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Half-life is a fundamental concept in the field of science, particularly in the study of radioactive decay. It is a measure of the time it takes for half of the atoms of a radioactive substance to undergo decay and transform into another element. This concept finds extensive application in various fields, such as chemistry, physics, geology, and medicine. Understanding how to calculate half-life is crucial for predicting the decay of radioactive materials, determining the age of fossils and archaeological artifacts, and even in cancer treatment. In this guide, we will explore the fundamentals of half-life and delve into the step-by-step process of calculating it. Whether you are a student, researcher, or simply curious about the topic, this guide will equip you with the knowledge and skills to effectively calculate half-life in different scenarios.
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For a substance in the process of decomposition, the time it takes for the amount of substance to be reduced by half is called the half-life or half-life. ^{[1] X Source of Research} Originally, this term was used to describe the decay of a radioactive substance such as uranium or plutonium, however, we can use the term for all substances. has an exponential or cyclic decomposition rate. The half-life of all substances can be calculated by the rate of decomposition, a value calculated based on the initial amount of the substance and the amount remaining after a specified period of time.
Steps
Understanding half-life
- In other words, when x{displaystyle x} increase, f(x){displaystyle f(x)} decreases and gradually approaches zero. This is the correlation used to describe the half-life. Considering the half-life case, we need a=first2,{displaystyle a={frac {1}{2}},} , thereforef(x+first)=first2f(x).{displaystyle f(x+1)={frac {1}{2}}f(x).}
- I will get f(t)=(first2)t{displaystyle f(t)=left({frac {1}{2}}right)^{t}}
- At this point, what we need to do is not simply put the values into the variable, but consider the actual half-life, in this case, a constant.
- Then we need to give the half-life tfirst/2{displaystyle t_{1/2}} into the exponential equation, however, care should be taken when performing this step. In physics, an exponential equation is an isotropic (direction independent) equation. We know that the amount of a substance depends on time, so we need to divide the quantity of the substance by the half-life – which is a constant with units of time – to get an isotropic quantity.
- Thus, we see that tfirst/2{displaystyle t_{1/2}} and t{displaystyle t} also have the same unit. Therefore, we get the equation given below.
- f(t)=(first2)ttfirst/2{displaystyle f(t)=left({frac {1}{2}}right)^{frac {t}{t_{1/2}}}}
- WOMEN(t)=WOMEN0(first2)ttfirst/2{displaystyle N(t)=N_{0}left({frac {1}{2}}right)^{frac {t}{t_{1/2}}}}
- Divide both sides of the expression by the original quantity of substance WOMEN0.{displaystyle N_{0}.}
- WOMEN(t)WOMEN0=(first2)ttfirst/2{displaystyle {frac {N(t)}{N_{0}}}=left({frac {1}{2}}right)^{frac {t}{t_{1/2}}}}
- Get the logarithm base first2{displaystyle {frac {1}{2}}} On both sides of the expression, we get a simpler expression that doesn’t contain the exponential function.
- logfirst/2(WOMEN(t)WOMEN0)=ttfirst/2{displaystyle log _{1/2}left({frac {N(t)}{N_{0}}}right)={frac {t}{t_{1/2}}}}
- Multiply both sides of the expression by tfirst/2{displaystyle t_{1/2}} , then divide both sides by the left side, we get the formula to calculate the half-life. The results will be in logarithmic form, which you can reduce to a regular numerical value using a calculator.
- tfirst/2=tlogfirst/2(WOMEN(t)WOMEN0){displaystyle t_{1/2}={frac {t}{log _{1/2}left({frac {N(t)}{N_{0}}}right)}}}
For example
- Solution : We have an initial quantity of substance WOMEN0=300 g,{displaystyle N_{0}=300{rm { g}},} The remaining amount is WOMEN=112 g.{displaystyle N=112{rm { g}}.} decomposition time ist=180 S{displaystyle t=180{rm { s}}} .
