Chapter 11, Problem 11.26EP

(a)

Interpretation Introduction

Interpretation:

If half-life of a radionuclide is 5.0 min, then how much fraction of the radionuclide will be present undecayed after 20 min has to be calculated.

Concept Introduction:

Radioactive nuclides undergo disintegration by emission of radiation.  All the radioactive nuclide do not undergo the decay at a same rate.  Some decay rapidly and others decay very slowly.  The nuclear stability can be quantitatively expressed by using the half-life.

The time required for half quantity of the radioactive substance to undergo decay is known as half-life.  It is represented as $t_{1/2}$.

The equation that relates amount of decayed radioactive material, amount of undecayed radioactive material and the time elapsed can be given as,

$\left(\begin{array}{l}\text{Amount}\text{of}\text{radionuclide}\\ \text{undecayed}\text{after}n\text{half}\text{lives}\end{array}\right)\text{= }\left(\begin{array}{l}\text{Original}\text{amount}\\ \text{of}\text{radionuclide}\end{array}\right)\text{ x }\left(\frac{\text{1}}{{\text{2}}^n}\right)$

Answer to Problem 11.26EP

Fraction of radionuclide that will remain after 20 min is 1/16.

Explanation of Solution

Half-life of the radionuclide is given as 5.0 min.  The number of half-lives can be calculated as shown below,

$\begin{array}{l}\text{20 min x }\left(\frac{\text{1 half-life}}{\text{5 min}}\right)\text{=}\text{n half-lives}\\ \text{=}\text{4}\text{half-lives}\end{array}$

The fraction of nuclide that remains after 20 min is calculated as shown below,

$\begin{array}{l}\left(\begin{array}{l}\text{Amount}\text{of}\text{radionuclide}\\ \text{undecayed}\text{after}n\text{half}\text{lives}\end{array}\right)\text{= }\left(\begin{array}{l}\text{Original}\text{amount}\\ \text{of}\text{radionuclide}\end{array}\right)\text{ x }\left(\frac{\text{1}}{{\text{2}}^n}\right)\\ \left(\frac{\text{1}}{{\text{2}}^n}\right)=\frac{\text{1}}{{\text{2}}^4}\\ =\frac{\text{1}}{\text{16}}\end{array}$

The fraction of sample that remains after 20 min is calculated as 1/16.

Conclusion

The fraction of the radionuclide sample that remains after 20 min is calculated.

(b)

Interpretation Introduction

Interpretation:

If half-life of a radionuclide is 5.0 min, then how much fraction of the radionuclide will be present undecayed after 30 min has to be calculated.

Concept Introduction:

Radioactive nuclides undergo disintegration by emission of radiation.  All the radioactive nuclide do not undergo the decay at a same rate.  Some decay rapidly and others decay very slowly.  The nuclear stability can be quantitatively expressed by using the half-life.

The time required for half quantity of the radioactive substance to undergo decay is known as half-life.  It is represented as $t_{1/2}$.

The equation that relates amount of decayed radioactive material, amount of undecayed radioactive material and the time elapsed can be given as,

$\left(\begin{array}{l}\text{Amount}\text{of}\text{radionuclide}\\ \text{undecayed}\text{after}n\text{half}\text{lives}\end{array}\right)\text{= }\left(\begin{array}{l}\text{Original}\text{amount}\\ \text{of}\text{radionuclide}\end{array}\right)\text{ x }\left(\frac{\text{1}}{{\text{2}}^n}\right)$

Answer to Problem 11.26EP

Fraction of radionuclide that will remain after 30 min is 1/64.

Explanation of Solution

Half-life of the radionuclide is given as 5.0 min.  The number of half-lives can be calculated as shown below,

$\begin{array}{l}\text{30 min x }\left(\frac{\text{1 half-life}}{\text{5 min}}\right)\text{=}\text{n half-lives}\\ \text{=}\text{6}\text{half-lives}\end{array}$

The fraction of nuclide that remains after 30 min is calculated as shown below,

$\begin{array}{l}\left(\begin{array}{l}\text{Amount}\text{of}\text{radionuclide}\\ \text{undecayed}\text{after}n\text{half}\text{lives}\end{array}\right)\text{= }\left(\begin{array}{l}\text{Original}\text{amount}\\ \text{of}\text{radionuclide}\end{array}\right)\text{ x }\left(\frac{\text{1}}{{\text{2}}^n}\right)\\ \left(\frac{\text{1}}{{\text{2}}^n}\right)=\frac{\text{1}}{{\text{2}}^6}\\ =\frac{\text{1}}{\text{64}}\end{array}$

The fraction of sample that remains after 30 min is calculated as 1/64.

