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The concept of half-life is a fundamental principle in various scientific disciplines, ranging from nuclear physics to chemistry to medicine. Whether you are studying radioactive decay, carbon dating, or the effectiveness of a drug, understanding how to calculate the half-life is essential. In simple terms, the half-life represents the time it takes for half of a given quantity to decay or diminish. While the calculation itself might seem complex and intimidating, with a clear understanding of the underlying principles and a step-by-step approach, anyone can confidently calculate the half-life of a substance. In this guide, we will explore the concept of half-life in detail and provide you with a comprehensive methodology to calculate it accurately.
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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] XSource 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=300g,{displaystyle N_{0}=300{rm { g}},} The remaining amount is WOMEN=112g.{displaystyle N=112{rm { g}}.} decomposition time ist=180S{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=180Slogfirst/2(112g300g)≈127S{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=20kg,{displaystyle N_{0}=20{rm { kg}},} the final quantity is WOMEN=0.1kg,{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(0.1kg20kg)≈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] XSource 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 a fundamental aspect of understanding its decay process and can be determined through various methods. The concept of half-life allows scientists to predict the rate at which radioactive or unstable substances decay, aiding in numerous fields such as medicine, archaeology, and environmental science. By using mathematical equations, radioactive decay graphs, or experimental data, scientists can accurately calculate the half-life of a substance. Understanding the half-life of a substance enables us to make informed decisions regarding the safety, effectiveness, and longevity of various materials and compounds. In summary, the calculation of half-life is a vital tool that allows us to gain insights into the decay processes of substances and is an essential component of various scientific disciplines.
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