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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 . Overneliance 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 \(^{10} \mathrm{C}\) 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 \(1.99 \times 10^{-25} \mathrm{kg}\left(^{12} \mathrm{C}\right)\) 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 \(^{13} \mathrm{C}\) 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?

Step by step solution

01

Understand the Problem

We need to calculate the strength of the magnetic field required to bend singly ionized carbon-12 isotopes in a semicircle of a given diameter and then find out changes for carbon-13 isotopes. The problem also asks us to find out the difference in paths of the two isotopes at the end of the semicircle.

02

Apply Centripetal Force Formula

The force acting on an ion moving in a magnetic field can be expressed as \( F = qvB \). Here, \( q \) is the charge, \( v \) is the velocity, and \( B \) is the magnetic field strength. This force must equal the centripetal force needed to bend the path of the ion, which is \( F_c = \frac{mv^2}{r} \).

03

Set Forces Equal

Since the magnetic force equals the centripetal force, we have \( qvB = \frac{mv^2}{r} \). Solving for \( B \), we get \( B = \frac{mv}{qr} \).

04

Use Given Values

For \(^{12}C\), use \( m = 1.99 \times 10^{-26} \text{ kg} \), \( v = 8.50 \times 10^3 \text{ m/s} \), and a semicircle of diameter 0.25 m, which gives a radius \( r = 0.125 \text{ m} \). The charge \( q \) for a singly ionized atom is \( 1.6 \times 10^{-19} \text{ C} \).

05

Calculate Magnetic Field for \(^{12}C\)

Substitute \( m = 1.99 \times 10^{-26} \text{ kg} \), \( v = 8.50 \times 10^3 \text{ m/s} \), \( r = 0.125 \text{ m} \), and \( q = 1.6 \times 10^{-19} \text{ C} \) into the equation \( B = \frac{mv}{qr} \). This yields \( B = \frac{1.99 \times 10^{-26} \times 8.50 \times 10^3}{1.6 \times 10^{-19} \times 0.125} \approx 0.084 \text{ T} \).

06

Determine Diameter for \(^{13}C\)

For \(^{13}C\), use the same formula to calculate the radius and diameter. Since they are moving with the same speed in the same magnetic field, the formula \( r = \frac{mv}{qB} \) will give \( r \). Use \( m = 2.16 \times 10^{-26} \text{ kg} \) for \(^{13}C\). The calculated radius will be \( r = \frac{2.16 \times 10^{-26} \times 8.50 \times 10^3}{1.6 \times 10^{-19} \times 0.084} \approx 0.136 \text{ m} \), and the diameter is \( 0.272 \text{ m} \).

07

Calculate Separation at Detector

The separation of the ions at the detector is given by the difference in their path diameters: \( 0.272 \text{ m} - 0.250 \text{ m} = 0.022 \text{ m} \) or \( 2.2 \text{ cm} \).

08

Conclusion about Observability

A separation of 2.2 cm is generally large enough to be detected by standard laboratory equipment.

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Key Concepts

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

Stable Isotope Ratio Analysis

Stable isotope ratio analysis (SIRA) is a fascinating technique used by scientists and archaeologists to study ancient diets and environmental conditions. This method involves examining the ratios of different isotopes within a sample. Each isotope is a variant of an element with a different number of neutrons, and they often have unique properties.
In the context of archaeological studies, isotopes like carbon-13 and carbon-12 are particularly useful. Plants, such as corn, tend to concentrate certain isotopes during photosynthesis. Corn, for example, absorbs more carbon-13 compared to most other plants. By measuring the ratio of carbon-13 to carbon-12 in human remains, archaeologists can infer the amount of corn in ancient diets.
This method helps reconstruct historical dietary patterns and even links them to health outcomes. Corn is deficient in lysine, a crucial amino acid, and its overconsumption can lead to nutritional deficiencies. By understanding ancient diets, we can learn much about the lifestyle and health of past populations.

Magnetic Field Strength

The concept of magnetic field strength is central to the operation of a mass spectrometer. In this device, a magnetic field is used to alter the path of charged particles, such as ions, by exerting a force on them. This force is proportional to the strength of the magnetic field, the charge of the ions, and their velocity.
Mathematically, the force experienced by an ion in a magnetic field is given by the equation: \[ F = qvB \] Here, \( F \) represents the force, \( q \) the charge of the ion, \( v \) its velocity, and \( B \) the magnetic field strength.
For ions to bend along a designated path, like a semicircle in a mass spectrometer, the magnetic field must exert sufficient force to provide the necessary centripetal force. This allows the separation of isotopes based on their mass, as different isotopes will curve differently under the influence of the magnetic field, thus enabling their identification and analysis.

Carbon Isotopes

Carbon isotopes are essential for a variety of scientific analyses, from studying climate change to investigating dietary habits of ancient populations. Naturally occurring carbon primarily exists in two stable isotopes: carbon-12 and carbon-13.
Carbon-12 is the most common isotope, comprising the vast majority of carbon found on Earth. It consists of six protons and six neutrons. On the other hand, carbon-13 is less abundant, having an extra neutron. Despite this, it is extremely valuable for scientific studies.
The ratio of carbon-13 to carbon-12 in materials can reveal significant information. In the case of plants, the ratio is influenced by the type of photosynthesis they conduct. This information is crucial in reconstructing ancient environments and diets, as different plants imprint distinct carbon isotope signatures.
In stable isotope ratio analysis, these differences in isotope composition become a powerful tool, shedding light on dietary consumption and agricultural practices throughout history.

Centripetal Force

Centripetal force is a fundamental concept in physics, describing the force necessary to keep an object moving in a circular path. When an object travels in a circle, it constantly changes direction, which requires a center-seeking force known as centripetal force.
In the context of a mass spectrometer, centripetal force ensures that charged particles like ions follow a curved path when exposed to a magnetic field. The equation for centripetal force is given by: \[ F_c = \frac{mv^2}{r} \] where \( m \) is the mass of the ion, \( v \) its velocity, and \( r \) the radius of the circular path.
Within a mass spectrometer, the magnetic field provides the necessary centripetal force to bend the ions into a semicircular path. As the ions have different masses, their paths will differ in radius. By analyzing these paths, scientists can determine the mass and, hence, identify different isotopes present in a sample. This application of centripetal force is crucial for separating and analyzing isotopes based on their mass.