In the mass spectrometer of the diagram, neon ions (q= +e...

In the mass spectrometer of the diagram, neon ions $(q=$ $+e$ come from the ion source and are accelerated through a potential difference $V$. The ions then pass through an aperture in a metal plate into a uniform magnetic field where they travel in semicircular paths until exiting into the detector. Neon ions having a mass of $20.0$ u leave the field at a distance of $50.0 \mathrm{~cm}$ from the aperture. At what distance from the aperture do neon ions having a mass of $22.0 \mathrm{u}$ leave the field?

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Key Concepts

Ion Acceleration in an Electric Field

When ions are accelerated through a potential difference, they gain kinetic energy equal to the work done by the electric field. This energy is given by qV, where q is the charge of the ion and V is the potential difference. This concept is critical because it determines the velocity of the ions, which in turn affects their motion in the magnetic field.

Motion of Charged Particles in a Magnetic Field

A charged particle entering a uniform magnetic field experiences a force perpendicular to its velocity, known as the Lorentz force, which causes it to follow a circular path. The radius of this path is derived by equating the magnetic force (qvB) with the centripetal force (mv^2/r), yielding r = (mv)/(qB). This relationship is essential for understanding how different masses result in different path radii.

Mass-to-Charge Ratio in Mass Spectrometry

Mass spectrometers function based on the separation of ions according to their mass-to-charge ratio (m/q). Since the radius of curvature in a magnetic field depends on both mass and velocity of the ion, ions with different masses but the same charge will trace different paths. This principle allows for the identification and analysis of different isotopes or species in a sample.

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