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 54

Chapter 42, Problem 54

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?

Verified step by step guidance

Determine the energy released per decay of ²²³Ra. Since ²²³Ra is an alpha emitter, the energy released per decay can be found in nuclear data tables or provided in the problem. Let this energy be denoted as \( E_{\text{decay}} \) (in joules).

Calculate the decay constant \( \lambda \) using the half-life formula: \( \lambda = \frac{\ln(2)}{T_{1/2}} \), where \( T_{1/2} \) is the half-life of ²²³Ra (11.43 days, converted to seconds).

Determine the initial number of nuclei \( N_0 \) in the 1.0 g cube of ²²³Ra. Use the formula \( N_0 = \frac{m}{M} \cdot N_A \), where \( m \) is the mass of the sample (1.0 g), \( M \) is the molar mass of ²²³Ra (223 g/mol), and \( N_A \) is Avogadro's number (\( 6.022 \times 10^{23} \) nuclei/mol).

Calculate the power output of the radium sample. The power is given by \( P = \lambda N_0 E_{\text{decay}} \), where \( \lambda \) is the decay constant, \( N_0 \) is the initial number of nuclei, and \( E_{\text{decay}} \) is the energy released per decay.

Determine the time required to heat the water to its boiling point. Use the formula \( Q = mc\Delta T \) to calculate the heat required to raise the temperature of the water from 18℃ to 100℃, where \( m \) is the mass of the water (100 mL = 100 g), \( c \) is the specific heat capacity of water (4.186 J/g·℃), and \( \Delta T \) is the temperature change (100℃ - 18℃). Then, divide the total heat \( Q \) by the power \( P \) to find the time: \( t = \frac{Q}{P} \).

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Key Concepts

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

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This can occur in various forms, including alpha, beta, and gamma decay. The rate of decay is characterized by the half-life, which is the time required for half of the radioactive substance to decay. Understanding this concept is crucial for calculating how much of the radium isotope remains after a certain period.

Half-Life

Half-life is a specific measure of the time it takes for half of a given quantity of a radioactive substance to decay. For ²²³Ra, the half-life is 11.43 days, meaning that after this period, only half of the original amount of radium will remain. This concept is essential for determining how much energy will be released over time as the radium decays, which directly affects the heating of the water.

Heat Transfer

Heat transfer is the process of thermal energy moving from one object or substance to another due to a temperature difference. In this scenario, the heat generated by the decaying radium will be transferred to the water, raising its temperature. Understanding the principles of heat transfer, including conduction and the specific heat capacity of water, is vital for calculating how long it will take for the water to reach its boiling point.

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Textbook Question

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?

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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

What energy (in MeV) alpha particle has a de Broglie wavelength equal to the diameter of a ²³⁸U nucleus?

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Textbook Question

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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What is the age in years of a bone in which the ¹⁴C/¹²C ratio is measured to be 1.65 x 10⁻¹³?

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Use the graph of binding energy to estimate the total energy released if three ⁴He nuclei fuse together to form a ¹²C nucleus.

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