An electron collides inelastically and head-on with a...

An electron collides inelastically and head-on with a mercury atom at rest. (a) If the separation of the first excited state and the ground state of the atom is exactly $4.9 \mathrm{eV}$ what is the minimum initial electron kinetic energy needed to raise the atom to its first excited state and also conserve momentum? Assume that the collision is completely inelastic. (b) What is the initial speed of the electron in this case? (c) What is the speed of the electron and atom after the collision? (d) What is the kinetic energy (in electron volts) of the electron after collision? Is the approximation that the electron loses all of its kinetic energy in an inelastic collision justified?

Key Concepts

Inelastic Collisions

In an inelastic collision, the colliding bodies do not conserve kinetic energy even though momentum is conserved. Some of the initial kinetic energy is converted into other forms of energy, such as internal energy or excitation energy in quantum systems. In completely inelastic collisions, the objects may stick together, making the analysis of the final kinetic energy more involved and especially relevant when energy is used to excite internal states.

Energy Quantization and Excited States

Many atomic and molecular systems have quantized energy levels, meaning that transitions between states (such as the ground state to an excited state) require specific amounts of energy. This concept is critical in understanding processes such as electronic excitations, where a minimum threshold energy (the energy difference between levels) must be provided to achieve a transition.

Conservation of Momentum

Momentum conservation is a fundamental principle in physics that holds true in all collision processes, irrespective of whether the collision is elastic or inelastic. In the context of collisions involving particles and atoms, conservation of momentum is used to relate the velocities before and after the collision, imposing constraints that are essential when calculating the minimum kinetic energy required or the final velocities of the objects involved.

Kinetic Energy and Speed Relationship

The relationship between kinetic energy and speed is given by the equation K = 1/2 m v^2, where m is the mass and v is the velocity of the particle. This formula is used to connect the kinetic energy of particles with their speeds, and it is particularly useful in determining the initial speed required for a particle to achieve a specific kinetic energy threshold necessary for exciting another particle or system.

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