- The formula for calculating the half-life after transformation is tfirst/2=tlogfirst/2(WOMEN(t)WOMEN0).{displaystyle t_{1/2}={frac {t}{log _{1/2}left({frac {N(t)}{N_{0}}}right)}}.} . We just need to substitute the values on the right side of the expression and do the calculation to get the half-life of the radioactive substance in question.
- tfirst/2=180 Slogfirst/2(112 g300 g)≈127 S {displaystyle {begin{aligned}t_{1/2}&={frac {180{rm { s}}}{log _{1/2}left({frac {112{rm { g}}}{300{ rm { g}}}}right)}}&approx 127{rm { s}}end{aligned}}}
- Check if the result obtained is reasonable or not. We see that 112 g is less than half of 300 g, so the substance is at least half decayed. Since 127 seconds < 180 seconds, which means that the substance has passed a half-life, the results we obtained here are reasonable.
- Solution : We know the initial quantity of substance is WOMEN0=20 kg,{displaystyle N_{0}=20{rm { kg}},} the final quantity is WOMEN=0.1 kg,{displaystyle N=0.1{rm { kg}},} The half-life of uranium-232 is tfirst/2=70year.{displaystyle t_{1/2}=70{text{ year}}.}
- Write a formula to calculate the half-life based on the half-life.
- t=(tfirst/2)logfirst/2(WOMEN(t)WOMEN0){displaystyle t=(t_{1/2})log _{1/2}left({frac {N(t)}{N_{0}}}right)}
- Substitute variables and calculate.
- t=(70year)logfirst/2(0.1 kg20 kg)≈535year{displaystyle {begin{aligned}t&=(70{text{year }})log _{1/2}left({frac {0.1{rm { kg}}}{20{rm { kg}}}}right) &approx 535{text{year}}end{aligned}}}
- Remember to always double-check that the results you get are reasonable.
Advice
- There is another way to calculate the half-life using an integer base. In this formula, WOMEN(t){displaystyle N(t)} and WOMEN0{displaystyle N_{0}} will reverse the position in the logarithmic function.
- tfirst/2=tlog2(WOMEN0WOMEN(t)){displaystyle t_{1/2}={frac {t}{log _{2}left({frac {N_{0}}{N(t)}}right)}}}
- The half-life is a probability-based estimate of the amount of time it takes for a substance to decay to half, not an exact calculation. For example, if there is only one atom of a substance left, it is unlikely that the atom will decay to half an atom after a half-life, but that number of atoms will be zero (zero) or 1 remaining. The larger the residual substance, the more accurate the calculation of the semiconductor period is due to the law of probability for extremely large numbers.
This article is co-authored by a team of editors and trained researchers who confirm the accuracy and completeness of the article.
The wikiHow Content Management team carefully monitors the work of editors to ensure that every article is up to a high standard of quality.
This article has been viewed 47,177 times.
For a substance in the process of decomposition, the time it takes for the amount of substance to be reduced by half is called the half-life or half-life. ^{[1] X Source of Research} Originally, this term was used to describe the decay of a radioactive substance such as uranium or plutonium, however, we can use the term for all substances. has an exponential or cyclic decomposition rate. The half-life of all substances can be calculated by the rate of decomposition, a value calculated based on the initial amount of the substance and the amount remaining after a specified period of time.
In conclusion, calculating the half-life of a substance is an essential process in various scientific fields such as medicine, chemistry, and environmental studies. By understanding the half-life, scientists can accurately predict the decay and stability of radioactive elements, determine dosage intervals for medications, and assess the rate of pollutant degradation in the environment. The calculation of half-life involves applying the exponential decay formula, which takes into account the initial amount of the substance, the decay constant, and the time. It is important to note that calculating half-life requires accurate data and assumptions, and various factors can influence the accuracy of the calculation. Regardless of these limitations, understanding the half-life is crucial for numerous scientific applications and contributes to our understanding of the behavior and properties of different substances.
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