Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

Calculate the mass of a sample of (initially pure) ${}^{40}K$ that has an initial decay rate of $1.70 \times 10^{5}$ disintegrations/s. The isotope has a half-life of $1.28 \times 10^{9} y$ .

The mass of a sample of ${}^{40}K$ is $0.66 g$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The given data

a) Initial decay rate of ${}^{40}K, R_{0} = 1.70 \times 10^{5} disintegration / s$ ,

b) Half-life of the isotope, $T_{1 / 2} = 1.28 \times 10^{9} or 4.04 \times 10^{16} s$

c) Molar mass of the sample, $A = 40 g / mol$

Step 2: Understanding the concept of decay and mass

The radioactive decay is due to the loss of the elementary particles from an unstable nucleus to convert them into a more stable one. From the concept of the decay rate, we can get the number of undecayed nuclei. Now, using this in the equation of finding the number of nuclei then determine unknown mass of the sample using its molar mass value and Avogadro number.

Formulae:

The rate of decay is as follows:

$R = (\frac{In 2}{T_{1 / 2}}) N$ ……. (i)

Here, $λ$ is the disintegration constant, N is the number of undecayed nuclei.

$T_{1 / 2}$ is the half-life of the substance, the number of atoms in a given mass of an atom.

$N = \frac{m}{A} N_{A} Here, N_{A} = 6.022 \times 10^{23} \frac{atoms}{mol}$ …… (ii)

Step 3: Calculate the mass of the potassium sample

Substituting value of number of undecayed nuclei from the equation (i) in equation (ii), determine the mass of the potassium sample as follows:

$m = (\frac{R T_{1 / 2}}{In 2}) (\frac{A}{N_{A}})$

Substitute the values and solve as:

$m = (\frac{(1.70 \times 10^{5} \frac{disintegration}{s}) (4.04 \times 10^{16} s)}{In 2}) (\frac{40 \frac{g}{mol}}{6.022 \times 10^{23} \frac{atoms}{mol}}) = 0.66 g$

Hence, the value of the mass is 0.66g.

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Fundamentals Of Physics

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Short Answer

Calculate the mass of a sample of (initially pure) ${}^{40}K$ that has an initial decay rate of $1.70 \times 10^{5}$ disintegrations/s. The isotope has a half-life of $1.28 \times 10^{9} y$ .

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of decay and mass

Step 3: Calculate the mass of the potassium sample

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Calculate the mass of a sample of (initially pure) K40 that has an initial decay rate of1.70×105 disintegrations/s. The isotope has a half-life of 1.28×109y.

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of decay and mass

Step 3: Calculate the mass of the potassium sample

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Calculate the mass of a sample of (initially pure) ${}^{40}K$ that has an initial decay rate of $1.70 \times 10^{5}$ disintegrations/s. The isotope has a half-life of $1.28 \times 10^{9} y$ .