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Last updated on August 2nd, 2025

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Math Formula for Half-Life

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In nuclear physics and chemistry, the half-life is a measure of the time it takes for half of a given amount of a radioactive substance to decay. Understanding the half-life formula is crucial for calculations in these fields. In this topic, we will learn the formula for half-life and its applications.

Math Formula for Half-Life for Indonesian Students
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List of Math Formulas for Half-Life

The half-life formula is used to determine the decay rate of radioactive substances. Let’s learn the formula to calculate the half-life.

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Math Formula for Half-Life

The half-life is the time required for half of a radioactive substance to decay.

 

It is calculated using the formula: [ t_{1/2} = frac{ln(2)}{lambda} ] where ( lambda ) is the decay constant.

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Importance of Half-Life Formula

In physics and chemistry, the half-life formula is essential for understanding radioactive decay and its applications.

 

Here are some important aspects of the half-life formula:

  • It helps in determining the stability of isotopes. 
     
  • It is used in dating archaeological finds through radiocarbon dating.
     
  • It assists in managing nuclear waste by predicting decay over time.
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Tips and Tricks to Memorize Half-Life Formula

Students often find mathematical formulas challenging.

 

Here are some tips and tricks to master the half-life formula:

  • Remember that half-life is about the time it takes for half of the material to decay.
     
  • Use the mnemonic "Half-Life is Half Gone" to remember that it measures time for half the decay.
     
  • Practice by calculating the half-life for different substances to become familiar with the formula.
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Real-Life Applications of Half-Life Formula

In real life, the half-life formula plays a major role in various fields.

 

Here are some applications of the half-life formula:

  • In medicine, to determine the dosage and timing of radioactive tracers in diagnostic imaging.
     
  • In archaeology, to date artifacts using carbon-14 dating.
     
  • In environmental science, to assess the long-term safety of nuclear waste disposal.
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Common Mistakes and How to Avoid Them While Using the Half-Life Formula

Students make errors when calculating half-life. Here are some mistakes and the ways to avoid them to master this concept.

Mistake 1

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Confusing decay constant with half-life

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Students sometimes confuse the decay constant ( lambda ) with the half-life ( t_{1/2} ).

 

To avoid this, remember that ( lambda ) is a measure of the decay rate, while ( t_{1/2} ) is the time for half the substance to decay.

Mistake 2

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Incorrect calculation of the natural logarithm

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Errors may occur when calculating the natural logarithm ((ln(2))).

 

To avoid these errors, ensure that calculators are set to the correct mode and double-check calculations.

Mistake 3

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Omitting units in calculations

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Students often forget to include units when calculating half-life, which leads to confusion.

 

Always include appropriate time units (e.g., seconds, minutes, years) in your answers.

Mistake 4

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Misinterpreting half-life as the total time for decay

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Half-life is sometimes mistakenly interpreted as the total time for complete decay.

 

Remember that half-life only indicates the time for half the material to decay, not the entire amount.

Mistake 5

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Ignoring the exponential nature of decay

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The exponential decay nature is sometimes overlooked.

 

Always consider the exponential decay formula when analyzing the decay process.

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Examples of Problems Using the Half-Life Formula

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Problem 1

If a radioactive substance has a decay constant of 0.693 per year, what is its half-life?

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The half-life is 1 year.

Explanation

Using the formula

( t_{1/2} = frac{ln(2)}{lambda} ), where ( ln(2) approx 0.693 ) and ( lambda = 0.693 ),

we get: [ t_{1/2} = frac{0.693}{0.693} = 1 { year} ]

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Problem 2

A radioactive isotope has a half-life of 5 years. What is the decay constant?

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The decay constant is approximately 0.139 per year.

Explanation

Using the formula ( t_{1/2} = frac{ln(2)}{lambda} ), we rearrange to find ( lambda ):

[ lambda = frac{ln(2)}{t_{1/2}} = frac{0.693}{5} approx 0.139 { per year} ]

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Problem 3

If a sample initially has 80 grams and after 10 years it has 20 grams, what is the half-life?

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The half-life is approximately 5 years.

Explanation

The substance decays from 80 grams to 20 grams, which means it has gone through two half-lives (80 to 40 to 20).

Since this process took 10 years, each half-life is ( frac{10}{2} = 5 ) years.

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Problem 4

A substance's half-life is 3 years. How much of a 100g sample remains after 6 years?

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25 grams remain.

Explanation

After one half-life (3 years), 50 grams remain.

After another half-life (6 years total), half of that remains, which is 25 grams.

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Problem 5

Calculate the half-life if a substance decays with a decay constant of 0.231 per day.

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The half-life is approximately 3 days.

Explanation

Using the formula ( t_{1/2} = frac{ln(2)}{lambda} ), where ( lambda = 0.231 ),

we get: [ t_{1/2} = frac{0.693}{0.231} approx 3 { days} ]

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FAQs on Half-Life Formula

1.What is the half-life formula?

The formula to find the half-life is: \( t_{1/2} = \frac{\ln(2)}{\lambda} \), where \( \lambda \) is the decay constant.

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2.How do you calculate the decay constant?

The decay constant \( \lambda \) can be calculated using the rearranged half-life formula: \( \lambda = \frac{\ln(2)}{t_{1/2}} \).

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3.What is the importance of half-life in radiocarbon dating?

Half-life is crucial in radiocarbon dating as it allows scientists to calculate the age of organic materials by measuring the remaining carbon-14.

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4.How long does it take for a substance to be fully decayed?

A substance theoretically never fully decays to zero, but after 5 to 7 half-lives, the remaining amount is negligible.

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5.Is the half-life the same for all isotopes?

No, the half-life varies among different isotopes, depending on their nuclear stability.

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Glossary for Half-Life Formula

  • Half-Life: The time required for half of a radioactive substance to decay.

 

  • Decay Constant (( lambda )): A parameter that describes the rate at which a substance decays.

 

  • Radioactive Decay: The process by which an unstable atomic nucleus loses energy by radiation.

 

  • Exponential Decay: A decrease at a rate proportional to the value at any time, characteristic of radioactive decay.

 

  • Radiocarbon Dating: A method for determining the age of an object containing organic material by measuring its carbon-14 content.
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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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