Half-Life Formula
- Introduction
- Understanding Half Life
- Half Life Formula
- Deriving Half Life Formula
- Applications of Half Life Formula
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
The half-life formula helps us figure out how long it takes for a substance to decay to half of its original amount. When things decay, they do so at different rates depending on how much is left. As the substance decreases in quantity, its decay rate also slows down. This makes it tricky to determine its lifespan accurately. That's why we use the half-life formula – it gives us the right measurements to understand how long a decaying material will last.
Understanding Half Life
Half-life is a term commonly used in nuclear physics to describe how long it takes for half of a substance to undergo change. It's like the time it takes for something to decrease by half. Imagine you have a jar of jelly beans. If every minute you take out half of the jelly beans, the half-life of jelly beans in your jar is one minute. So, if you start with `100` jelly beans, after one minute you'd have `50`, after two minutes you'd have `25`, and so on. This concept helps scientists understand how quickly atoms decay. For example, in medicine, it's used to know how long a drug stays active in your body.
Half Life Formula
Half-life refers to the time it takes for half of a sample to react, reducing its initial value by half. This concept is widely used in nuclear physics, particularly in describing the rate of radioactive decay of atoms. The formula for calculating half-life of a substance is:
\(t_{1/2} = \frac{0.693}{\lambda}\)
Here, \(t_{1/2} =\) half-life, and \(\lambda =\) decay constant.
This formula is crucial for understanding the decay of radioactive substances, as it helps determine the time it takes for half of a radioactive sample to decay. Knowing about half-lives is essential for assessing the safety of handling radioactive materials.
The concept of half-life is also commonly employed to describe any form of exponential decay. Exponential decay can be described using any of the three formulas:
\( N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \)
\( N(t) = N_0 e^{-\frac{t}{\tau}} \)
\( N(t) = N_0 e^{-\lambda t} \)
Where:
\( N_0 \) refers to the initial quantity of the substance that will decay, measured in grams, moles, number of atoms, etc.
\( N(t) \) is the quantity that still remains and its decay has not taken place after a time \( t \).
\( t_{1/2} \) represents the half-life of the decaying quantity.
\( \tau \) is a positive number and is the mean lifetime of the decaying quantity.
\( \lambda \) is a positive number and is the decay constant of the decaying quantity.
The relationship between the three parameters, \( t_{1/2} \), \( \tau \), and \( \lambda \) is as follows:
\( t_{1/2} = \frac{\ln(2)}{\lambda} = \tau \ln(2) \)
where \( \ln(2) \) represents the natural logarithm of `2` (approximately `0.693`).
Deriving Half Life Formula
Let the decay constant, also called the constant of proportionality of any decay be \( \lambda \). Then, the following differential equation can be written:
\( \frac{dN}{dt} = -\lambda N \)
Here, \( N \) is the quantity/amount of the substance at any time \( t \). Therefore,
\( \frac{dN}{N} = -\lambda dt \)
Integrating both sides,
\( \int \frac{dN}{N} = \int -\lambda dt \)
\( \ln(N) \Big|_{N_0}^{N} = -\lambda t \Big|_{0}^{t} \)
\( \ln(N) - \ln(N_0) = -\lambda t \)
\( \ln\left(\frac{N}{N_0}\right) = -\lambda t \)
\( \ln\left(\frac{N}{N_0}\right) = -\lambda t \quad \Rightarrow \quad (i) \)
At half-life, the value of \( N \) reduces to half of the initial value. Thus,
\( N = \frac{N_0}{2} \)
Putting this value in equation `(i)`,
\( \ln\left(\frac{1}{2}\right) = -\lambda t_{1/2} \)
\( -\lambda t_{1/2} = -\ln(2) \)
\( t_{1/2} = \frac{\ln(2)}{\lambda} \)
Since \( \ln(2) \approx 0.693 \),
\( t_{1/2} = \frac{0.693}{\lambda} \)
Applications of Half Life Formula
`1`. Radiometric Dating: Half-life formula is extensively used in radiometric dating to determine the age of archaeological artifacts, fossils, and geological formations. By measuring the remaining ratio of radioactive isotopes to their stable decay products, scientists can calculate the elapsed time since the material was formed.
