# 31.5 Half-life and activity  (Page 4/16)

 Page 4 / 16
$\text{1 Ci}=3\text{.}\text{70}×{\text{10}}^{\text{10}}\phantom{\rule{0.25em}{0ex}}\text{Bq,}$

or $3\text{.}\text{70}×{\text{10}}^{\text{10}}$ decays per second. A curie is a large unit of activity, while a becquerel is a relatively small unit. $\text{1 MBq}=\text{100 microcuries}\phantom{\rule{0.25em}{0ex}}\left(\mu \text{Ci}\right)$ . In countries like Australia and New Zealand that adhere more to SI units, most radioactive sources, such as those used in medical diagnostics or in physics laboratories, are labeled in Bq or megabecquerel (MBq).

Intuitively, you would expect the activity of a source to depend on two things: the amount of the radioactive substance present, and its half-life. The greater the number of radioactive nuclei present in the sample, the more will decay per unit of time. The shorter the half-life, the more decays per unit time, for a given number of nuclei. So activity $R$ should be proportional to the number of radioactive nuclei, $N$ , and inversely proportional to their half-life, ${t}_{1/2}$ . In fact, your intuition is correct. It can be shown that the activity of a source is

$R=\frac{0\text{.}\text{693}N}{{t}_{1/2}}$

where $N$ is the number of radioactive nuclei present, having half-life ${t}_{1/2}$ . This relationship is useful in a variety of calculations, as the next two examples illustrate.

## How great is the ${}^{\text{14}}\text{C}$ Activity in living tissue?

Calculate the activity due to ${}^{\text{14}}\text{C}$ in 1.00 kg of carbon found in a living organism. Express the activity in units of Bq and Ci.

Strategy

To find the activity $R$ using the equation $R=\frac{0\text{.}\text{693}N}{{t}_{1/2}}$ , we must know $N$ and ${t}_{1/2}$ . The half-life of ${}^{\text{14}}\text{C}$ can be found in Appendix B , and was stated above as 5730 y. To find $N$ , we first find the number of ${}^{\text{12}}\text{C}$ nuclei in 1.00 kg of carbon using the concept of a mole. As indicated, we then multiply by $1\text{.}3×{\text{10}}^{-\text{12}}$ (the abundance of ${}^{\text{14}}\text{C}$ in a carbon sample from a living organism) to get the number of ${}^{\text{14}}\text{C}$ nuclei in a living organism.

Solution

One mole of carbon has a mass of 12.0 g, since it is nearly pure ${}^{\text{12}}\text{C}$ . (A mole has a mass in grams equal in magnitude to $A$ found in the periodic table.) Thus the number of carbon nuclei in a kilogram is

$N\left({}^{12}\text{C}\right)=\frac{6.02×{\text{10}}^{\text{23}}\phantom{\rule{0.25em}{0ex}}{\text{mol}}^{–1}}{12.0 g/mol}×\text{(1000 g)}=\text{5.02}×{\text{10}}^{\text{25}}\text{.}$

So the number of ${}^{\text{14}}\text{C}$ nuclei in 1 kg of carbon is

$N\left({}^{\text{14}}\text{C}\right)=\left(5.02×{\text{10}}^{\text{25}}\right)\left(1.3×{\text{10}}^{\text{−12}}\right)=6.52×{\text{10}}^{\text{13}}\text{.}$

Now the activity $R$ is found using the equation $R=\frac{0\text{.}\text{693}N}{{t}_{1/2}}$ .

Entering known values gives

$R=\frac{0\text{.}\text{693}\left(6\text{.}\text{52}×{\text{10}}^{\text{13}}\right)}{\text{5730 y}}=7\text{.}\text{89}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{y}}^{–1},$

or $7\text{.}\text{89}×{\text{10}}^{9}$ decays per year. To convert this to the unit Bq, we simply convert years to seconds. Thus,

$R=\left(\text{7.89}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{y}}^{–1}\right)\frac{1.00 y}{3\text{.}\text{16}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{s}}=\text{250 Bq,}$

or 250 decays per second. To express $R$ in curies, we use the definition of a curie,

$R=\frac{\text{250 Bq}}{3.7×{\text{10}}^{\text{10}}\phantom{\rule{0.25em}{0ex}}\text{Bq/Ci}}=6.76×{\text{10}}^{-9}\phantom{\rule{0.25em}{0ex}}\text{Ci.}$

Thus,

$R=6.76\phantom{\rule{0.25em}{0ex}}\text{nCi.}$

Discussion

Our own bodies contain kilograms of carbon, and it is intriguing to think there are hundreds of ${}^{\text{14}}\text{C}$ decays per second taking place in us. Carbon-14 and other naturally occurring radioactive substances in our bodies contribute to the background radiation we receive. The small number of decays per second found for a kilogram of carbon in this example gives you some idea of how difficult it is to detect ${}^{\text{14}}\text{C}$ in a small sample of material. If there are 250 decays per second in a kilogram, then there are 0.25 decays per second in a gram of carbon in living tissue. To observe this, you must be able to distinguish decays from other forms of radiation, in order to reduce background noise. This becomes more difficult with an old tissue sample, since it contains less ${}^{\text{14}}\text{C}$ , and for samples more than 50 thousand years old, it is impossible.

A weather vane is some sort of directional arrow parallel to the ground that may rotate freely in a horizontal plane. A typical weather vane has a large cross-sectional area perpendicular to the direction the arrow is pointing, like a “One Way” street sign. The purpose of the weather vane is to indicate the direction of the wind. As wind blows pa
If a prism is fully imersed in water then the ray of light will normally dispersed or their is any difference?
the same behavior thru the prism out or in water bud abbot
Ju
If this will experimented with a hollow(vaccum) prism in water then what will be result ?
Anurag
What was the previous far point of a patient who had laser correction that reduced the power of her eye by 7.00 D, producing a normal distant vision power of 50.0 D for her?
What is the far point of a person whose eyes have a relaxed power of 50.5 D?
Jaydie
What is the far point of a person whose eyes have a relaxed power of 50.5 D?
Jaydie
A young woman with normal distant vision has a 10.0% ability to accommodate (that is, increase) the power of her eyes. What is the closest object she can see clearly?
Jaydie
29/20 ? maybes
Ju
In what ways does physics affect the society both positively or negatively
how can I read physics...am finding it difficult to understand...pls help
try to read several books on phy don't just rely one. some authors explain better than other.
Ju
And don't forget to check out YouTube videos on the subject. Videos offer a different visual way to learn easier.
Ju
hope that helps
Ju
I have a exam on 12 february
what is velocity
Jiti
the speed of something in a given direction.
Ju
what is a magnitude in physics
Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
What is meant by dielectric charge?
what happens to the size of charge if the dielectric is changed?
omega= omega not +alpha t derivation
u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
Abrar
do to be peaceful with any body
the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
Saeed
define variable velocity
displacement in easy way.