Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

Answers without the blur.

Just sign up for free and you're in.

Short Answer

The uranium ore mined today contains only 0.72% of fissionable $^{235} U$ , too little to make reactor fuel for thermal-neutron fission. For this reason, the mined ore must be enriched with $^{235} U$ . Both role="math" localid="1661753958684" $^{235} U (T_{1 / 2} = 7.0 \times 10^{8} y)$ and $^{238} U (T_{1 / 2} = 4.5 \times 10^{9} y)$ are radioactive. How far back in time would natural uranium ore have been a practical reactor fuel, with a $\frac{^{235} U}{^{238} U}$ ratio of 3.0%?

The natural uranium ore would have been a practical reactor fuel back in $1.7 \times 10^{9} y$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Write the given data:

The ratio of the uranium deposits initially, $\frac{N_{5} (0)}{N_{8} (0)} = 0.03$
The ratio of the uranium deposits now, $\frac{N_{5} (t)}{N_{8} (t)} = 0.0072$
Half-life of, $^{235} U, T_{1 / 2 (^{235} U)} = 7.0 \times 10^{8} y$
Half-life of, $^{238} U, T_{1 / 2 (^{238} U)} = 4.5 \times 10^{9} y$

Step 2: Determine the concept of decay equation

Being a highly unstable radioactive nuclide, the uranium isotopes undergo a decay process. This decay results in the disintegration of the deposits. Now, using the decay equation, and determine the required value of the time taken for the decay. With a shorter half-life, uranium-235 has a greater decay rate than uranium-238. Thus, if the ore contains only 0.72% of uranium-235 today, then the concentration must be higher than in the far distant past.

Formulae:

The amount of undecayed nuclei in a sample as follows:

$N = N_{0} e^{- λ t}$ …… (i)

The disintegration constant is as follows:

$λ = \frac{\ln 2}{T_{1 / 2}}$ ……. (ii)

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

Step 3: Calculate the time taken of decay

Now, using equation (ii) in equation (i) and substitute the given data as per required, the time taken for the initially present uranium deposits to decay can be given as follows:

$\frac{N_{5} (t)}{N_{8} (t)} = \frac{N_{5} (0)}{N_{8} (0)} e^{- (\frac{\ln 2}{T_{1 / 2, 235},} - \frac{\ln 2}{T_{1 / 2, 238},}) t}$

Substitute the values and solve as:

$t = \frac{T_{1 / 2, 235} T_{1 / 2, 238}}{(T_{1 / 2, 235} T_{1 / 2, 238}) l n 2} \ln [(\frac{N_{5} (t)}{N_{8} (t)}) (\frac{N_{8} (t)}{N_{5} (t)})] t = \frac{(7.0 \times 10^{8} y) (4.5 \times 10^{9} y)}{((4.5 \times 10^{9} y) - (7.0 \times 10^{8} y)) \ln 2} \ln [(0.0072) {(0.03)}^{- 1}] t = 1.7 \times 10^{9} y$

Hence, the required value of the time is $1.7 \times 10^{9} y$ .

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Physics

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: Write the given data:

Step 2: Determine the concept of decay equation

Step 3: Calculate the time taken of decay

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Select your language

Fundamentals Of Physics

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: Write the given data:

Step 2: Determine the concept of decay equation

Step 3: Calculate the time taken of decay

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Work Energy and Power

Radiation

Astrophysics

Physics of Motion

Scientific Method Physics

Fluids

Torque and Rotational Motion

Nuclear Physics

Fields in Physics

Measurements

Space Physics

Electricity

Physical Quantities and Units

Medical Physics

Electrostatics

Further Mechanics and Thermal Physics

Mechanics and Materials

Engineering Physics

Energy Physics

Force

Particle Model of Matter

Waves Physics

Magnetism

Modern Physics

Atoms and Radioactivity

Dynamics

Electricity and Magnetism

Conservation of Energy and Momentum

Fundamentals of Physics

Kinematics Physics

Circular Motion and Gravitation

Linear Momentum

Rotational Dynamics

Oscillations

Translational Dynamics

Geometrical and Physical Optics

Thermodynamics

Magnetism and Electromagnetic Induction

Electromagnetism

Electric Charge Field and Potential

Turning Points in Physics