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Chapter 42: Q. 50 (page 1237)

A $115 m C i$ radioactive tracer is made in a nuclear reactor. When it is delivered to a hospital hours later its activity is $95 m C i$ . The lowest usable level of activity is $10 m C i$ .
A. What is the tracer鈥檚 half-life?
B. For how long after delivery is the sample usable?

Short Answer

Expert verified

(a) Half-life of tracer is: $t_{\frac{1}{2}} = 58 h r$

(b) Max time with the sample being usable $t = 188.5 h r$

Step by step solution

01

Part(a): Step 1: Given information

We have been given that

$R_{o} = 115 mCi$

$t = 16 hr$

$R = 95 mCi$

02

Part(a) Step 2: Solving

Using the formula $R = R_{o} {(\frac{1}{2})}^{t / t_{1 / 2}}$

Substituting the given values $0.95 mCi = 1.15 mCi {(\frac{1}{2})}^{16 hr / t_{1 / 2}}$

Taking Natural log $\ln [\frac{0.95 mCi}{1.15 mCi}] = [\frac{16 hr}{t_{1 / 2}}] \ln (\frac{1}{2})$

Solving $t_{1 / 2} = \frac{16 hr}{0.27536} = 58.105 h r 鈮� 58 h r$

03

Part(b) Step 1: Given information:

We need to find out the value of t for the sample.

04

Part (b) Step 2: Solving:

Using $R = R_{o} {(\frac{1}{2})}^{t / t_{1 / 2}}$ $(Where is R = 0.10 mCi)$

Substituting the given values $0.10 mCi = 0.95 mCi {(\frac{1}{2})}^{t / 58 hr}$

Taking natural log $\ln (\frac{0.10}{0.95}) = (\frac{t}{58 hr}) \ln (\frac{1}{2})$

Solving $3.25 = (\frac{t}{58 hr})$

$t = 188.5 hr$

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Most popular questions from this chapter

The uranium isotope ${}^{238}U$ is naturally present at low levels in many soils. One of the nuclei in the decay series of ${}^{238}U$ is the radon isotope ${}^{222}R n$ , which decays by emitting a localid="1650487447445" $5.50 M e V$ alpha particle with localid="1650487458663" $t_{1 / 2} = 3.82$ days. Radon is a gas, and it tends to seep from soil into basements. The Environmental Protection Agency recommends that homeowners take steps to remove radon, by pumping in fresh air, if the radon activity exceeds $4 p C i p e r l i t e r$ of air.

a. How many ${}^{222}R n$ atoms are there in $1 m^{3}$ of air if the activity is $4 \frac{p C i}{L}$ ?

b. The range of alpha particles in air is $鈮� 3 c m$ . Suppose we model a person as a $180 - c m$ -tall, $25 - c m$ -diameter cylinder with a mass of $65 k g$ . Only decays within $3 c m$ of the cylinder can cause exposure, and only $鈮� 50 %$ , of the decays direct the alpha particle toward the person. Determine the dose in $m S v p e r y e a r$ for a person who spends the entire year in a room where the activity is $4 p C i / L$ .

c. Does the EPA recommendation seem appropriate? Why?

Nucleus A decays into nucleus B with a half-life of 10 s. At

t = 0 s, there are 1000 A nuclei and no B nuclei. At what time

will there be 750 B nuclei?

a. Is the binding energy of a nucleus with $A = 200$ more than, less than, or equal to the binding energy of a nucleus with $A = 60$ ? Explain.

b. Is a nucleus with $A = 200$ more tightly bound, less tightly bound, or bound equally tightly as a nucleus with $A = 60$ ? Explain.

How many grays of gamma-ray photons cause the same bio-

logical damage as 30 Gy of alpha radiation?

What kind of decay, if any, can occur for the nuclei in

FIGURE Q42.7?

See all solutions

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