Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

The isotope ${}^{238}U$ decays to ${}^{206}P b$ with a half-life of $4.47 \times 10^{9} Y$ . Although the decay occurs in many individual steps, the first step has by far the longest half-life; therefore, one can often consider the decay to go directly to lead. That is, ${}^{238}U \to {}^{206}P b + v a r i o u s d e c a y p r o d u c t s$
A rock is found to contain 4.20mg of ${}^{238}U$ and 2.135mg of ${}^{206}P b$ . Assume that the rock contained no lead at formation, so all the lead now present arose from the decay of uranium. How many atoms of (a) ${}^{238}U$ and (b) ${}^{206}P b$ does the rock now contain? (c) How many atoms of ${}^{238}U$ did the rock contain at formation? (d) What is the age of the rock?

a) The rock now contains $1.06 \times 10^{19}$ number of ${}^{238}U$ atoms.

b) The rock now contains $6.24 \times 10^{18}$ number of ${}^{206}P b$ atoms.

c) The rock at formation contained $1.684 \times 10^{19}$ number of ${}^{238}U$ atoms.

d) The age of the rock is $2.97 \times 10^{9} y$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Write the given data

a) Half-life ${}^{238}U$ of $T_{1 / 2} = 4.47 \times 10^{9} y$ ,

b) Mass of ${}^{238}U$ that the rock contains, $m_{u} = 4.20 m g$

c) Mass of ${}^{206}P b$ that the rock contains, $m_{p b} = 2.135 m g$

d) The rock contains no lead at the formation.

Step 2: Determine the concept of decay and formulas:

The rock at the time of formation was having only uranium and thus with every decay of one uranium atom, we get one lead atom, thus, the initial atoms present were that of uranium which is the combination of the present amount of combined uranium and lead atoms present in the rock.

The disintegration constant as:

$λ = \frac{\ln 2}{T_{\frac{1}{2}}}$ …… (i)

Here, $T_{\frac{1}{2}}$ is the half-life of the substance.

The undecayed sample remaining after a given time is as follows:

$N = N_{0} e^{- λ t}$ …… (ii)

The number of undecayed atoms in a given mass of a substance is as follows:

$N = \frac{m}{A} N_{A}$ …… (iii)

Here, A is the molar mass of the substance and $N_{A} = 6.022 \times 10^{23} \frac{a t o m s}{m o l}$ .

Step 3: a) Determine the number of uranium atoms present in the rock now

Using the given in equation (iii), the number of ${}^{238}U$ atoms present in the rock now can be given as follows:

$N_{u} = \frac{4.20 \times 10^{- 3} g}{238 \frac{g}{m o l}} (6.022 \times 10^{23} \frac{a t o m s}{m o l}) = 1.06 \times 10^{19}$

Hence, the number of uranium atoms is $1.06 \times 10^{19}$ .

Step 4: b) Determine the number of lead atoms present in the rock now

Using the given in equation (iii), the number of ${}^{206}P b$ atoms present in the rock now can be given as follows:

$N_{P b} = \frac{2.135 \times 10^{- 3} g}{206 \frac{g}{m o l}} (6.022 \times 10^{23} \frac{a t o m s}{m o l}) = 6.24 \times 10^{18}$

Hence, the number of lead atoms is $6.24 \times 10^{18}$ .

Step 5: c) Determine the uranium atoms at the formation of the rock

If no lead was lost, there was originally one uranium atom for each lead atom formed by decay, in addition to the uranium atoms that did not yet decay. Thus, the original number of uranium atoms was

$N_{U 0} = N_{U} + N_{P b}$

Substitute the values and solve as:

$N_{U 0} = 1.06 \times 10^{19} + 6.24 \times 10^{18} = 1.684 \times 10^{19}$

Hence, the number of uranium atoms is $1.684 \times 10^{19}$ .

Step 6: d) Calculate the age of the rock

Now, the age of the rock can be calculated by substituting the disintegration constant value of equation (i) in equation (ii) as follows: (considering the uranium decay that is the decay of the uranium atoms)

$N_{U} = N_{U 0} e^{- (\frac{\ln 2}{T_{1 / 2}}) t} t = \frac{T_{\frac{1}{2}}}{\ln 2} \ln (\frac{N_{U}}{N_{U 0}}) t = - \frac{4.47 \times 10^{9} y}{\ln 2} \ln (\frac{1.06 \times 10^{19}}{1.684 \times 10^{19}}) t = 2.97 \times 10^{9} y$

Hence, the age of the rock is $2.97 \times 10^{9} y$ .

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Short Answer

Step by Step Solution

Step 1: Write the given data

Step 2: Determine the concept of decay and formulas:

Step 3: a) Determine the number of uranium atoms present in the rock now

Step 4: b) Determine the number of lead atoms present in the rock now

Step 5: c) Determine the uranium atoms at the formation of the rock

Step 6: d) Calculate the age of the rock

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: Write the given data

Step 2: Determine the concept of decay and formulas:

Step 3: a) Determine the number of uranium atoms present in the rock now

Step 4: b) Determine the number of lead atoms present in the rock now

Step 5: c) Determine the uranium atoms at the formation of the rock

Step 6: d) Calculate the age of the rock

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