# 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 decay—releasing alpha, beta, or gamma emissions—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.