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BIO Ancient meat eating. The amount of meat in prehistoric diets can be determined by measuring the ratio of the isotopes nitrogen- 15 to nitrogen- 14 in bone from human remains. Carnivores concentrate \({ }^{15} \mathrm{~N},\) so this ratio tells archaeologists how much meat was consumed by ancient people. Suppose you use a velocity selector to obtain singly ionized (missing one electron) nitrogen atoms of speed \(8.50 \mathrm{~km} / \mathrm{s}\) and bend them along a semicircle within a uniform magnetic field. If the \({ }^{14} \mathrm{~N}\) atoms travel along a semicircle with a diameter of \(29.2 \mathrm{~cm},\) find the separation of the \({ }^{14} \mathrm{~N}\) and \({ }^{15} \mathrm{~N}\) isotopes at the detector. The measured masses of these isotopes are \(2.32 \times 10^{-26} \mathrm{~kg}\left({ }^{14} \mathrm{~N}\right)\) and \(2.49 \times 10^{-26} \mathrm{~kg}\left({ }^{15} \mathrm{~N}\right)\)

Step by step solution

01

Understanding the Problem

We need to find the separation between the paths of nitrogen-14 and nitrogen-15 isotopes when subjected to a magnetic field after being ionized and traveling at a certain speed. The separation will help us quantify differences that relate to their isotopic ratio which is indicative of meat consumption.

02

Formula for Radius in Magnetic Field

The formula for the radius of a charged particle's path in a magnetic field is \( r = \frac{mv}{qB} \), where \( m \) is the mass, \( v \) is the velocity, \( q \) is the charge, and \( B \) is the magnetic field strength. Since the particles are singly ionized, \( q = 1.6 \times 10^{-19} \, C \).

03

Calculate the Radius for \(^{14}\mathrm{N}\)

The given diameter of the semicircle for \(^{14}\mathrm{N}\) is 29.2 cm, so the radius \( r_{14} \) is 14.6 cm or 0.146 m. Using the formula, \( B = \frac{mv}{qr} \). Substituting the mass and velocity for \(^{14}\mathrm{N}\), we solve for \( B \).

04

Use B to Calculate Radius for \(^{15}\mathrm{N}\)

Now, taking the mass of \(^{15}\mathrm{N} = 2.49 \times 10^{-26} \mathrm{kg} \), calculate the radius \( r_{15} \) using \( r = \frac{mv}{qB} \) with the previously found \( B \).

05

Find Separation at the Detector

The separation at the detector is the difference in diameters of the semicircles traced by \(^{14}\mathrm{N}\) and \(^{15}\mathrm{N}\) isotopes, which is \( 2(r_{15} - r_{14}) \). Compute this to find the separation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nitrogen Isotopes

When studying chemical elements, isotopes reveal crucial details about atomic structure. Isotopes are versions of an element with the same number of protons but different numbers of neutrons. For nitrogen, the two most common isotopes we often hear about are Nitrogen-14 ( ^{14} ) and Nitrogen-15 ( ^{15} ).
What makes these isotopes interesting is their application in various scientific fields. In archeology, for example, analyzing the ratio of Nitrogen-15 to Nitrogen-14 in prehistoric human remains can lead to insights about ancient diets.

• Nitrogen-14 is the more abundant isotope, making up the bulk of nitrogen on Earth.
• Nitrogen-15, though less common, is a key indicator used in isotope analysis to understand dietary patterns.

When carnivores consume meat, Nitrogen-15 becomes concentrated, much more than in plant-based diets. By measuring this concentration, scientists can infer how much meat was consumed by ancient populations.

Magnetic Field

A magnetic field is an invisible field around a magnetic object or a moving electric charge that exerts a force on other nearby magnets and moving electrons. In isotope analysis, magnetic fields play a crucial role.
The essence of a magnetic field in these analyses is its ability to influence the trajectory of charged particles. Consider charged particles moving through a magnetic field; they follow a curved path because of the Lorentz force, which is perpendicular to their motion. This phenomenon is useful when separating isotopes, such as Nitrogen-14 and Nitrogen-15, based on their mass differences.
In practice, a velocity selector aligns all the ions to move at the same speed before they enter the magnetic field. The isotopes then follow semicircular paths based on their mass. This separation allows scientists to detect differences and perform studies on them.

Mass Spectrometry

Mass spectrometry is a powerful analytical technique used to identify isotopes and quantify their amounts in a sample. It works by ionizing chemical species and sorting the ions based on their mass-to-charge ratio.
In our scenario, mass spectrometry aids in detecting and analyzing nitrogen isotopes. It provides the framework by which isotopes are separated in a magnetic field.

• A sample is first ionized, often by knocking off an electron to make it positively charged.
• These ions are accelerated and deflected in magnetic and electric fields based on their mass-to-charge ratio.
• Ultimately, they reach a detector that measures the degree of deflection to distinguish between different isotopes.

Mass spectrometry has revolutionized isotope analysis, lending insights invaluable in fields like archeology, forensics, and environmental science.

Prehistoric Diets

Unlocking the dietary habits of ancient civilizations can be akin to solving a historical puzzle. The study of prehistoric diets often involves a multifaceted approach including isotope analysis.
By examining the nitrogen isotope ratios in bone collagen, researchers can estimate the types of food consumed. Since nitrogen is a key component in protein, the relative amounts of nitrogen isotopes reflect meat versus plant consumption.

• Higher levels of Nitrogen-15 suggest a diet rich in meat.
• Conversely, lower levels point to more plant-based foods.

These dietary preferences not only convey information about ancient nutrition but also shed light on social structures, trading patterns, and even cultural practices of prehistoric societies.