Half-life

Science and society, grade 11 (3 units) Lesson 4 of 16Learning

Half-life

Half-life

A radioactive substance breaks down by itself, and its amount decays exponentially. The half-life is the time in which half of the amount breaks down. It is fixed for each substance: in each half-life the amount halves, whatever the amount.

In medicine iodine-131 is used to treat the thyroid gland.
Its half-life is 8 days.
A patient was given 80 milligrams.

Days	0	8	16	24	32
Iodine-131 (mg)	80	40	20	10	5

Every 8 days the amount is multiplied by $\displaystyle \frac{1}{2}$ .
The unit of time here is the half-life, 8 days – not a day.

The formula by half-lives

To work out the amount, count how many half-lives have gone by – $\displaystyle n$ – and multiply by $\displaystyle \frac{1}{2}$ $\displaystyle n$ times:

\displaystyle n=\frac{t}{T}

\displaystyle M_t=M_0\cdot\left(\frac{1}{2}\right)^{n}

$\displaystyle t$ – the time gone by; $\displaystyle T$ – the half-life, in the same units.

How much of 80 mg of iodine-131 is left after 40 days?

$\displaystyle {n}={\frac{40}{8}}={5}$ .
$\displaystyle {M}={80\cdot\left(\frac{1}{2}\right)^{5}}={\frac{80}{32}}={2.5}$ mg.

Carbon-14 dating

Every living thing holds a fixed amount of carbon-14. Once it dies, the carbon-14 decays, with a half-life of about 5,730 years. From what is left of it in a bone or in wood, we estimate how long ago the creature lived.

An ancient bone holds $\displaystyle \frac{1}{8}$ of the carbon-14 it once had.

$\displaystyle {\frac{1}{8}}={\left(\frac{1}{2}\right)^{3}}$ , so 3 half-lives have gone by.
$\displaystyle {3\cdot5730}={17190}$ : the bone is about 17,000 years old.

Caffeine: a number of half-lives that is not whole

The half-life of caffeine in an adult’s body is about 5 hours.
Dana drank a cup of coffee with 100 mg of caffeine at eight in the morning.

At six in the evening 10 hours have gone by: $\displaystyle {n}={\frac{10}{5}}={2}$ , and $\displaystyle {100\cdot\left(\frac{1}{2}\right)^{2}}={25}$ mg are left.
At three in the afternoon 7 hours have gone by: $\displaystyle {n}={\frac{7}{5}}={1.4}$ – not a whole number. The power is worked out on a calculator:

100\cdot0.5^{1.4}\approx37.9

At three about 37.9 mg of caffeine are left in her body – less than 50 (after 5 hours) and more than 25 (after 10 hours).

Exercise 1

The half-life of iodine-131 is 8 days.
A hospital has 60 mg of iodine-131.

a

How many mg will be left after 24 days?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

How many mg will be left after 40 days?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

An energy drink holds 160 mg of caffeine.
The half-life of caffeine in the body is 5 hours.

How much caffeine will be left in the body 15 hours after the drink?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

A piece of wood from an ancient site holds $\displaystyle \frac{1}{16}$ of the carbon-14 it once had.
The half-life of carbon-14 is 5,730 years.

How many years ago was the tree cut down?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

Noam had a drink with 200 mg of caffeine.
The half-life of caffeine in the body is 5 hours.

a

How many half-lives go by in 8 hours?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

How much caffeine will be left in Noam’s body after 8 hours? Round to a whole number.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

The half-life of a medicine in the body is 6 hours.
The doctor said: “After a day less than 10% of the medicine will be left in the body.”

a

A day is 4 half-lives. What part of the medicine will be left?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

Was the doctor right? Explain.

Formulas: science and society, grade 11 (3 units)

Growth and decay

An exponential process – $\displaystyle M_0$ the amount at the start, $\displaystyle q$ the factor the amount is multiplied by in each unit of time, $\displaystyle M_t$ the amount after $\displaystyle t$ units of time
$\displaystyle M_t=M_0\cdot q^{t}$
The growth factor – a growth of $\displaystyle p$ percent in each unit of time
$\displaystyle q=1+\frac{p}{100}$
The decay factor – a decay of $\displaystyle p$ percent in each unit of time
$\displaystyle q=1-\frac{p}{100}$
Decay by half-life – $\displaystyle n$ the number of half-lives gone by
$\displaystyle M_t=M_0\cdot\left(\frac{1}{2}\right)^{n}$
The number of half-lives – $\displaystyle t$ the time gone by, $\displaystyle T$ the half-life
$\displaystyle n=\frac{t}{T}$
Linear growth – the amount grows by $\displaystyle d$ in each unit of time
$\displaystyle M_t=M_0+d\cdot t$

Measures of spread

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The range – the greatest datum minus the least
$\displaystyle x_{\max}-x_{\min}$
The deviation of the datum $\displaystyle x_i$ from the mean
$\displaystyle x_i-\bar{x}$
The standard deviation of $\displaystyle n$ data
$\displaystyle \sigma=\sqrt{\frac{(x_1-\bar{x})^2+(x_2-\bar{x})^2+\dots+(x_n-\bar{x})^2}{n}}$
The standard deviation from a frequency table – $\displaystyle f_1, \dots, f_k$ the frequencies of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \sigma=\sqrt{\frac{f_1(x_1-\bar{x})^2+f_2(x_2-\bar{x})^2+\dots+f_k(x_k-\bar{x})^2}{f_1+f_2+\dots+f_k}}$

Probability

The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
A path of a tree diagram – the product of the probabilities on its branches
$\displaystyle P=p_1\cdot p_2\cdot p_3$
“At least once” in $\displaystyle n$ independent experiments – $\displaystyle p$ the probability that it happens in each
$\displaystyle 1-(1-p)^{n}$
Conditional probability – the probability of $\displaystyle A$ when $\displaystyle B$ is known to have happened; $\displaystyle A\cap B$ – both $\displaystyle A$ and $\displaystyle B$
$\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}$
The probability of “both $\displaystyle B$ and $\displaystyle A$ ”
$\displaystyle P(A\cap B)=P(B)\cdot P(A|B)$
Independent events – knowing that $\displaystyle B$ happened does not change the probability of $\displaystyle A$
$\displaystyle P(A|B)=P(A)$
The expected value – $\displaystyle x_1, \dots, x_k$ the possible values, $\displaystyle p_1, \dots, p_k$ their probabilities
$\displaystyle E=x_1p_1+x_2p_2+\dots+x_kp_k$