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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 $R$ .
In a $3\times {\text{10}}^{9}$ -year-old rock that originally contained some ${}^{\text{238}}\text{U}$ , which has a half-life of $4.5\times {\text{10}}^{9}$ 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 $\text{BE=}\left[\text{ZM}\left({}^{1}\text{H}\right)+{\text{Nm}}_{n}\right]{c}^{2}-m\left({}^{A}\mathrm{X}\right){c}^{2}$ , 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?
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 $1\text{.}\text{85}\times {\text{10}}^{7}$ Bq. (a) What is the present activity in mCi? (b) How long ago did it actually have a 4.00-mCi activity?
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