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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
All textbooksKnight Calc 5th EditionCh 42: Nuclear PhysicsProblem 49
Chapter 42, Problem 49

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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Determine the initial ratio of ¹⁴C/¹²C in a living organism. This is typically known to be approximately 1.3 × 10⁻¹². This value will serve as the initial concentration (N₀) of ¹⁴C relative to ¹²C.
Use the radioactive decay formula: N=N0e⁻λt, where N is the current ratio (1.65 × 10⁻¹³), N₀ is the initial ratio (1.3 × 10⁻¹²), λ is the decay constant, and t is the time (age of the bone).
Calculate the decay constant λ using the half-life of ¹⁴C, which is approximately 5730 years. The relationship is λ=ln(2)T1/2, where T₁/₂ is the half-life.
Rearrange the decay formula to solve for time t: t=ln(NN0)⁻λ. Substitute the known values of N, N₀, and λ into this equation.
Simplify the expression to calculate the age of the bone in years. Ensure that the units are consistent throughout the calculation.

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Key Concepts

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

Radiocarbon Dating

Radiocarbon dating is a method used to determine the age of organic materials by measuring the amount of carbon-14 (¹⁴C) remaining in a sample. Since ¹⁴C is radioactive and decays over time, scientists can calculate the time elapsed since the death of the organism by comparing the ratio of ¹⁴C to stable carbon-12 (¹²C) isotopes.

Half-Life of Carbon-14

The half-life of carbon-14 is approximately 5,730 years, which is the time it takes for half of the original amount of ¹⁴C in a sample to decay into nitrogen-14. This property is crucial for radiocarbon dating, as it allows researchers to estimate the age of a sample based on the remaining ¹⁴C content.
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Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In the context of radiocarbon dating, the amount of ¹⁴C in a sample decreases exponentially over time, which can be mathematically modeled to determine the age of the sample based on the measured ¹⁴C/¹²C ratio.
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Related Practice
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

You learned in Chapter 41 that the binding energy of the electron in a hydrogen atom is 13.6 eV. By how much does the mass decrease when a hydrogen atom is formed from a proton and an electron? Give your answer both in atomic mass units and as a percentage of the mass of the hydrogen atom.

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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

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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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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Textbook Question

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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