In a mass spectrometer, germanium atoms have radii of...

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

Key Concepts

Mass Spectrometry

Mass spectrometry is an analytical technique used to separate and identify isotopes or molecules based on their mass?to?charge ratio. In such instruments, ions are generated and then accelerated by an electric field before being deflected by a magnetic field. The resulting paths have radii of curvature that depend on the mass and charge of the ions, making it possible to determine their masses with high accuracy.

Ion Acceleration by Electric Fields

Ions in a mass spectrometer are accelerated by a fixed voltage, which imparts them with a defined kinetic energy. This acceleration ensures that all ions, regardless of their mass, start with the same kinetic energy. Because the kinetic energy is proportional to the velocity squared and inversely proportional to the mass, this process is crucial in ensuring that differences in the observed radius of curvature are due solely to differences in mass.

Magnetic Deflection and Circular Motion

After acceleration, ions enter a magnetic field where the Lorentz force causes them to move in circular arcs. The radius of this arc is determined by the balance between the centripetal force required for circular motion and the magnetic force acting on the moving charged ion. This relationship allows the instrument to separate ions based on their masses, as heavier ions will have larger radii of curvature than lighter ones under the same conditions.

Proportional Relationship between Radius and Atomic Mass

Under the operating conditions in a mass spectrometer, ions accelerated through the same voltage and subjected to the same magnetic field exhibit a relationship where the radius of curvature is proportional to the square root of the mass. Consequently, squaring the radius provides a value that is proportional to the mass. This scaling relationship is key to determining the relative masses of isotopes when one isotope's mass is known, as the masses of other isotopes can be calculated by comparing the squares of their radii.

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