`2`. Medical Imaging: In nuclear medicine, half-life formula is applied to calculate the decay rate of radioactive tracers used in medical imaging procedures such as positron emission tomography (PET) scans. This helps in determining the optimal time for imaging while ensuring patient safety.
`3`. Radioactive Waste Management: Half-life formula plays a crucial role in managing radioactive waste. By understanding the decay rates of radioactive isotopes present in nuclear waste, engineers can estimate the time required for the material to reach a safe level of radioactivity, thus aiding in the design of disposal methods.
`4`. Industrial Applications: Industries utilize half-life formula in various processes such as quality control, material testing, and sterilization. For instance, in sterilization processes involving irradiation, knowing the half-life of the radiation source ensures that the sterilization process is effective while minimizing radiation exposure risks.
`5`. Environmental Monitoring: Environmental scientists use half-life formula to study the behavior of radioactive contaminants in natural ecosystems. By measuring the decay rates of radioactive isotopes in environmental samples, researchers can assess the extent of contamination and track the dispersion of pollutants over time.
`6`. Food Preservation: Half-life formula is applied in food irradiation processes to extend the shelf life of perishable items. By irradiating food products with controlled doses of radiation from isotopes with known half-lives, harmful bacteria and pests can be eliminated, thereby increasing food safety and reducing food wastage.
Solved Examples
Example `1`: We have a radioactive substance with a decay constant \( \lambda \) of \( 0.05 \, \text{yr}^{-1} \). Calculate its half-life using the equation for half-life.
Solution:
Using the equation for half-life:
\( t_{1/2} = \frac{0.693}{\lambda} \)
Substituting \( \lambda = 0.05 \) into the formula:
\( t_{1/2} = \frac{0.693}{0.05} = 13.86 \, \text{years} \)
Therefore, the half-life of the substance is \( 13.86 \) years.
Example `2`: Consider a radioactive isotope with a half-life of \( 5 \) days. If initially, there are \( 800 \) atoms, what will be the number of atoms left after \( 15 \) days?
Solution:
Using the half-life formula:
\( N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \)
Substituting the given values:
\( N(15) = 800 \times \left(\frac{1}{2}\right)^{\frac{15}{5}} \)
\( N(15) = 800 \times \left(\frac{1}{2}\right)^{3} \)
\( N(15) = 800 \times \frac{1}{8} \)
\( N(15) = 100 \)
So, after `15` days, there will be `100` atoms remaining.
Example `3`: A radioactive substance has a half-life of \( 10 \) years. Calculate its decay constant (\( \lambda \)).
Solution:
We know the half-life (\( t_{1/2} \)) and we can rearrange the equation for half-life to solve for \( \lambda \) as follows:
\( t_{1/2} = \frac{0.693}{\lambda} \)
Plugging in \( t_{1/2} = 10 \) years:
\( 10 = \frac{0.693}{\lambda} \)
Solving for \( \lambda \):
\( \lambda = \frac{0.693}{10} = 0.0693 \, \text{years}^{-1} \)
Therefore, the decay constant of the substance is \( 0.0693 \) per year.
Example `4`: Suppose you have a sample of a radioactive element with an initial quantity of \( 1000\) atoms and a decay constant of \(\ 0.01\) per year. Calculate the number of atoms remaining after \(10\) years.
Solution:
Given: \(N_0 = 1000\), \(\lambda = 0.01\), \(t = 10\)
We use the formula \(N(t) = N_0 e^{-\lambda t}\) to find the number of atoms remaining after \(t\) years.
