Van der Waals critical constants calculator
Turn the two van der Waals constants of a gas into its critical point: a nitrogen-like gas with a of 1.382 and b of 0.0387 L/mol has a critical volume of 0.1161 L/mol, with critical temperature and pressure alongside. Free in your browser.
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- Your data stays on your device
Run — free
Worked example
- A
- 1.382
- B
- 0.0387
- Gas constant
- 0.0831446262
- Critical temperature
- 127.259178618
- Critical pressure
- 34.1760879656
- Critical volume
- 0.1161
- Tc
- 127.259178618
The van der Waals equation of state corrects the ideal gas law with two substance-specific constants: a, which captures attractive intermolecular forces that lower the observed pressure, and b, which accounts for the finite volume occupied by the molecules themselves. At the critical point the isotherm has a horizontal inflection, and solving that condition yields closed-form critical constants: critical temperature Tc equals 8a over 27 R b, critical pressure Pc equals a over 27 b squared, and critical molar volume Vc equals 3b. This calculator accepts a and b (and an optional gas constant R), returns Tc, Pc, Vc, and the universal van der Waals compressibility factor Zc of three-eighths, and rejects a zero excluded-volume constant with a clear validation error so you never divide by zero.
How van der Waals critical constants are derived
Start from the molar van der Waals equation written as pressure plus a over Vm squared, times Vm minus b, equals R T. At the critical point the isotherm on a pressure–volume diagram has both a first and a second derivative with respect to volume equal to zero: the horizontal inflection that marks the end of the two-phase region.
Differentiating once and twice with respect to molar volume, setting those derivatives to zero at the critical state, and solving the resulting algebraic system eliminates pressure and temperature and leaves a unique critical volume Vc equals three times b. Substituting that volume back into the derivative conditions fixes the critical pressure as Pc equals a divided by twenty-seven b squared. Finally the original equation of state at that critical volume and pressure yields the critical temperature Tc equals eight a divided by twenty-seven R b.
Because every step is algebraic, the critical constants are exact functions of a, b, and R with no iterative root finding. The ratio Zc equals Pc times Vc over R times Tc collapses to the universal van der Waals value three-eighths for any substance whose a and b are used consistently. Entries must be finite; NaN and Infinity are rejected.
A zero or near-zero b fails with a message that critical formulas divide by powers of b, so clients can fix the table value rather than guessing. Non-positive a is rejected because attraction must be positive for a physical critical point above absolute zero.
What you send and what you get back
Send two required fields and one optional field. The field a is the van der Waals attraction constant as a finite positive number, for example one point three eight two in litre-squared bar per mole squared for a nitrogen-like gas. The field b is the excluded-volume constant as a finite positive number greater than zero, for example zero point zero three eight seven litres per mole; b equal to zero is rejected because Pc and Tc divide by powers of b.
If you work in litre-bar or litre-atmosphere tables, pass a matching R so temperature comes out in kelvin. Formula strings document each expression for homework write-ups. Classic textbook checks include nitrogen, carbon dioxide, and water tables of a and b from physical chemistry handbooks.
Where critical-constant calculators show up in real work
Physical chemistry and chemical engineering courses constantly assign problems that convert tabulated van der Waals a and b into Tc, Pc, and Vc, then compare those predictions against experimental critical data; a single trusted oracle that returns every critical constant beside the formula string speeds grading and demo setup. Process engineers estimating corresponding-states reduced variables Tr equals T over Tc and Pr equals P over Pc need quick critical anchors before applying generalized compressibility charts or cubic EOS packages.
Research software that fits a and b from laboratory isotherms often cross-checks the implied critical point against handbook values, and a hard rejection when b is zero prevents silent infinity in downstream reduced-temperature loops. Data-table maintainers who republish a and b in mixed unit systems can batch-verify that Zc remains three-eighths after unit conversion, catching transposition errors in a or b.
If you later need compressibility factor Z at an arbitrary state, ideal-gas molar volume, or full cubic root finding for subcritical isotherms, treat those as separate search intents and landings rather than overloading this van der Waals critical constants calculator page.
What you can do with it
Corresponding-states setup
Anchor reduced temperature and pressure calculations with consistent critical constants before using generalized compressibility charts or cubic EOS packages.
FAQ
Do I need to provide the gas constant R?
No. R defaults to 8.314462618 J/(mol·K) for SI units.
What is Zc for a van der Waals gas?
Zc = Pc·Vc/(R·Tc) equals 3/8 exactly when a, b, and R are dimensionally consistent. The response includes the computed Zc for verification.
What does it cost?
The browser tool is free and runs locally.
What happens if the constant b is zero?
The calculator rejects it, because the critical pressure and temperature are divided by powers of b. Zero or negative values of a are refused too, since the attraction constant must be positive for a real critical point.
Which units should a and b be in?
Any consistent set that matches the gas constant. With the default constant, use pascal times cubic metre squared per mole squared and cubic metres per mole; with litres and bar, set the constant to 0.08314462618.
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/critical-constants \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":1.382,"b":0.0387}'const res = await fetch("https://api.kit.forhosting.com/chem/critical-constants", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 1.382,
"b": 0.0387
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/chem/critical-constants",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 1.382,
"b": 0.0387
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/chem/critical-constants", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":1.382,"b":0.0387}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":1.382,"b":0.0387}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/chem/critical-constants", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 1.382,
"b": 0.0387
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "chem.critical_constants",
"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.
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
How van der Waals critical constants are derived
The entire pipeline is deterministic floating-point arithmetic: no random seeds, no wall-clock timestamps, no network calls, and no model drift, so identical a, b, and R always produce identical JSON.
What you send and what you get back
The optional field gas_constant is the universal gas constant R used only in Tc; when omitted it defaults to eight point three one four four six two six one eight joules per mole kelvin for SI units where a is in pascal metre to the sixth per mole squared and b is in cubic metres per mole. Missing fields, non-numeric strings, NaN, Infinity, non-positive a, non-positive b, and non-positive R fail with invalid_input and are not charged. On success the response echoes cleaned a, b, and gas_constant, flags whether R was defaulted, returns critical_temperature, critical_pressure, and critical_volume together with short aliases Tc, Pc, and Vc, and reports Zc which should be three-eighths when units are consistent. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance.
Where critical-constant calculators show up in real work
Because this capability is a pure parse engine, API pricing is the published base of $0.002 per successful item and failed validation does not consume balance.
Integration use cases
Homework and exam answer keys
Generate Tc, Pc, and Vc from textbook a and b so teachers and auto-graders share one van der Waals critical-point ground truth with Zc = 3/8.
Table QA and unit checks
Bulk-verify that published a and b pairs imply a finite critical point and Zc near 3/8, catching zero-b rows and unit mismatches with hard invalid_input.
Technical questions
What are van der Waals critical constants?
They are Tc = 8a/(27Rb), Pc = a/(27b²), and Vc = 3b — the critical temperature, pressure, and molar volume implied by the van der Waals equation of state for given a, b, and R.
What happens if b is zero?
The request fails with invalid_input. Critical pressure and temperature divide by powers of b, so b = 0 makes the formulas undefined; the error message states that b is zero.