31.5 Half-life and activity  (Page 3/16)

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How old is the shroud of turin?

Calculate the age of the Shroud of Turin given that the amount of ${}^{\text{14}}\text{C}$ found in it is 92% of that in living tissue.

Strategy

Knowing that 92% of the ${}^{\text{14}}\text{C}$ remains means that $N/{N}_{0}=0\text{.}\text{92}$ . Therefore, the equation $N={N}_{0}{e}^{-\mathrm{\lambda t}}$ can be used to find $\mathrm{\lambda t}$ . We also know that the half-life of ${}^{\text{14}}\text{C}$ is 5730 y, and so once $\mathrm{\lambda t}$ is known, we can use the equation $\lambda =\frac{0\text{.}\text{693}}{{t}_{1/2}}$ to find $\lambda$ and then find $t$ as requested. Here, we postulate that the decrease in ${}^{\text{14}}\text{C}$ is solely due to nuclear decay.

Solution

Solving the equation $N={N}_{0}{e}^{-\mathrm{\lambda t}}$ for $N/{N}_{0}$ gives

$\frac{N}{{N}_{0}}={e}^{-\mathrm{\lambda t}}\text{.}$

Thus,

$0\text{.}\text{92}={e}^{-\mathrm{\lambda t}}\text{.}$

Taking the natural logarithm of both sides of the equation yields

$\text{ln}\phantom{\rule{0.25em}{0ex}}0\text{.}\text{92}=\mathrm{–\lambda t}$

so that

$-0\text{.}\text{0834}=-\mathrm{\lambda t}\text{.}$

Rearranging to isolate $t$ gives

$t=\frac{0\text{.}\text{0834}}{\lambda }\text{.}$

Now, the equation $\lambda =\frac{0\text{.}\text{693}}{{t}_{1/2}}$ can be used to find $\lambda$ for ${}^{\text{14}}\text{C}$ . Solving for $\lambda$ and substituting the known half-life gives

$\lambda =\frac{0\text{.}\text{693}}{{t}_{1/2}}=\frac{0\text{.}\text{693}}{\text{5730 y}}\text{.}$

We enter this value into the previous equation to find $t$ :

$t=\frac{0\text{.}\text{0834}}{\frac{0\text{.}\text{693}}{\text{5730 y}}}=\text{690 y.}$

Discussion

This dates the material in the shroud to 1988–690 = a.d. 1300. Our calculation is only accurate to two digits, so that the year is rounded to 1300. The values obtained at the three independent laboratories gave a weighted average date of a.d. $\text{1320}±\text{60}$ . The uncertainty is typical of carbon-14 dating and is due to the small amount of ${}^{\text{14}}\text{C}$ in living tissues, the amount of material available, and experimental uncertainties (reduced by having three independent measurements). It is meaningful that the date of the shroud is consistent with the first record of its existence and inconsistent with the period in which Jesus lived.

There are other forms of radioactive dating. Rocks, for example, can sometimes be dated based on the decay of ${}^{\text{238}}\text{U}$ . The decay series for ${}^{\text{238}}\text{U}$ ends with ${}^{\text{206}}\text{Pb}$ , so that the ratio of these nuclides in a rock is an indication of how long it has been since the rock solidified. The original composition of the rock, such as the absence of lead, must be known with some confidence. However, as with carbon-14 dating, the technique can be verified by a consistent body of knowledge. Since ${}^{\text{238}}\text{U}$ has a half-life of $4\text{.}5×{\text{10}}^{9}$ y, it is useful for dating only very old materials, showing, for example, that the oldest rocks on Earth solidified about $3\text{.}5×{\text{10}}^{9}$ years ago.

Activity, the rate of decay

What do we mean when we say a source is highly radioactive? Generally, this means the number of decays per unit time is very high. We define activity     $R$ to be the rate of decay    expressed in decays per unit time. In equation form, this is

$R=\frac{\Delta N}{\Delta t}$

where $\text{Δ}N$ is the number of decays that occur in time $\text{Δ}t$ . The SI unit for activity is one decay per second and is given the name becquerel    (Bq) in honor of the discoverer of radioactivity. That is,

$1\phantom{\rule{0.25em}{0ex}}\text{Bq}=\text{1 decay/s.}$

Activity $R$ is often expressed in other units, such as decays per minute or decays per year. One of the most common units for activity is the curie    (Ci), defined to be the activity of 1 g of ${}^{\text{226}}\text{Ra}$ , in honor of Marie Curie’s work with radium. The definition of curie is