Conclusion

The fraction of the radionuclide sample that remains after 30 min is calculated.

(c)

Interpretation Introduction

Interpretation:

If half-life of a radionuclide is 5.0 min, then how much fraction of the radionuclide will be present undecayed after 3 half-lives has to be calculated.

Concept Introduction:

Radioactive nuclides undergo disintegration by emission of radiation.  All the radioactive nuclide do not undergo the decay at a same rate.  Some decay rapidly and others decay very slowly.  The nuclear stability can be quantitatively expressed by using the half-life.

The time required for half quantity of the radioactive substance to undergo decay is known as half-life.  It is represented as $t_{1/2}$.

The equation that relates amount of decayed radioactive material, amount of undecayed radioactive material and the time elapsed can be given as,

$\left(\begin{array}{l}\text{Amount}\text{of}\text{radionuclide}\\ \text{undecayed}\text{after}n\text{half}\text{lives}\end{array}\right)\text{= }\left(\begin{array}{l}\text{Original}\text{amount}\\ \text{of}\text{radionuclide}\end{array}\right)\text{ x }\left(\frac{\text{1}}{{\text{2}}^n}\right)$

Answer to Problem 11.26EP

Fraction of radionuclide that will remain after 3 half-lives is 1/8.

Explanation of Solution

Given number of half-lives is 3 half-lives.

The fraction of nuclide that remains after 3 half-lives is calculated as shown below,

$\begin{array}{l}\left(\begin{array}{l}\text{Amount}\text{of}\text{radionuclide}\\ \text{undecayed}\text{after}n\text{half}\text{lives}\end{array}\right)\text{= }\left(\begin{array}{l}\text{Original}\text{amount}\\ \text{of}\text{radionuclide}\end{array}\right)\text{ x }\left(\frac{\text{1}}{{\text{2}}^n}\right)\\ \left(\frac{\text{1}}{{\text{2}}^n}\right)=\frac{\text{1}}{{\text{2}}^3}\\ =\frac{\text{1}}{8}\end{array}$

The fraction of sample that remains after 3 half-lives is calculated as 1/8.

Conclusion

The fraction of the radionuclide sample that remains after 3 half-lives is calculated.

(d)

Interpretation Introduction

Interpretation:

If half-life of a radionuclide is 5.0 min, then how much fraction of the radionuclide will be present undecayed after 8 half-lives has to be calculated.

Concept Introduction:

Radioactive nuclides undergo disintegration by emission of radiation.  All the radioactive nuclide do not undergo the decay at a same rate.  Some decay rapidly and others decay very slowly.  The nuclear stability can be quantitatively expressed by using the half-life.

The time required for half quantity of the radioactive substance to undergo decay is known as half-life.  It is represented as $t_{1/2}$.  Half-life for a radionuclide is constant.

The equation that relates amount of decayed radioactive material, amount of undecayed radioactive material and the time elapsed can be given as,

$\left(\begin{array}{l}\text{Amount}\text{of}\text{radionuclide}\\ \text{undecayed}\text{after}n\text{half}\text{lives}\end{array}\right)\text{= }\left(\begin{array}{l}\text{Original}\text{amount}\\ \text{of}\text{radionuclide}\end{array}\right)\text{ x }\left(\frac{\text{1}}{{\text{2}}^n}\right)$

Answer to Problem 11.26EP

Fraction of radionuclide that will remain after 8 half-lives is 1/256.

Explanation of Solution

Given number of half-lives is 8 half-lives.

The fraction of nuclide that remains after 8 half-lives is calculated as shown below,

$\begin{array}{l}\left(\begin{array}{l}\text{Amount}\text{of}\text{radionuclide}\\ \text{undecayed}\text{after}n\text{half}\text{lives}\end{array}\right)\text{= }\left(\begin{array}{l}\text{Original}\text{amount}\\ \text{of}\text{radionuclide}\end{array}\right)\text{ x }\left(\frac{\text{1}}{{\text{2}}^n}\right)\\ \left(\frac{\text{1}}{{\text{2}}^n}\right)=\frac{\text{1}}{{\text{2}}^8}\\ =\frac{\text{1}}{256}\end{array}$

The fraction of sample that remains after 8 half-lives is calculated as 1/256.

Conclusion

The fraction of the radionuclide sample that remains after 8 half-lives is calculated.