Substitute the given values:
\(N(10) = 1000 \times e^{-0.01 \times 10}\)
\(N(10) = 1000 \times e^{-0.1}\)
Using the value of \(e^{-0.1} \approx 0.9048\):
\(N(10) \approx 1000 \times 0.9048\)
\(N(10) \approx 904.8\)
So, after `10` years, there are approximately `904.8` atoms remaining.
Example `5`: A radioactive substance has a mean lifetime of `10` years. If you start with `100` grams of this substance, how much will remain after `30` years?
Solution:
Given:
- \( N_0 = 100 \) grams (initial amount)
- \( \tau = 10 \) years (mean lifetime)
- \( t = 30 \) years (time elapsed)
Using the equation for half-life:
\( N(t) = N_0 e^{-\frac{t}{\tau}} \)
Substituting the given values:
\( N(30) = 100 \times e^{-\frac{30}{10}} \)
\( N(30) = 100 \times e^{-3} \)
\( N(30) \approx 100 \times 0.0498 \)
\( N(30) \approx 4.98 \) grams
So, approximately `4.98` grams of the substance will remain after `30` years.
Practice Problems
Q`1`. A radioactive substance has a decay constant \( \lambda \) of \( 0.02 \, \text{yr}^{-1} \). What is its half-life (\( t_{1/2} \))?
- \( 20.65 \) years
- \( 34.65 \) years
- \( 17.325 \) years
- \( 6.93 \) years
Answer: b
Q`2`. If a radioactive material has a half-life of \( 5 \) hours, what is its decay constant (\( \lambda \))?
- \( 0.1386 \) hours\(^{-1}\)
- \( 0.03465 \) hours\(^{-1}\)
- \( 0.0693 \) hours\(^{-1}\)
- \( 0.2079 \) hours\(^{-1}\)
Answer: a
Q`3`. A laboratory starts with `100` grams of a radioactive substance. If the substance decays with a half-life of `20` years, how much of the substance will remain after `60` years?
- `6.25` grams
- `12.5` grams
- `25` grams
- `50` grams
Answer: b
Q`4`. The decay constant (\(\lambda\)) of a radioactive substance is `0.02` per year. If the initial quantity of the substance is `100` grams, what will be the quantity remaining after `10` years?
- \(75.3\) grams
- \(81.87\) grams
- \(90.12\) grams
- \(97.78\) grams
Answer: b
Q`5`. A medication has a half-life of `6` hours in the human body. If a patient takes `200` milligrams of the medication, how much of it will remain in their system after `18` hours?
- `25` milligrams
- `5.5` milligrams
- `11` milligrams
- `9.96` milligrams
Answer: d
Frequently Asked Questions
Q`1`. What does the half-life formula represent?
Answer: The half-life formula (\( t_{1/2} = \frac{0.693}{\lambda} \)) represents the time it takes for the quantity of a radioactive substance to reduce to half of its initial value, where \( \lambda \) is the decay constant.
Q`2`. How is the half-life formula derived?
Answer: The half-life formula is derived from the exponential decay equation for radioactive decay, where the decay rate (\( \lambda \)) is inversely proportional to the half-life (\( t_{1/2} \)).
Q`3`. What is the significance of the decay constant (\( \lambda \)) in the half-life formula?
Answer: The decay constant (\( \lambda \)) represents the rate at which the radioactive substance undergoes decay. It is a fundamental parameter in the half-life formula and determines the speed of decay.
Q`4`. Can the half-life formula be used for non-radioactive decay processes?
Answer: No, the half-life formula is specific to radioactive decay processes governed by exponential decay. It is not applicable to non-radioactive decay processes.
Q`5`. How does the half-life formula aid in determining the safety of radioactive materials?
Answer: By calculating the half-life of a radioactive substance, scientists can estimate the time it takes for the material to decay to a safe level of radioactivity. This information is essential for ensuring the safe handling, storage, and disposal of radioactive materials.