# 31.5 Half-life and activity  (Page 7/16)

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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?

(a) Calculate the activity $R$ in curies of 1.00 g of ${}^{\text{226}}\text{Ra}$ . (b) Discuss why your answer is not exactly 1.00 Ci, given that the curie was originally supposed to be exactly the activity of a gram of radium.

(a) 0.988 Ci

(b) The half-life of ${}^{\text{226}}\text{Ra}$ is now better known.

Show that the activity of the ${}^{\text{14}}\text{C}$ in 1.00 g of ${}^{\text{12}}\text{C}$ found in living tissue is 0.250 Bq.

Mantles for gas lanterns contain thorium, because it forms an oxide that can survive being heated to incandescence for long periods of time. Natural thorium is almost 100% ${}^{\text{232}}\text{Th}$ , with a half-life of $1\text{.}\text{405}×{\text{10}}^{\text{10}}\phantom{\rule{0.25em}{0ex}}\text{y}$ . If an average lantern mantle contains 300 mg of thorium, what is its activity?

$1.22×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{Bq}$

Cow’s milk produced near nuclear reactors can be tested for as little as 1.00 pCi of ${}^{\text{131}}\text{I}$ per liter, to check for possible reactor leakage. What mass of ${}^{\text{131}}\text{I}$ has this activity?

(a) Natural potassium contains ${}^{\text{40}}\text{K}$ , which has a half-life of $1\text{.}\text{277}×{\text{10}}^{9}$ y. What mass of ${}^{\text{40}}\text{K}$ in a person would have a decay rate of 4140 Bq? (b) What is the fraction of ${}^{\text{40}}\text{K}$ in natural potassium, given that the person has 140 g in his body? (These numbers are typical for a 70-kg adult.)

(a) 16.0 mg

(b) 0.0114%

There is more than one isotope of natural uranium. If a researcher isolates 1.00 mg of the relatively scarce ${}^{\text{235}}\text{U}$ and finds this mass to have an activity of 80.0 Bq, what is its half-life in years?

${}^{\text{50}}\text{V}$ has one of the longest known radioactive half-lives. In a difficult experiment, a researcher found that the activity of 1.00 kg of ${}^{\text{50}}\text{V}$ is 1.75 Bq. What is the half-life in years?

$1.48×{\text{10}}^{\text{17}}\phantom{\rule{0.25em}{0ex}}\text{y}$

You can sometimes find deep red crystal vases in antique stores, called uranium glass because their color was produced by doping the glass with uranium. Look up the natural isotopes of uranium and their half-lives, and calculate the activity of such a vase assuming it has 2.00 g of uranium in it. Neglect the activity of any daughter nuclides.

A tree falls in a forest. How many years must pass before the ${}^{\text{14}}\text{C}$ activity in 1.00 g of the tree’s carbon drops to 1.00 decay per hour?

$5.6×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}y$

What fraction of the ${}^{\text{40}}\text{K}$ that was on Earth when it formed $4\text{.}5×{\text{10}}^{9}$ years ago is left today?

A 5000-Ci ${}^{\text{60}}\text{Co}$ source used for cancer therapy is considered too weak to be useful when its activity falls to 3500 Ci. How long after its manufacture does this happen?

2.71 y

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