# Half-Life Calculator

## Calculate the half-life, and quantity of a substance step by step

This calculator will calculate the half-life, initial quantity, quantity remained, and time, with steps shown.

The Half-Life Calculator is a tool designed to explain the concepts of radioactive decay. It is not only for mastering half-life calculations but also for determining the initial and remaining amounts of a substance or its half-life.

## How to Use the Half-Life Calculator?

### Input

Of the four values (starting quantity, remaining quantity, half-life, time), enter any three. The fourth will be calculated. Depending on what you want to determine:

- To find the half-life, input the remaining amount, the time to achieve that remaining amount, and the starting amount.
- To discover how much of the substance remains, leave this field empty and fill in the remaining fields.
- To determine the initial amount, leave its field blank and ensure other required details are filled in.

### Calculation

Click the "Calculate" button for results. The tool will quickly provide them based on the data you've entered.

### Result

The calculator will display the initial quantity of the substance, the remaining quantity, the half-life, or the time, depending on your inputs.

## What Is Half-Life?

Radioactive substances consist of both stable and unstable atomic nuclei. While stable nuclei remain unchanged, the unstable ones undergo radioactive decayâ€”releasing alpha, beta, or gamma emissionsâ€”and eventually transform into stable forms.

The half-life is the time required for half of a quantity of a radioactive substance to decay. In other words, it's the duration for half of the initial number of radioactive atoms in a sample to decay or transform into another atom or isotope.

Different substances have different half-lives. For instance, consider carbon-10 with its short half-life of just 19 seconds, which ensures that it does not occur in nature. In contrast, uranium-233 has a much longer half-life, approximately 160,000 years.

**Formula**

The primary formula used to determine the half-life of a radioactive substance is

$$t_h=\frac{t\ln(2)}{\ln\left(\frac{N_0}{N(t)}\right)},$$where:

- $$$t_h$$$ is the half-life of the substance.
- $$$t$$$ stands for time.
- $$$\ln$$$ is the natural logarithm.
- $$$N_0$$$ is the initial amount of the substance.
- $$$N(t)$$$ is the amount of the substance after the time $$$t$$$.

For example, suppose you have 100 grams of a radioactive isotope. After 25 seconds, there will be 70 grams left. What would be its half-life?

Using the formula, we have that

$$t_h=\frac{25\ln(2)}{\ln\left(\frac{100}{70}\right)}\approx48.58$$This means that after 48.58 seconds, half of the initial amount of the radioactive isotope would have decayed.

## What Is the Difference Between Half-Life and Decay Rate?

The distinction between half-life and decay rate lies in their definitions and measurements.

**Half-Life:** This is the duration required for half of a given sample of a radioactive substance to decay. It represents a time-based measure, indicating how long it takes for a substance to halve its radioactivity.

**Decay Rate:** The decay constant $$$\lambda$$$ is the probability per unit of time that a single atom will decay. It measures the speed or frequency of radioactive decay in a substance. The formula that relates the half-life and the decay constant is $$$t_h\lambda=\ln(2)$$$.

While both half-life and decay rate refer to the radioactive decay of a substance, they approach the concept from different perspectives: one measures the time (duration) required for half of the substance to decay, while the other quantifies the speed or frequency of this decay.

## Why Choose Our Half-Life Calculator?

### Precision & Accuracy

Our tool uses advanced algorithms to ensure you receive the most accurate results, eliminating potential calculation errors.

### User-Friendly Interface

With an intuitive layout and simple input methods, even those unfamiliar with half-life calculations can navigate and use the tool with ease.

### Speed

Get instant results. Without manual calculations or waiting, our calculator processes your data in no time, providing immediate output.

### Versatility

Whether you're a student, a professional, or just someone curious about radioactive decay, our calculator caters to a wide range of users and needs.

### Step-by-Step Explanations

Alongside the calculator, we provide explanatory content and examples to help users understand the concept of half-life and its implications better.

### FAQ

#### What is half-life?

Half-life is a term used to describe the time it takes for half of the radioactive atoms in a sample to decay. It measures the rate at which a substance loses its radioactivity. Different substances have different half-lives, ranging from fractions of a second to millions of years.

#### What is the half-life of carbon?

Carbon has several isotopes, but the most commonly referred to in terms of half-life is carbon-14 (C-14). Carbon-14 has a half-life of approximately 5,730 years. This property makes it valuable for archaeological and geological dating (radiocarbon dating).

#### What is the half-life of radium?

Radium has several isotopes with different half-lives. The most common isotope, radium-226 (Ra-226), has a half-life of about 1600 years. It decays into radon gas through alpha decay.

#### How does temperature affect the half-life of a substance?

While half-life is generally considered a constant for a specific isotope, extreme conditions like very high temperatures might influence the decay rate of some substances. However, for most practical purposes and in standard conditions, temperature doesn't significantly impact half-life.