How do you calculate nuclear half life?

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How do you calculate nuclear half life?

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Answer 1 · 2014-03-13T22:08:41

You calculate the half-life from the amount of material that disappears in a given time.

Half-life ( $t_{½}$ $t_½$ ) is the time required for the nuclei to decay to half of the original amount.

Radioactive nuclei decay according to the equation:

Equation 1: $\frac{N_{t}}{N_{0}} = e^{— λ t}$ $N_t /N_0 = e^(—λt)$ , where

$N_{0}$ $N_0$ is the initial number of nuclei at time $t = 0$ $t = 0$ .

$N_{t}$ $N_t$ is the number of nuclei that remain after time $t$ $t$ . We can also use any number that is proportional to the number of nuclei, such as mass or disintegration counts.

$λ$ $λ$ is a constant called the decay constant. Each nucleus has its own decay constant.

The equation for half-life is

Equation 2: $t_{½} = \frac{ln 2}{λ}$ $t_½ = ln2/λ$

We can combine these two equations to get

Equation 3: $\frac{N_{t}}{N_{0}} = {0.5}^{\frac{t}{t_{½}}}$ $N_t/N_0 = 0.5^(t/t_½)$

EXAMPLE:

A 50 g sample of radium–226 decays to 5.7 g after 5000 years. What is the half-life of radium–226?

Solution:

Let’s use Equation 3:

$\frac{N_{t}}{N_{0}} = {0.5}^{\frac{t}{t_{½}}}$ $N_t/N_0 = 0.5^(t/t_½)$

$\frac{5.7 g}{50 g} = {0.5}^{\frac{5000 y r}{t_{½}}}$ $(5.7 g)/(50 g) = 0.5^((5000 yr)/t_½)$

$0.114 = {0.5}^{\frac{5000 y r}{t_{½}}}$ $0.114 = 0.5^((5000 yr)/t_½)$

Take the natural logarithm of each side

$ln 0.114 = \frac{5000 y r}{t_{½}} \times ln 0.5$ $ln0.114 = (5000 yr)/t_½ × ln0.5$

$- 2.17 = \frac{5000 y r}{t_{½}} \times (- 0.693)$ $-2.17 = (5000 yr)/t_½ × (-0.693)$

$t_{½} = (\frac{5000 y r \times 0.693}{2.17})$ $t_½ = ((5000 yr × 0.693)/2.17)$ = 1600 yr

The half-life of radium–226 is 1600 yr.

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How do you calculate nuclear half life?

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