How to Calculate Half-Life

X

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.

Table of Contents

Understanding half-life

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/8/8d/Calculate-Half-Life-Step-1-Version-4.jpg/v4-728px-Calculate-Half-Life- Step-1-Version-4.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/8/8d/Calculate-Half-Life-Step-1-Version-4.jpg/ v4-728px-Calculate-Half-Life-Step-1-Version-4.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:” <div class=”mw-parser-output”></div>”}

About exponential decomposition. The exponential decay process follows the formula

f

(

x

)

=

a

x

,

{displaystyle f(x)=a^{x},}

f(x)=a^{{x}},

in there

|

a

|

<

first.

{displaystyle |a|<1.}

|a|<1.

In other words, when $x$ ${displaystyle x}$ $x$ increase, $f$ $($ $x$ $)$ ${displaystyle f(x)}$ $f(x)$ decreases and gradually approaches zero. This is the correlation used to describe the half-life. Considering the half-life case, we need $a$ $=$ $first$ $2$ $,$ ${displaystyle a={frac {1}{2}},}$ $a={frac {1}{2}},$ , therefore $f$ $($ $x$ $+$ $first$ $)$ $=$ $first$ $2$ $f$ $($ $x$ $)$ $.$ ${displaystyle f(x+1)={frac {1}{2}}f(x).}$ $f(x+1)={frac {1}{2}}f(x).$

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/b/bc/Calculate-Half-Life-Step-2-Version-3.jpg/v4-728px-Calculate-Half-Life- Step-2-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/b/bc/Calculate-Half-Life-Step-2-Version-3.jpg/ v4-728px-Calculate-Half-Life-Step-2-Version-3.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:” <div class=”mw-parser-output”></div>”}

Rewrite the formula as a half-cycle. This half-life equation does not depend on the variable

x

,

{displaystyle x,}

x,

which depends on time

t

.

{displaystyle t.}

t.

I will get $f$ $($ $t$ $)$ $=$ $($ $first$ $2$ $)$ $t$ ${displaystyle f(t)=left({frac {1}{2}}right)^{t}}$ $f(t)=left({frac {1}{2}}right)^{{t}}$
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 $t$ $first$ $/$ $2$ ${displaystyle t_{1/2}}$ $t_{{1/2}}$ 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 $t$ $first$ $/$ $2$ ${displaystyle t_{1/2}}$ $t_{{1/2}}$ and $t$ ${displaystyle t}$ $t$ also have the same unit. Therefore, we get the equation given below.
$f$ $($ $t$ $)$ $=$ $($ $first$ $2$ $)$ $t$ $t$ $first$ $/$ $2$ ${displaystyle f(t)=left({frac {1}{2}}right)^{frac {t}{t_{1/2}}}}$ $f(t)=left({frac {1}{2}}right)^{{{frac {t}{t_{{1/2}}}}}}$

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/6/6a/Calculate-Half-Life-Step-3-Version-3.jpg/v4-728px-Calculate-Half-Life- Step-3-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/6/6a/Calculate-Half-Life-Step-3-Version-3.jpg/ v4-728px-Calculate-Half-Life-Step-3-Version-3.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:” <div class=”mw-parser-output”></div>”}

Take into account the amount of the starting substance. Equation

f

(

t

)

{displaystyle f(t)}

f(t)

We are considering is a correlation equation used to determine the percentage of the substance remaining after a period of time compared to the original amount. Just add the original amount

WOMEN

0

{displaystyle N_{0}}

N_{{0}}

In the above equation, we will get the formula for calculating the half-life of a substance.

$WOMEN$ $($ $t$ $)$ $=$ $WOMEN$ $0$ $($ $first$ $2$ $)$ $t$ $t$ $first$ $/$ $2$ ${displaystyle N(t)=N_{0}left({frac {1}{2}}right)^{frac {t}{t_{1/2}}}}$ $N(t)=N_{{0}}left({frac {1}{2}}right)^{{{frac {t}{t_{{1/2}}}}}}$

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/9/94/1425718-4.jpg/v4-728px-1425718-4.jpg”,”bigUrl”:”https:// www.wikihow.com/images/thumb/9/94/1425718-4.jpg/v4-728px-1425718-4.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,” bigHeight”:546,”licensing”:”<div class=”mw-parser-output”></div>”}

Find the half-life. Usually, the above expression includes all the variables we need to determine the half-life. However, if the substance in question is an unknown radioactive substance, it is possible to determine the mass before and after a period of time, but not its half-life. Thus, we can expand the half-life in terms of measurable variables. This is just a way to transform the expression so that you can easily determine the value to look for. The steps of the transformation process are as follows:

Divide both sides of the expression by the original quantity of substance WOMEN0.{displaystyle N_{0}.} $N_{{0}}.$
- $WOMEN$ $($ $t$ $)$ $WOMEN$ $0$ $=$ $($ $first$ $2$ $)$ $t$ $t$ $first$ $/$ $2$ ${displaystyle {frac {N(t)}{N_{0}}}=left({frac {1}{2}}right)^{frac {t}{t_{1/2}}}}$ ${frac {N(t)}{N_{{0}}}}=left({frac {1}{2}}right)^{{{frac {t}{t_{{1/2}}}} }}$
Get the logarithm base first2{displaystyle {frac {1}{2}}} ${frac {1}{2}}$ On both sides of the expression, we get a simpler expression that doesn’t contain the exponential function.
- $log$ $first$ $/$ $2$ $($ $WOMEN$ $($ $t$ $)$ $WOMEN$ $0$ $)$ $=$ $t$ $t$ $first$ $/$ $2$ ${displaystyle log _{1/2}left({frac {N(t)}{N_{0}}}right)={frac {t}{t_{1/2}}}}$ $log _{{1/2}}left({frac {N(t)}{N_{{0}}}}right)={frac {t}{t_{{1/2}}}}$
Multiply both sides of the expression by tfirst/2{displaystyle t_{1/2}} $t_{{1/2}}$ , 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.
- $t$ $first$ $/$ $2$ $=$ $t$ $log$ $first$ $/$ $2$ $($ $WOMEN$ $($ $t$ $)$ $WOMEN$ $0$ $)$ ${displaystyle t_{1/2}={frac {t}{log _{1/2}left({frac {N(t)}{N_{0}}}right)}}}$ $t_{{1/2}}={frac {t}{log _{{1/2}}left({frac {N(t)}{N_{{0}}}right)}}$

