Dividing both sides by
and taking the natural logarithm yields
which reduces to
Thus, if we know the half-life
T1/2 of a radioactive substance, we can find its decay constant. The
lifetime
of a radioactive substance is defined as the average amount of time that a nucleus exists before decaying. The lifetime of a substance is just the reciprocal of the decay constant, written as
The
activityA is defined as the magnitude of the decay rate, or
The infinitesimal change
dN in the time interval
dt is negative because the number of parent (undecayed) particles is decreasing, so the activity (
A ) is positive. Defining the initial activity as
, we have
Thus, the activity
A of a radioactive substance decreases exponentially with time (
[link] ).
(a) A plot of the activity as a function of time (b) If we measure the activity at different times, we can plot ln
A versus
t , and obtain a straight line.
Decay constant and activity of strontium-90
The half-life of strontium-90,
, is 28.8 y. Find (a) its decay constant and (b) the initial activity of 1.00 g of the material.
Strategy
We can find the decay constant directly from
[link] . To determine the activity, we first need to find the number of nuclei present.
Solution
The decay constant is found to be
The atomic mass of
is 89.91 g. Using Avogadro’s number
atoms/mol, we find the initial number of nuclei in 1.00 g of the material:
From this, we find that the activity
at
for 1.00 g of strontium-90 is
Expressing
in terms of the half-life of the substance, we get
Therefore, the activity is halved after one half-life. We can determine the decay constant
by measuring the activity as a function of time. Taking the natural logarithm of the left and right sides of
[link] , we get
This equation follows the linear form
. If we plot ln
A versus
t , we expect a straight line with slope
and
y -intercept
(
[link] (b)). Activity
A is expressed in units of
becquerels (Bq), where one
. This quantity can also be expressed in 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
. The relationship between the Bq and Ci is
What is
Activity in living tissue?
Approximately
of the human body by mass is carbon. Calculate the activity due to
in 1.00 kg of carbon found in a living organism. Express the activity in units of Bq and Ci.
Strategy
The activity of
is determined using the equation
, where
λ is the decay constant and
is the number of radioactive nuclei. The number of
nuclei in a 1.00-kg sample is determined in two steps. First, we determine the number of
nuclei using the concept of a mole. Second, we multiply this value by
(the known abundance of
in a carbon sample from a living organism) to determine the number of
nuclei in a living organism. The decay constant is determined from the known half-life of
(available from
[link] ).
Solution
One mole of carbon has a mass of 12.0 g, since it is nearly pure
. Thus, the number of carbon nuclei in a kilogram is
The number of
nuclei in 1 kg of carbon is therefore
Now we can find the activity
A by using the equation
Entering known values gives us
or
decays per year. To convert this to the unit Bq, we simply convert years to seconds. Thus,
or 250 decays per second. To express
A in curies, we use the definition of a curie,
Thus,
Significance
Approximately
of the human body by weight is carbon. Hundreds of
decays take place in the human body every second. Carbon-14 and other naturally occurring radioactive substances in the body compose a person’s background exposure to nuclear radiation. As we will see later in this chapter, this activity level is well below the maximum recommended dosages.
Polarization is the process of transforming unpolarized light into polarized light.
types of polarization
1. linear polarization.
2. circular polarization.
3. elliptical polarization.
Eze
Describe what you would see when looking at a body whose temperature is increased from 1000 K to 1,000,000 K
In physics and mechanics, torque is the rotational
equivalent of linear force. It is also referred to as the
moment, moment of force, rotational force or turning
effect, depending on the field of study.
Teka
Torque refers to the rotational force. i.e Torque = Force × radius.
Arun
Torque is the rotational equivalent of force .
Specifically, it is a force exerted at a distance
from an object's axis of rotation. In the same way
that a force applied to an object will cause it to
move linearly, a torque applied to an object will
cause it to rotate around a pivot point.
Teka
Torque is the rotational equivalence of force . So,
a net torque will cause an object to rotate with an
angular acceleration. Because all rotational
motions have an axis of rotation, a torque must
be defined about a rotational axis. A torque is a
force applied to a point on an object about the
axis
Teka
When a missle is shot from one spaceship towards another, it leaves the first at 0.950c and approaches the other at 0.750c. what is the relative velocity of the two shipd
can someone help explain why v2/c2 is =1/2
Using The Lorentz Transformation For Time
Spacecraft S′ is on its way to Alpha Centauri when Spacecraft S passes it at relative speed c /2. The captain of S′ sends a radio signal that lasts 1.2 s according to that ship’s clock. Use the Lorentz transformati
