Ch 42: Nuclear Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 42: Nuclear Physics

Problem 63

Chapter 42, Problem 63

All the very heavy atoms found in the earth were created long ago by nuclear fusion reactions in a supernova, an exploding star. The debris spewed out by the supernova later coalesced into the gases from which the sun and the planets of our solar system were formed. Nuclear physics suggests that the uranium isotopes ²³⁵U and ²³⁸U should have been created in roughly equal numbers. Today, 99.28% of uranium is ²³⁸U and only 0.72% is ²³⁵U. How long ago did the supernova occur?

Verified step by step guidance

Step 1: Understand the problem. The question involves radioactive decay, specifically the decay of uranium isotopes ²³⁵U and ²³⁸U. The problem asks us to determine the time elapsed since the supernova, based on the current ratio of these isotopes. This requires using the concept of half-life and the radioactive decay formula.

Step 2: Write the radioactive decay formula for each isotope. The number of atoms remaining after a time t is given by:

N_{t} = N_{0} \cdot e^{- λ t}

Here, Nt is the number of atoms at time t, N0 is the initial number of atoms, λ is the decay constant, and t is the time elapsed.

Step 3: Relate the decay constants to the half-lives of the isotopes. The decay constant λ is related to the half-life T1/2 by the formula:

λ = \frac{\ln (2)}{T_{1/2}}

The half-lives of ²³⁵U and ²³⁸U are approximately 7.04 × 10⁸ years and 4.47 × 10⁹ years, respectively.

Step 4: Use the ratio of the isotopes to set up an equation. The current ratio of ²³⁵U to ²³⁸U is given as 0.72% to 99.28%. Initially, the isotopes were created in equal amounts, so the ratio of their remaining quantities is:

\frac{N_{t}}{N_{t}} = \frac{e^{- λ_{235} t}}{e^{- λ_{238} t}}

Simplify this to:

e^{- (λ_{235} - λ_{238}) t} = \frac{0.72}{99.28}

Step 5: Solve for t. Take the natural logarithm of both sides to isolate t:

t = \frac{\ln (\frac{0.72}{99.28})}{-}

Substitute the values of the decay constants (calculated from the half-lives) and solve for t to find the time elapsed since the supernova.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Nuclear Fusion

Nuclear fusion is the process by which two light atomic nuclei combine to form a heavier nucleus, releasing energy in the process. This reaction powers stars, including our sun, and is responsible for the creation of heavier elements in stellar environments. In the context of supernovae, fusion occurs in the cores of massive stars, leading to the formation of elements like uranium before they are expelled into space during an explosion.

Isotopes

Isotopes are variants of a particular chemical element that have the same number of protons but different numbers of neutrons, resulting in different atomic masses. For uranium, the two main isotopes are ²³⁵U and ²³⁸U, with ²³⁵U being fissile and used in nuclear reactors and weapons. Understanding the ratio of these isotopes helps scientists determine the age of uranium deposits and the processes that formed them.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation, leading to the transformation into a different element or isotope. This decay occurs at a predictable rate, characterized by the half-life, which is the time required for half of a sample of a radioactive substance to decay. By measuring the current ratio of uranium isotopes and knowing their half-lives, scientists can estimate the time since the supernova that produced them.

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Textbook Question

There is evidence that low-energy x rays have an RBE slightly greater than 1. Suppose that 10 keV photons with an RBE of 1.2 are used to make a chest x ray. A 60 kg person receives a 0.30 mSv dose from a chest x ray that exposes 25% of the patient's body. How many x ray photons are absorbed in the patient's body?

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Textbook Question

It might seem strange that in beta decay the positive proton, which is repelled by the positive nucleus, remains in the nucleus while the negative electron, which is attracted to the nucleus, is ejected. To understand beta decay, let's analyze the decay of a free neutron that is at rest in the laboratory. We'll ignore the antineutrino and consider the decay n → p⁺ + e⁻. The analysis requires the use of relativistic energy and momentum, from Chapter 36. What is the total kinetic energy, in MeV, of the proton and electron?

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Textbook Question

It might seem strange that in beta decay the positive proton, which is repelled by the positive nucleus, remains in the nucleus while the negative electron, which is attracted to the nucleus, is ejected. To understand beta decay, let's analyze the decay of a free neutron that is at rest in the laboratory. We'll ignore the antineutrino and consider the decay n → p⁺ + e⁻. The analysis requires the use of relativistic energy and momentum, from Chapter 36. Write the equation that expresses the conservation of relativistic momentum for this decay. Let v represent speed, rather than velocity, then write any minus signs explicitly.

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The radium isotope ²²³Ra, an alpha emitter, has a half-life of 11.43 days. You happen to have a 1.0 g cube of ²²³Ra, so you decide to use it to boil water for tea. You fill a well-insulated container with 100 mL of water at 18℃ and drop in the cube of radium. How long will it take the water to boil?

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A sample contains radioactive atoms of two types, A and B. Initially there are five times as many A atoms as there are B atoms. Two hours later, the numbers of the two atoms are equal. The half-life of A is 0.50 hour. What is the half-life of B?

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