier impurity under equal conditions. Developers call it from scripts to seed deterministic test vectors for chemistry dashboards, to normalize user-entered molar masses before plotting rate ratios, or to satisfy programming-challenge statements that demand the Graham ratio rather than a hand-waved estimate.
Contest and homework practice often needs a fast, offline source of rate1 over rate2 so solutions can be checked without graphing calculators under time pressure. Product and education tools use the evaluation when a learner types two molar masses and the UI must show the relative effusion rate plus the inverse comparison.
What you can do with it
Homework and exam checks
Compute rate1/rate2 = √(M2/M1) for textbook molar masses and verify which gas effuses faster before lab write-ups.
Tutoring and kinetics software
Normalize two user-entered molar masses into a Graham ratio for plotters, step helpers, and interactive lessons.
FAQ
What does the Graham's law calculator cost?
It is free in your browser on this page.
How is the relative rate of effusion defined?
rate1/rate2 = √(M2/M1), where M1 and M2 are the molar masses of gas 1 and gas 2 in the same units. The lighter gas has the larger effusion rate.
What happens if a molar mass is zero?
Zero or negative molar mass is invalid for Graham's law because the ratio places mass under a square root.
Does this tool look up molar mass from a chemical formula?
No. You supply numerical molar masses (for example 2.016 and 32). Use a molar-mass tool if you need formula-to-mass conversion first.
Why are results rounded to a precision?
Default precision is 10 decimal places; you can set precision from 0 to 15.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/chem/rate-of-effusion \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"molar_mass_1":2.016,"molar_mass_2":32}'const res = await fetch("https://api.kit.forhosting.com/chem/rate-of-effusion", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"molar_mass_1": 2.016,
"molar_mass_2": 32
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/chem/rate-of-effusion",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"molar_mass_1": 2.016,
"molar_mass_2": 32
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/chem/rate-of-effusion", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"molar_mass_1":2.016,"molar_mass_2":32}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"molar_mass_1":2.016,"molar_mass_2":32}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/chem/rate-of-effusion", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"molar_mass_1": 2.016,
"molar_mass_2": 32
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "chem.rate_of_effusion",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_abs_value | 1000000000000000 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
What Graham's law means and what the response contains
Graham's law of effusion links how quickly a gas escapes through a tiny opening to how heavy its molecules are. Written as rate one over rate two equals the square root of M two over M one, it says that the lighter gas always effuses faster when temperature and pressure are the same, because root-mean-square molecular speed scales with the inverse square root of molar mass. Effusion here means passage through a pinhole into vacuum or a much lower-pressure region; the same inverse-square-root dependence appears in classical kinetic theory for diffusion rates under similar idealizations. This calculator returns rate_ratio as rate of gas one relative to gas two, rate2_over_rate1 as the inverse comparison, mass_ratio as M two divided by M one, and echoes the normalized molar_mass_1 and molar_mass_2 values after rounding. The method tag is grahams_law, precision records the decimal width used for display, formula restates rate1 over rate2 equals sqrt of M2 over M1, and unit_note reminds you that both molar masses must share the same units and that the rate ratio is dimensionless. There is no lookup of molar masses from chemical formulas, no temperature or pressure correction, and no symbolic CAS path: the engine either evaluates a finite floating-point ratio 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, and any molar mass that is zero or negative, because Graham's law places molar mass under a square root in the denominator path and non-positive mass is physically and mathematically invalid.
How M1 and M2 are read and how the rate ratio stays reproducible
Units 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 ratio 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 effusion ratio. The preferred fields are molar_mass_1 for M1 and molar_mass_2 for M2, but aliases M1, m1, mm1, mass1, and molarMass1 for the first gas and M2, m2, mm2, mass2, and molarMass2 for the second 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 make the law undefined and negative mass has no meaning for a pure substance. 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 square-root ratio yourself.
Where relative effusion rates fit in study, lab work, and software
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 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.
Integration use cases
Deterministic gas-law fixtures
Generate reproducible rate_ratio and mass_ratio values for unit tests, property checks, and CI assertions.