Half-Life Calculator for Radioactive Decay & More

How to Use Our Half-Life Calculator

This calculator is designed to be flexible. You can solve for any one of the four variables in the half-life formula. Simply enter the three values you know, and the calculator will find the missing one.

• Initial Quantity ($N\_0$): This is the starting amount of the substance at time zero. You can use any unit, such as grams, kilograms, or number of atoms.

• Remaining Quantity ($N(t)$): This is the amount of the substance that is left after the elapsed time. It must be in the same unit as the initial quantity.

• Half-Life ($T\_1/2$): This is the time it takes for exactly half of the substance to decay. The time unit for half-life (e.g., seconds, days, years) must be the same as the time unit for the elapsed time.

• Time Elapsed (t): This is the total duration of the decay process. It must be in the same unit of time as the half-life.

Understanding Your Results

The calculator’s result will be the missing value you were looking for, whether it’s an amount of substance or a period of time. The calculation is based on the universal formula for exponential decay.

The Half-Life Formula

Half-life calculations are governed by the following formula:

N(t)=N_0times(0.5)(t/T_1/2)

Let’s break down what each part of this formula represents:

• $N(t)$ is the Remaining Quantity of the substance after a time t has passed.

• $N\_0$ is the Initial Quantity of the substance.

• $T\_1/2$ is the Half-Life of the substance.

• t is the Time Elapsed.

The core of the formula is the exponent (t / T_½). This part of the equation calculates how many half-lives have occurred. For instance, if 20 days have passed for a substance with a 10-day half-life, 20 / 10 = 2 half-lives have occurred. The formula then raises 0.5 to this power (0.5^2 = 0.25), meaning 25% of the substance remains.

The Decay Process Over Time

The concept of half-life means that the amount of substance that decays in each period is less than the last. The decay is exponential, not linear.

Frequently Asked Questions

What is half-life used for?

Half-life is a fundamental concept used in many scientific fields:

• Nuclear Physics: To describe the rate of radioactive decay for isotopes like Uranium-238 or Carbon-14.

• Pharmacology & Medicine: To determine how long a drug remains in the body. This helps doctors decide on dosing frequency. For example, a drug with a short half-life needs to be taken more often than one with a long half-life.

• Environmental Science: To measure the persistence of harmful chemicals, like pesticides, in the environment.

• Archaeology: To date ancient organic materials through a process called radiocarbon dating.

Does half-life only apply to radioactive decay?

No. While it is most famously associated with radioactive isotopes, the concept of half-life describes any process that undergoes exponential decay. This includes the breakdown of drugs in the body, the decay of biological matter, and even some financial models.

What is carbon dating?

Carbon dating, or radiocarbon dating, is a method for determining the age of an object containing organic material. All living things absorb carbon, including a small amount of a radioactive isotope called Carbon-14 (14C). When an organism dies, it stops absorbing carbon, and the 14C it contains begins to decay into Nitrogen-14 with a half-life of approximately 5,730 years. By measuring the ratio of remaining 14C to the stable Carbon-12 in an artifact, scientists can calculate how many half-lives have passed and thus determine the object’s age.

How is half-life used in medicine?

In medicine, half-life refers to the time it takes for the concentration of a drug in the bloodstream to be reduced by 50%. This is crucial for several reasons:

• Dosing Schedule: A doctor needs to know a drug’s half-life to prescribe it effectively. A drug with a 4-hour half-life may need to be taken every 4-6 hours, while one with a 24-hour half-life can be taken once a day.

• Reaching Steady State: It takes about 4 to 5 half-lives for a drug to reach a “steady state” in the body, where the amount of drug administered is equal to the amount being eliminated.

• Elimination Time: It also takes about 4 to 5 half-lives for a drug to be almost completely cleared from the system after the last dose.

Can a substance ever fully decay to zero?

Theoretically, no. The half-life model describes an exponential decay process that approaches zero but never technically reaches it. After each half-life, half of the remaining substance decays. This means there will always be some infinitesimally small amount left. In practice, however, after a sufficient number of half-lives (usually around 10), the remaining amount is so minuscule that it is considered negligible or undetectable.

What is the difference between half-life and mean lifetime?

• Half-Life ($T\_1/2$): The time it takes for half of the atoms in a sample to decay.

• Mean Lifetime ($tau$): The average lifetime of all the atoms in the sample. They are closely related by a simple formula: Mean Lifetime (τ) = Half-Life / ln(2), which means the mean lifetime is approximately 1.44 times longer than the half-life.

Concrete Example: Medical Isotope Decay

Technetium-99m (Tc-99m) is a widely used medical isotope for diagnostic imaging. It has a half-life of 6 hours. A hospital prepares a dose of 200 millicuries (mCi) at 8:00 AM. A patient’s appointment is delayed, and they don’t receive the dose until 2:00 PM. What is the remaining activity of the dose?

1. Identify the knowns:

   • Initial Quantity ($N\_0$): 200 mCi

   • Half-Life ($T\_1/2$): 6 hours

   • Time Elapsed (t): 6 hours (from 8 AM to 2 PM)

2. Use the Calculator: Enter the three known values to solve for the Remaining Quantity ($N(t)$).

3. Result: The calculator shows a Remaining Quantity of 100 mCi.

4. Conclusion: Because exactly one half-life has passed, the activity of the dose has been reduced by half. The medical staff would need to account for this to ensure the patient receives an effective diagnostic dose.

Does the half-life of a substance ever change?

For radioactive isotopes, the half-life is an intrinsic and unchangeable property of that specific nucleus. It is not affected by external factors like temperature, pressure, or chemical environment. This predictability is what makes it so reliable for applications like carbon dating.

How do scientists determine the half-life of a substance?

Scientists determine half-life by measuring the activity of a sample over time. They use radiation detectors (like a Geiger counter) to count the number of decay events per unit of time. By plotting this activity on a graph versus time, they can observe the exponential decay curve and mathematically determine the time it took for the activity to drop to half its initial value.

Can you calculate the age of a rock using half-life?

Yes, but not with Carbon-14. Carbon dating only works for organic matter and is only accurate up to about 50,000 years. To date rocks, geologists use other radioactive isotopes with much longer half-lives, a process called radiometric dating. For example, the decay of Potassium-40 to Argon-40 (half-life of 1.25 billion years) or Uranium-238 to Lead-206 (half-life of 4.47 billion years) is used to determine the age of rocks and the Earth itself.

The mathematics behind half-life involves exponents and logarithms. If you’re interested in the underlying calculations, you may find our Exponent Calculator and Logarithm Calculator helpful for exploring these concepts further.