Problem 33 - University Physics with Modern Physics

Question

The amount of meat in prehistoric diets can be determined by measuring the ratio of the isotopes \(^{15}\)N to \(^{14}\)N in bone from human remains. Carnivores concentrate \(^{15}\)N, so this ratio tells archaeologists how much meat was consumed. For a mass spectrometer that has a path radius of 12.5 cm for \(^{12}\)C ions (mass 1.99 \(\times\) 10\(^{-26}\) kg), find the separation of the \(^{14}\)N 1mass 2.32 \(\times\) 10\(^{-26}\) kg2 and 15N (mass 2.49 \(\times\) 10\(^{-26}\) kg) isotopes at the detector.

Step-by-Step Solution

Answer

The separation of the isotopes at the detector is 1.12 cm.

Step 1: Understanding the Mass Spectrometer Function

A mass spectrometer uses the magnetic force to separate isotopes based on their mass and charge. Ions with different masses will deflect by different amounts when subjected to a magnetic field, leading to different path radii. Understanding this principle is fundamental to solving the problem.

Step 2: Formula for Radius of Ion Path

The formula for the radius of the path of an ion in a magnetic field is given by \( r = \frac{mv}{qB} \), where \( m \) is the mass, \( v \) is the velocity, \( q \) is the charge, and \( B \) is the magnetic field. The radius is directly proportional to the mass of the ion.

Step 3: Determine the Velocity and Charge for Isotopes

Assume that both isotopes are singly charged ions (\( q = 1.6 \times 10^{-19} \) C). Typically, in a mass spectrometer, ions are accelerated to the same velocity. For simplicity, we analyze relative changes due to mass, and that involves only their masses and fixed velocity and charge in this context.

Step 4: Calculate the Radius for \(^{14}\)N

Given that \( r_{12}C = 12.5 \) cm for \(^{12}\)C with mass \( m_{12C} = 1.99 \times 10^{-26} \) kg, calculate \( v \) using \( r_{12C} = \frac{m_{12C}v}{qB} \), given we assume \( B \) and \( q \) stay consistent across similar ions.

Step 5: Apply Mass Relation to Equations

Find radius for \(^{14}\)N: given \( m_{14N} = 2.32 \times 10^{-26} \) kg, \( r_{14N} = 12.5 \times \frac{m_{14N}}{m_{12C}} \ = 12.5 \times \frac{2.32 \times 10^{-26}}{1.99 \times 10^{-26}} \approx 14.53 \) cm.

Step 6: Calculate the Radius for \(^{15}\)N

Similarly, for \(^{15}\)N, mass \( m_{15N} = 2.49 \times 10^{-26} \) kg,\( r_{15N} = 12.5 \times \frac{m_{15N}}{m_{12C}} = 12.5 \times \frac{2.49 \times 10^{-26}}{1.99 \times 10^{-26}} \approx 15.65 \) cm.

Step 7: Calculate the Separation at the Detector

The separation between \(^{14}\)N and \(^{15}\)N at the detector is the difference in their radii: \( \text{Separation} = r_{15N} - r_{14N} = 15.65 \text{ cm} - 14.53 \text{ cm} = 1.12 \text{ cm}. \)

Key Concepts

Isotope Separation

Isotope separation is a fascinating process, crucial for scientific investigations in fields like chemistry and archaeology. Mass spectrometry, a powerful analytical technique, often employs this process to differentiate isotopes of the same element. The primary factor distinguishing isotopes is their mass difference, which can be exceptionally small. However, mass spectrometers exploit the distinct paths isotopes take under the influence of a magnetic field to achieve separation.

Here's how it works:

• An isotope, which is an atom of the same element but with a different number of neutrons, is ionized.
• The ions are then accelerated through a magnetic field, where their paths diverge according to their mass-to-charge ratios.
• Ions with different masses follow different radius paths, leading to separation at the detector.

Isotope separation not only identifies different isotopes but can also measure their abundance. This capacity allows scientists to explore a wide range of applications, such as determining the composition of meteorites or verifying the authenticity of historical artifacts. In our mass spectrometry example, the separation was used to deduce prehistoric dietary practices by analyzing nitrogen isotope ratios in bone remains.

Prehistoric Diet Analysis

Prehistoric diet analysis is an exciting area of study that helps archaeologists understand how ancient populations lived and what they consumed. By examining specific isotopes, like nitrogen isotopes in human or animal remains, researchers can gain insights into dietary habits. This method offers a window into history, providing clues about socio-cultural practices and survival strategies of early humans.

Let's focus on nitrogen isotopes:

• The ratio of isotopes, such as \(^{15}\)N to \(^{14}\)N, reveals details about the individual's typical diet.
• Carnivorous diets show higher ratios of \(^{15}\)N because meat consumption inherently concentrates this nitrogen isotope.
• Analyzing these ratios from bones can determine the relative amount of meat in the diet of historical populations.

This approach not only aids in determining the proportion of animals versus plants that were consumed but also reflects on economic, environmental, and cultural aspects of prehistoric communities. By employing this isotope analysis, we can reconstruct diets and understand their evolution over time.

Magnetic Field Effects

Magnetic fields have a significant impact on particles, particularly ions in a mass spectrometer. They serve as an essential tool in the separation of isotopes by curvature of their paths. Understanding magnetic field effects is crucial for interpreting data from mass spectrometers.

Here's a breakdown of these effects:

• When an ion enters a magnetic field region, it experiences a magnetic force perpendicular to its velocity and the magnetic field direction.
• This force causes the ion to move in a circular path, with the radius determined by its mass-to-charge ratio \(r = \frac{mv}{qB}\).
• Heavier ions, which have greater mass, will have larger radii compared to lighter ions, which will have smaller radii.

Through these principles, magnetic fields enable the deflection and spatial separation of ions based on mass differences. This separation is measurable at the detector in a mass spectrometer, providing valuable data for isotope analysis, among other applications. This property of magnetic fields is foundational in analytical techniques that address questions spanning from chemistry to archaeological studies.