Second-order concentration calculator
Starting at 0.5 mol/L with a second-order rate constant of 0.1 L/(mol·s), 0.333333333333 mol/L remain after 10 s, or 0.666666666667 of the start. Enter the initial concentration, k and the elapsed time. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Initial reactant concentration [A]₀
- 0.5
- K
- 0.1
- Time
- 10
- Concentration
- 0.3333333333
- Remaining fraction
- 0.6666666667
- Reciprocal
- 3
- Reciprocal initial
- 2
Second-order chemical kinetics with rate equal to k times concentration of A squared appears in many elementary bimolecular steps and in effective rate laws fit from batch data. The integrated form links initial concentration, the observed rate constant, and elapsed time to the amount of reactant that remains, so students and process chemists can answer how much is left without re-deriving the reciprocal plot every time.
What second-order remaining concentration means and how to choose inputs
For a reaction that follows rate equals k times concentration of A squared, the differential equation integrates to one over concentration of A at time t equals one over the initial concentration plus k times t. Solving for the remaining concentration gives concentration of A at time t equals initial concentration divided by one plus k times initial concentration times t.
That identity is the heart of this calculator: send a strictly positive initial concentration in the same units you use for the rate law, a non-negative second-order rate constant in concentration inverse times time inverse, and a non-negative elapsed time in the matching time unit. If k is in liters per mole per second and initial concentration is in moles per liter, time must be in seconds and the returned concentration is again in moles per liter.
The engine does not convert units for you, which keeps the contract honest and free of silent scale mistakes. Negative time is invalid input because elapsed reaction time cannot run backward under this model; zero time correctly returns the initial concentration unchanged. Zero rate constant freezes the concentration at the initial value, which is mathematically consistent even if it is rarely the physical case of interest.
What you can do with it
Lab notebooks and batch planning
Process chemists paste fitted second-order k and planned reaction time to estimate how much reactant remains without re-deriving the integrated rate law in each spreadsheet.
FAQ
What happens if time is negative?
Elapsed reaction time must be non-negative; negative time is rejected and not charged.
What units should I use for k and concentration?
Use any consistent set: if [A]_0 is in mol/L and time is in s, k must be in L·mol⁻¹·s⁻¹. The calculator does not convert units automatically.
Is the second-order concentration calculator free?
The browser tool runs free on your device.
Does this handle second-order reactions of the form rate = k[A][B]?
This product implements the integrated law for rate = k[A]² (or equal initial concentrations of two reactants with the same form). Mixed second-order cases with unequal starting amounts need a different identity.
How can I check from my data that a reaction is second order?
Plot one over concentration against time, and a second-order reaction gives a straight line with slope k. In the example, one over concentration rises from 2 at the start to 3 at the end of the reaction time.
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/second-order-concentration \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"initial_concentration":0.5,"k":0.1,"time":10}'const res = await fetch("https://api.kit.forhosting.com/chem/second-order-concentration", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"initial_concentration": 0.5,
"k": 0.1,
"time": 10
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/chem/second-order-concentration",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"initial_concentration": 0.5,
"k": 0.1,
"time": 10
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/chem/second-order-concentration", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"initial_concentration":0.5,"k":0.1,"time":10}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"initial_concentration":0.5,"k":0.1,"time":10}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/chem/second-order-concentration", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"initial_concentration": 0.5,
"k": 0.1,
"time": 10
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "chem.second_order_concentration",
"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
This capability applies the textbook relation concentration of A at time t equals initial concentration divided by one plus k times initial concentration times t, rejects negative time with a clear invalid input error, and returns structured JSON that homework checkers, lab notebooks, and lightweight process tools can pin in automated tests.
What second-order remaining concentration means and how to choose inputs
Missing fields, non-finite numbers, empty strings, non-positive initial concentration, and negative k all fail with invalid_input. Identical inputs always produce identical JSON because the path is pure arithmetic with no network, no random seed, and no wall clock.
Reading the response fields and common classroom workflows
Typical classroom drills include verifying that concentration falls faster at higher initial concentration for second-order kinetics, computing remaining reactant after a given batch time, and contrasting this behavior with first-order half-lives that do not depend on starting amount. The formula field always restates concentration of A at t equals initial over one plus k times initial times t, and method is always second_order_integrated_rate_law for downstream branching. Pair this endpoint with first-order half-life tools when the next question is order identification, or with rate-law calculators when you still need an instantaneous rate from k and current concentrations. Validation failures never consume balance after the engine rejects the request, so exploratory calls with negative time or missing k stay free to fix.
API usage, pricing, lab pipelines, and practical limits
Call the free in-browser second-order concentration calculator for lecture demos, homework checks, and quick mental verification without installing a kinetics package. Keep magnitudes within ordinary floating-point ranges; extremely large products of k, initial concentration, and time drive remaining concentration toward zero in a controlled way, while tiny products leave concentration near the start. Send a JSON body with the required fields initial_concentration, k, and time; aliases a0, A0, c0, rate_constant, t, elapsed, and related names are accepted so pipelines can reuse field names from kinetic fits and spreadsheets. Authors shipping SDKs can map concentration and remaining_fraction into UI tables next to concentration–time plots so students see the same identity written in textbooks. This product does not estimate k from multi-point data, does not solve mixed second-order forms with two different reactants at unequal stoichiometry, and does not apply Arrhenius temperature corrections—those are different search intents. When you already hold A zero, k, and t and only need remaining second-order concentration, this endpoint is the direct, charge-transparent path from inputs to remaining amount.
Integration use cases
Homework and kinetics drills
Students compute remaining [A] after a stated batch time for rate = k[A]², compare remaining fractions at different A₀, and get a clear invalid_input error when they submit negative time.
Teaching platforms and CI fixtures
Tutoring apps and automated graders call a stable prepaid endpoint so worked examples and unit tests share one deterministic contract for second-order remaining concentration.
Technical questions
How do I calculate remaining concentration for a second-order reaction?
Send JSON with initial_concentration (or a0), k (second-order rate constant), and time (or t). The response concentration is [A]_t = [A]_0 / (1 + k·[A]_0·t).