For example

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/4/49/Calculate-Half-Life-Step-5-Version-2.jpg/v4-728px-Calculate-Half-Life- Step-5-Version-2.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/4/49/Calculate-Half-Life-Step-5-Version-2.jpg/ v4-728px-Calculate-Half-Life-Step-5-Version-2.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:” <div class=”mw-parser-output”></div>”}

Example 1. Within 180 seconds, an unidentified radioactive substance decays from its initial mass of 300 g to 112 g. What is the half-life of this substance?

Solution : We have an initial quantity of substance $WOMEN$ $0$ $=$ $300$ $g$ $,$ ${displaystyle N_{0}=300{rm { g}},}$ $N_{{0}}=300{{rm { g}}},$ The remaining amount is $WOMEN$ $=$ $112$ $g$ $.$ ${displaystyle N=112{rm { g}}.}$ $N=112{{rm { g}}}.$ decomposition time is $t$ $=$ $180$ $S$ ${displaystyle t=180{rm { s}}}$ $t=180{{rm { s}}}$ .
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)}}.} $t_{{1/2}}={frac {t}{log _{{1/2}}left({frac {N(t)}{N_{{0}}}right)}}.$ . 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.
- $t$ $first$ $/$ $2$ $=$ $180$ $S$ $log$ $first$ $/$ $2$ $($ $112$ $g$ $300$ $g$ $)$ $\approx$ $127$ $S$ ${displaystyle {begin{aligned}t_{1/2}&={frac {180{rm { s}}}{log _{1/2}left({frac {112{rm { g}}}{300{ rm { g}}}}right)}}&approx 127{rm { s}}end{aligned}}}$ ${begin{aligned}t_{{1/2}}&={frac {180{{rm { s}}}}{log _{{1/2}}left({frac {112{{rm { g}) }}}{300{{rm { g}}}}}right)}}&approx 127{{rm { s}}}end{aligned}}$
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.

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/5/52/Calculate-Half-Life-Step-6.jpg/v4-728px-Calculate-Half-Life-Step-6. jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/5/52/Calculate-Half-Life-Step-6.jpg/v4-728px-Calculate-Half-Life-Step- 6.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output”></div> “}

Example 2. A nuclear reactor produces 20 kg of uranium-232. If the half-life of uranium-232 is about 70 years, how long will it take for this uranium-232 to drop to 0.1 kg?

Solution : We know the initial quantity of substance is $WOMEN$ $0$ $=$ $20$ $k$ $g$ $,$ ${displaystyle N_{0}=20{rm { kg}},}$ $N_{{0}}=20{{rm { kg}}},$ the final quantity is $WOMEN$ $=$ $0.1$ $k$ $g$ $,$ ${displaystyle N=0.1{rm { kg}},}$ $N=0.1{{rm { kg}}},$ The half-life of uranium-232 is $t$ $first$ $/$ $2$ $=$ $70$ $year$ $.$ ${displaystyle t_{1/2}=70{text{ year}}.}$ $t_{{1/2}}=70{{text{year}}}.$
Write a formula to calculate the half-life based on the half-life.
- $t$ $=$ $($ $t$ $first$ $/$ $2$ $)$ $log$ $first$ $/$ $2$ $($ $WOMEN$ $($ $t$ $)$ $WOMEN$ $0$ $)$ ${displaystyle t=(t_{1/2})log _{1/2}left({frac {N(t)}{N_{0}}}right)}$ $t=(t_{{1/2}})log _{{1/2}}left({frac {N(t)}{N_{{0}}}}right)$
Substitute variables and calculate.
$t$ $=$ $($ $70$ $year$ $)$ $log$ $first$ $/$ $2$ $($ $0.1$ $k$ $g$ $20$ $k$ $g$ $)$ $\approx$ $535$ $year$ ${displaystyle {begin{aligned}t&=(70{text{year }})log _{1/2}left({frac {0.1{rm { kg}}}{20{rm { kg}}}}right) &approx 535{text{year}}end{aligned}}}$ ${begin{aligned}t&=(70{{text{year }}})log _{{1/2}}left({frac {0.1{{rm { kg}}}}{20{{rm { kg}) }}}}right)&approx 535{{text{year}}}end{aligned}}$
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)} $N(t)$ and WOMEN0{displaystyle N_{0}} $N_{{0}}$ will reverse the position in the logarithmic function.
- $t$ $first$ $/$ $2$ $=$ $t$ $log$ $2$ $($ $WOMEN$ $0$ $WOMEN$ $($ $t$ $)$ $)$ ${displaystyle t_{1/2}={frac {t}{log _{2}left({frac {N_{0}}{N(t)}}right)}}}$ $t_{{1/2}}={frac {t}{log _{{2}}left({frac {N_{{0}}}{N(t)}}right)}}$
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.

Steps

Understanding half-life

For example

Advice

Related Posts