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Ch. 27 - Magnetism
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 27 - MagnetismProblema 56
Capitolo 26, Problema 56

In a mass spectrometer, germanium atoms have radii of curvature equal to 21.0, 21.6, 21.9, 22.2, and 22.8 cm. The largest radius corresponds to an atomic mass of 76 u. What are the atomic masses of the other isotopes?

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Understand the relationship between the radius of curvature in a mass spectrometer and the mass of the isotope. The radius of curvature \( r \) is proportional to the mass \( m \) of the isotope, given by the equation \( r = \frac{mv}{qB} \), where \( v \) is the velocity of the particle, \( q \) is the charge, and \( B \) is the magnetic field. Since \( v \), \( q \), and \( B \) are constant for all isotopes in this problem, the ratio of radii corresponds directly to the ratio of masses.
Set up a proportionality equation to relate the radii and masses of the isotopes. For example, \( \frac{r_1}{r_2} = \frac{m_1}{m_2} \), where \( r_1 \) and \( r_2 \) are the radii of curvature, and \( m_1 \) and \( m_2 \) are the corresponding masses of the isotopes.
Use the given data: the largest radius \( r_5 = 22.8 \ \text{cm} \) corresponds to an atomic mass of \( m_5 = 76 \ \text{u} \). For each of the other radii \( r_1 = 21.0 \ \text{cm} \), \( r_2 = 21.6 \ \text{cm} \), \( r_3 = 21.9 \ \text{cm} \), and \( r_4 = 22.2 \ \text{cm} \), calculate the corresponding masses using the proportionality \( m = m_5 \cdot \frac{r}{r_5} \).
Substitute the values of \( r \) and \( r_5 \) into the proportionality equation for each isotope. For example, for the first isotope, \( m_1 = 76 \cdot \frac{21.0}{22.8} \). Repeat this calculation for \( r_2, r_3, \) and \( r_4 \).
Express the results for the atomic masses of the isotopes in atomic mass units (u). These values will correspond to the calculated masses for \( r_1, r_2, r_3, \) and \( r_4 \), based on the proportionality relationship.

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Mass Spectrometry

Mass spectrometry is an analytical technique used to measure the mass-to-charge ratio of ions. In this process, atoms or molecules are ionized and accelerated through a magnetic or electric field, causing them to follow curved paths. The radius of curvature is influenced by the mass and charge of the ions, allowing for the determination of their mass based on their trajectory.
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Mass Spectrometers

Isotopes

Isotopes are variants of a chemical element that have the same number of protons but different numbers of neutrons, resulting in different atomic masses. For example, germanium has several isotopes, each with a unique mass. The mass spectrometer can separate these isotopes based on their mass-to-charge ratios, enabling the identification of their respective atomic masses.

Radius of Curvature

The radius of curvature in a mass spectrometer is a measure of the path that an ion takes as it moves through a magnetic or electric field. It is directly related to the mass and charge of the ion; heavier ions or those with lower charge will have larger radii. By analyzing the radii of curvature of different isotopes, one can infer their atomic masses relative to a known isotope.
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Calculating Radius of Nitrogen
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