Half Life

Pronunciation: /ˈhæfˌlaɪf/ ?

The half life of a substance is the time it takes for 1/2 of the substance to decay, metabolize, or be used up. For example, if 1/2 of a drug is metabolized in 3 hours, after 6 hours 1/4 of the drug is left and 3/4 of the drug has been used:

(1/2 · 1/2 = 1/4)
After 9 hours, 1/8 of the drug is left:
(1/2 · 1/2 = 1/4)
The half life formula is an exponential equation:
D=D0*(1/2)&(t/h)
where D is the remaining substance after time t, D0 is the initial amount of the substance, h is the half life of the substance, and t is the elapsed time.

Graph of Half Life Function

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Manipulative 1: Graph of half life. Created with GeoGebra.

Click on the blue points on the sliders in manipulative 1 and drag them to change the figure. The red points mark when 1/2, 1/4, and 1/8 of the substance has decayed.

Example

The half life of a radioactive substance is 3 hours. The initial amount is 3 grams. How long before only 0.6 grams is left?

StepEquationDescription
1 D=D0*(1/2)&(t/h) Start with the half-life formula.
2 0.6=3*(1/2)^(t/3) Fill the values into the formula. The initial amount D0 = 3. The half-life h = 3. The amount left after 3 hours D = 0.6. Solve for t.
3 0.6/3=3/3*(1/2)^(t/3) implies 0.2=(1/2)^(t/3). Use the multiplication property of equality to multiply both sides of the equation by 1/3.
4 log base 1/2 of 0.2 = t/3. Use the logarithm to convert the equation from exponential form to logarithmic form. The definition of a logarithm is logab = c if and only if ac = b. In this case a = 1/2, b = 0.2 and c = t/3.
5 3*log base 1/2 of 0.2 = t. Use the multiplication property of equality to multiply both sides of the equation by 3.
6 3*(log base 10 of 0.2)/(log base 10 of 1/2) = t. Use the change of base formula to convert the logarithm to base 10. The change of base formula is log base a of x = (log base b of x )/(log base b of a).. In this case, a = 1/2, x = 0.2 and b = 10.
7 3*-0.69897/-0.30103 is approximately t. Substitute the values of the logarithms into the equation. log100.2 ≈ -0.69897. log100.5 ≈ -0.30103.
8 t ≈ 6.9658 Calculate the approximate value of t. It will take about 7 hours for only 0.6 grams to be left.
Table 2: Half life example.

Cite this article as:


Half Life. 2010-10-25. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/h/halflife.html.

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Revision History


2010-10-25: Expanded example and added description to example. (McAdams, David.)
2010-02-08: Added "References" (McAdams, David.)
2008-11-26: Change equations to images. Added graph (McAdams, David.)
2008-08-07: Change equations to Hot_Eqn (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

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