# 10.7 Half-life and activity  (Page 6/15)

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$R={R}_{0}{e}^{-\mathrm{\lambda t}}\text{,}$

where ${R}_{0}$ is the activity at $t=0$ . This equation shows exponential decay of radioactive nuclei. For example, if a source originally has a 1.00-mCi activity, it declines to 0.500 mCi in one half-life, to 0.250 mCi in two half-lives, to 0.125 mCi in three half-lives, and so on. For times other than whole half-lives, the equation $R={R}_{0}{e}^{-\mathrm{\lambda t}}$ must be used to find $R$ .

## Section summary

• Half-life ${t}_{1/2}$ is the time in which there is a 50% chance that a nucleus will decay. The number of nuclei $N$ as a function of time is
$N={N}_{0}{e}^{-\mathrm{\lambda t}},$
where ${N}_{0}$ is the number present at $t=0$ , and $\lambda$ is the decay constant, related to the half-life by
$\lambda =\frac{0\text{.}\text{693}}{{t}_{1/2}}.$
• One of the applications of radioactive decay is radioactive dating, in which the age of a material is determined by the amount of radioactive decay that occurs. The rate of decay is called the activity $R$ :
$R=\frac{\text{Δ}N}{\text{Δ}t}.$
• The SI unit for $R$ is the becquerel (Bq), defined by
$\text{1 Bq}=\text{1 decay/s.}$
• $R$ is also expressed in terms of curies (Ci), where
$1\phantom{\rule{0.25em}{0ex}}\text{Ci}=3\text{.}\text{70}×{\text{10}}^{\text{10}}\phantom{\rule{0.25em}{0ex}}\text{Bq.}$
• The activity $R$ of a source is related to $N$ and ${t}_{1/2}$ by
$R=\frac{0\text{.}\text{693}N}{{t}_{1/2}}.$
• Since $N$ has an exponential behavior as in the equation $N={N}_{0}{e}^{-\mathrm{\lambda t}}$ , the activity also has an exponential behavior, given by
$R={R}_{0}{e}^{-\mathrm{\lambda t}},$
where ${R}_{0}$ is the activity at $t=0$ .

## Conceptual questions

In a $3×{\text{10}}^{9}$ -year-old rock that originally contained some ${}^{\text{238}}\text{U}$ , which has a half-life of $4.5×{\text{10}}^{9}$ years, we expect to find some ${}^{\text{238}}\text{U}$ remaining in it. Why are ${}^{\text{226}}\text{Ra}$ , ${}^{\text{222}}\text{Rn}$ , and ${}^{\text{210}}\text{Po}$ also found in such a rock, even though they have much shorter half-lives (1600 years, 3.8 days, and 138 days, respectively)?

Does the number of radioactive nuclei in a sample decrease to exactly half its original value in one half-life? Explain in terms of the statistical nature of radioactive decay.

Radioactivity depends on the nucleus and not the atom or its chemical state. Why, then, is one kilogram of uranium more radioactive than one kilogram of uranium hexafluoride?

Explain how a bound system can have less mass than its components. Why is this not observed classically, say for a building made of bricks?

Spontaneous radioactive decay occurs only when the decay products have less mass than the parent, and it tends to produce a daughter that is more stable than the parent. Explain how this is related to the fact that more tightly bound nuclei are more stable. (Consider the binding energy per nucleon.)

To obtain the most precise value of BE from the equation $\text{BE=}\left[\text{ZM}\left({}^{1}\text{H}\right)+{\text{Nm}}_{n}\right]{c}^{2}-m\left({}^{A}X\right){c}^{2}$ , we should take into account the binding energy of the electrons in the neutral atoms. Will doing this produce a larger or smaller value for BE? Why is this effect usually negligible?

How does the finite range of the nuclear force relate to the fact that $\text{BE}/A$ is greatest for $A$ near 60?

## Problems&Exercises

Data from the appendices and the periodic table may be needed for these problems.

An old campfire is uncovered during an archaeological dig. Its charcoal is found to contain less than 1/1000 the normal amount of ${}^{\text{14}}\text{C}$ . Estimate the minimum age of the charcoal, noting that ${2}^{\text{10}}=\text{1024}$ .

57,300 y

A ${}^{\text{60}}\text{Co}$ source is labeled 4.00 mCi, but its present activity is found to be $1\text{.}\text{85}×{\text{10}}^{7}$ Bq. (a) What is the present activity in mCi? (b) How long ago did it actually have a 4.00-mCi activity?

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
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