Calculate Concentration, Rate Constant, or Time for First Order Reactions
Final Concentration ([A]t)
Initial Concentration ([A]0)
Rate Constant (k)
Time Elapsed (t)
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function calculateRateLaw() {
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var a0 = parseFloat(document.getElementById('initialConc').value);
var at = parseFloat(document.getElementById('finalConc').value);
var k = parseFloat(document.getElementById('rateConstant').value);
var t = parseFloat(document.getElementById('timeElapsed').value);
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// Formula: [A]t = [A]0 * e^(-kt)
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resultValue = a0 * Math.exp(-k * t);
resultText = "Final Concentration ([A]t): " + resultValue.toPrecision(4) + " M";
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// Formula: [A]0 = [A]t / e^(-kt)
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resultValue = at / Math.exp(-k * t);
resultText = "Initial Concentration ([A]0): " + resultValue.toPrecision(4) + " M";
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// Formula: k = -ln([A]t / [A]0) / t
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resultValue = -(Math.log(at / a0)) / t;
resultText = "Rate Constant (k): " + resultValue.toPrecision(4) + " time-1";
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// Formula: t = -ln([A]t / [A]0) / k
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resultValue = -(Math.log(at / a0)) / k;
resultText = "Time Elapsed (t): " + resultValue.toPrecision(4) + " units";
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Understanding the First Order Integrated Rate Law
In the field of chemical kinetics, determining how the concentration of reactants changes over time is crucial for understanding reaction mechanisms and predicting product yields. A first-order reaction is a chemical reaction where the rate is directly proportional to the concentration of only one reactant.
Common examples of first-order reactions include radioactive decay, the thermal decomposition of certain compounds, and many drug metabolism processes in the body (pharmacokinetics).
The Mathematical Formula
The differential rate law for a first-order reaction is Rate = -d[A]/dt = k[A]. Integrating this equation from time t=0 to time t gives us the Integrated Rate Law:
ln([A]t) = -kt + ln([A]0)
This can also be rearranged into the exponential form used by this calculator:
[A]t = [A]0 × e-kt
Where:
[A]t: The concentration of reactant A at time t.
[A]0: The initial concentration of reactant A at time t=0.
k: The first-order rate constant (units of time-1, e.g., s-1 or min-1).
t: The time elapsed.
How to Use This Calculator
This tool is designed to help students, chemists, and researchers solve for any of the four variables involved in the first-order integrated rate law equation.
Select the variable to solve for: Use the dropdown menu to choose whether you need to find the Final Concentration, Initial Concentration, Rate Constant, or Time.
Enter known values: Input the values for the remaining three variables. Ensure your units are consistent (e.g., if k is in s-1, time should be in seconds).
Calculate: Press the button to generate the result.
Half-Life Calculation
A unique characteristic of first-order reactions is that their half-life (t1/2) is constant and independent of the initial concentration. The relationship is derived from the integrated rate law:
t1/2 = ln(2) / k ≈ 0.693 / k
If you know the half-life, you can easily find the rate constant k to use in the calculator above.
Real-World Applications
Pharmacokinetics: Doctors use first-order kinetics to determine drug dosage schedules. Most drugs are eliminated from the bloodstream in a first-order process, meaning a constant fraction of the drug is removed per unit of time.
Nuclear Chemistry: Radioactive decay follows first-order kinetics strictly. This calculator can be used to determine the remaining amount of a radioactive isotope after a certain period, which is essential for carbon dating and nuclear medicine.