Questions & Answers

what is a wave
Williams Reply
wave means. A field of study
aondohemba
what are Atoms
aondohemba
is the movement back and front or up and down
sani
how ?
aondohemba
wave is a disturbance that transfers energy through matter or space with little or no associated mass.
lots
A wave is a motion of particles in disturbed medium that carry energy from one midium to another
conist
an atom is the smallest unit( particle) of an element that bares it's chemical properties
conist
what is electromagnetic induction?
conist
How is the de Broglie wavelength of electrons related to the quantization of their orbits in atoms and molecules?
Larissa Reply
How do you convert 0.0045kgcmÂ³ to the si unit?
EDYKING Reply
how many state of matter do we really have like I mean... is there any newly discovered state of matter?
Falana Reply
I only know 5: •Solids •Liquids •Gases •Plasma •Bose-Einstein condensate
Thapelo
Alright Thank you
Falana
Which one is the Bose-Einstein
James
can you explain what plasma and the I her one you mentioned
Olatunde
u can say sun or stars are just the state of plasma
Mohit
but the are more than seven
Issa
list it out I wanna know
Cristal
what the meaning of continuum
Akhigbe Reply
What state of matter is fire
Thapelo Reply
fire is not in any state of matter...fire is rather a form of energy produced from an oxidising reaction.
Xenda
Isn`t fire the plasma state of matter?
Walter
all this while I taught it was plasma
Victor
How can you define time?
Thapelo Reply
Time can be defined as a continuous , dynamic , irreversible , unpredictable quantity .
Tanaya
unpredictable? but I can say after one o'clock its going to be two o'clock predictably!
Victor
how can we define vector
mahmud
I would define it as having a magnitude (size)with a direction. An example I can think of is a car traveling at 50m/s (magnitude) going North (direction)
Hanzo
as for me guys u would say time is quantity that measures how long it takes for a specific condition to happen e.g how long it takes for the day to end or how it takes for the travelling car to cover a km.
conist
what is the relativity of physics
Paul Reply
How do you convert 0.0045kgcm³ to the si unit?
flint
What is the formula for motion
Anthony Reply
V=u+at V²=u²-2as
flint
S=ut+½at
flint
they are eqns of linear motion
King
S=Vt
Thapelo
v=u+at s=ut+at^\2 v^=u^+2as where ^=2
King
hi
Mehadi
hello
King
Explain dopplers effect
Jennifer Reply
Not yet learnt
Bob
Explain motion with types
Bob
Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.
Alabi Reply
Scalar quantity Because acceleration has only magnitude
Bob
acleration is vectr quatity it is found in a spefied direction and it is product of displcemnt
bhat
its a scalar quantity
Paul
velocity is speed and direction. since velocity is a part of acceleration that makes acceleration a vector quantity. an example of this is centripetal acceleration. when you're moving in a circular patter at a constant speed, you are still accelerating because your direction is constantly changing.
Josh
acceleration is a vector quantity. As explained by Josh Thompson, even in circular motion, bodies undergoing circular motion only accelerate because on the constantly changing direction of their constant speed. also retardation and acceleration are differentiated by virtue of their direction in
fitzgerald
respect to prevailing force
fitzgerald
What is the difference between impulse and momentum?
Manyo
Momentum is the product of the mass of a body and the change in velocity of its motion. ie P=m(v-u)/t (SI unit is kgm/s). it is literally the impact of collision from a moving body. While Impulse is the product of momentum and time. I = Pt (SI unit is kgm) or it is literally the change in momentum
fitzgerald
Or I = m(v-u)
fitzgerald
the tendency of a body to maintain it's inertia motion is called momentum( I believe you know what inertia means) so for a body to be in momentum it will be really hard to stop such body or object..... this is where impulse comes in.. the force applied to stop the momentum of such body is impulse..
Pelumi
Calculation of kinetic and potential energy
dion Reply
K.e=mv² P.e=mgh
Malia
K is actually 1/2 mv^2
Josh
what impulse is given to an a-particle of mass 6.7*10^-27 kg if it is ejected from a stationary nucleus at a speed of 3.2*10^-6ms²? what average force is needed if it is ejected in approximately 10^-8 s?
John
speed=velocity÷time velocity=speed×time=3.2×10^-6×10^-8=32×10^-14m/s impulse [I]=∆momentum[P]=mass×velocity=6.7×10^-27×32×10^-14=214.4×10^-41kg/ms force=impulse÷time=214.4×10^-41÷10^-8=214.4×10^-33N. dats how I solved it.if wrong pls correct me.
Melody
what is sound wave
Nworu Reply
sound wave is a mechanical longitudinal wave that transfers energy from one point to another
Ogor
its a longitudnal wave which is associted wth compresion nad rearfractions
bhat
what is power
PROMISE Reply
it's also a capability to do something or act in a particular way.
Kayode
Newton laws of motion
Mike
power also known as the rate of ability to do work
Slim
power means capabilty to do work p=w/t its unit is watt or j/s it also represents how much work is done fr evry second
bhat
what does fluorine do?
Cheyanne Reply
strengthen and whiten teeth.
Gia

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