Half Life Rate Constant Calculator

Half-Life and Rate Constant Calculator (First-Order)

Instantly convert between half-life (t_1/2) and the reaction rate constant (k) for first-order processes.

1. Calculate Rate Constant (k) from Half-Life

Enter value in any time unit (seconds, years, etc.)

2. Calculate Half-Life (t_1/2) from Rate Constant

Enter value in inverse time units (e.g., s⁻¹, year⁻¹)

function calculateRateConstant() { var t12Input = document.getElementById('halfLifeInput').value; var resultDiv = document.getElementById('kResult'); if (t12Input === "" || isNaN(t12Input) || parseFloat(t12Input) <= 0) { resultDiv.innerHTML = "Please enter a valid positive value for Half-Life."; return; } var t12 = parseFloat(t12Input); // The relationship for first-order kinetics is k = ln(2) / t1/2 // ln(2) is approximately 0.69314718056 var k = Math.log(2) / t12; // Display result with precision for scientific values resultDiv.innerHTML = "The Rate Constant (k) is: " + k.toPrecision(6) + " (inverse time units)"; } function calculateHalfLife() { var kInput = document.getElementById('rateConstantInput').value; var resultDiv = document.getElementById('t12Result'); if (kInput === "" || isNaN(kInput) || parseFloat(kInput) <= 0) { resultDiv.innerHTML = "Please enter a valid positive value for the Rate Constant."; return; } var k = parseFloat(kInput); // The relationship for first-order kinetics is t1/2 = ln(2) / k var t12 = Math.log(2) / k; // Display result with precision resultDiv.innerHTML = "The Half-Life (t_1/2) is: " + t12.toPrecision(6) + " (time units)"; }

Understanding Half-Life and Rate Constants in First-Order Kinetics

In chemistry, physics, and pharmacology, understanding how quickly a substance decays or reacts is crucial. For processes that follow first-order kinetics, there is a direct, constant relationship between the half-life of the substance and its rate constant. This calculator allows you to easily convert between these two fundamental parameters.

What is First-Order Half-Life (t_1/2)?

The half-life of a reaction is the time required for the concentration of a reactant to decrease to exactly half of its initial value. In a first-order reaction, the half-life is constant; it does not depend on the initial concentration of the substance. Whether you start with 100 grams or 10 grams, it takes the same amount of time for half of it to decay.

Common examples include radioactive decay of isotopes (like Carbon-14 dating) and the metabolism of many drugs in the human body.

What is the Rate Constant (k)?

The rate constant, denoted as k, is a proportionality coefficient that connects the rate of a chemical reaction at a specific temperature to the concentration of the reactant. For first-order reactions, the units of k are inverse time (e.g., seconds⁻¹, minutes⁻¹, hours⁻¹, or years⁻¹). A higher value of k indicates a faster reaction or decay process.

The Mathematical Relationship

For any first-order process, the relationship between the half-life (t_1/2) and the rate constant (k) is fixed by the natural logarithm of 2 ($\ln(2)$). The approximate value of $\ln(2)$ is 0.693.

The formulas used in this calculator are:

• To find the Rate Constant: $k = \frac{\ln(2)}{t_{1/2}} \approx \frac{0.693}{t_{1/2}}$
• To find the Half-Life: $t_{1/2} = \frac{\ln(2)}{k} \approx \frac{0.693}{k}$

Example Calculation: Carbon-14 Dating

A classic example of first-order decay is Carbon-14, used in radiocarbon dating. The accepted half-life of Carbon-14 is approximately 5,730 years.

To find the decay rate constant (k) for Carbon-14, we use the formula:

$k = \frac{0.693}{5730 \text{ years}}$

$k \approx 0.0001209 \text{ year}^{-1}$

This means that approximately 0.012% of a Carbon-14 sample decays every year. By inputting 5730 into the first section of the calculator above, you will obtain this rate constant value.