Determining Diet. One method for determining the amount of...

Determining Diet. One method for determining the amount of corn in early Native American diets is the stable isotope ratio analysis (SIRA) technique. As corn photosynthesizes, it concentrates the isotope carbon-13, whereas most other plants concentrate carbon-12. Overreliance on corn consumption can then be correlated with certain diseases, because corn lacks the essential amino acid lysine. Archaeologists use a mass spectrometer to separate the 12$\mathrm { C }$ and $^ { 13 } \mathrm { C }$ isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing one electron) atoms of speed $8.50 \mathrm { km } / \mathrm { s } ,$ and you want to bend them within a uniform magnetic field in a semicircle of diameter 25.0$\mathrm { cm }$ for the 12$\mathrm { C }$ . The measured masses of these isotopes are $1.99 \times 10 ^ { - 26 } \mathrm { kg } \left( ^ { 12 } \mathrm { C } \right)$ and $2.16 \times 10 ^ { - 26 } \mathrm { kg } \left( ^ { 13 } \mathrm { C } \right) .$ (a) What strength of magnetic field is required? (b) What is the diameter of the 13 C semicircle? (c) What is the separation of the $^ { 12 } \mathrm { C }$ and $^ { 13 } \mathrm { C }$ ions at the detector at the end of the semicircle? Is this distance large enough to be easily observed?

Determining Diet. One method for determining the amount of corn in early Native American diets is the stable isotope ratio analysis (SIRA) technique. As corn photosynthesizes, it
concentrates the isotope carbon-13, whereas most other plants concentrate carbon-12. Overreliance on corn consumption can then be
correlated with certain diseases, because corn lacks the essential amino acid lysine. Archaeologists use a mass spectrometer to separate the 12$\mathrm { C }$ and $^ { 13 } \mathrm { C }$ isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing
one electron) atoms of speed $8.50 \mathrm { km } / \mathrm { s } ,$ and you want to bend
them within a uniform magnetic field in a semicircle of diameter 25.0$\mathrm { cm }$ for the 12$\mathrm { C }$ . The measured masses of these isotopes are
$1.99 \times 10 ^ { - 26 } \mathrm { kg } \left( ^ { 12 } \mathrm { C } \right)$ and $2.16 \times 10 ^ { - 26 } \mathrm { kg } \left( ^ { 13 } \mathrm { C } \right) .$ (a) What strength of magnetic field is required? (b) What is the diameter of
the 13 C semicircle? (c) What is the separation of the $^ { 12 } \mathrm { C }$ and $^ { 13 } \mathrm { C }$
ions at the detector at the end of the semicircle? Is this distance
large enough to be easily observed?

Key Concepts

Stable Isotope Ratio Analysis

Stable isotope ratio analysis (SIRA) is an analytical technique used to measure the ratios of stable isotopes, often to study processes such as diet, metabolism, or environmental conditions. In archaeology, analyzing the ratios of carbon-13 to carbon-12 found in human remains can provide insights into the relative consumption of different types of plants, like corn versus other sources, because of the different photosynthetic pathways that preferentially incorporate one isotope over the other.

Isotope Separation in Magnetic Fields

Isotope separation in a magnetic field takes advantage of the small differences in mass between isotopes, which result in slightly different radii of curvature when the ions are subjected to the Lorentz force. The separation of these ions at a detector allows for their identification and quantification. While the differences in radii (and hence the separation at the detector) may be very small, careful calibration and design of the instrument can make these differences observable.

Mass Spectrometry

Mass spectrometry is a technique used to separate ions based on their mass?to?charge ratio. In such instruments, ions are accelerated by an electric field and then deflected by a magnetic field, with the curvature of their path depending on the momentum and charge of the particles. This makes it possible to identify different isotopes and molecular fragments, which is essential in fields ranging from chemistry to archaeology.

Lorentz Force and Circular Motion

The Lorentz force is the force exerted on a charged particle moving through a magnetic field and is given by F = qvB, where q is the charge, v is the velocity, and B is the strength of the magnetic field. When this force acts perpendicular to the velocity of the particle, it causes the particle to move in a circular path. Understanding this concept allows one to calculate the radius of the circular motion, which is crucial in determining the deflection of ions in a mass spectrometer.

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