Final answer:
The kinetic energy needed by an electron to excite an atom can be found by summing the electron's classical kinetic and potential energies, with potential energy considering the electric field of the nucleus. This sum represents the total energy of the electron and can be expressed in terms of the mass of the electron, the charge of the electron, and other given constants.
Step-by-step explanation:
To find the kinetic energy (KE) that an electron must have to excite an atom, we can derive the expression using principles from physics. The total energy (En) of an electron in an atom is the sum of its kinetic energy and potential energy (PE).
In classical physics, for non-relativistic speeds, kinetic energy is given by KE = (1/2)me v², where me is the mass of the electron and v is its velocity.
The potential energy of the electron due to the electric field of the nucleus is PE = -kZqe/rn, where k is Coulomb's constant, Z is the atomic number, qe is the charge of the electron, and rn is the distance of the electron from the nucleus.
Combining these, we can represent the total energy of the orbiting electron as the sum of KE and PE: En = KE + PE. Since we are asked to find KE to excite the atom, the energy needed would be to move the electron to a higher energy level.
This equation shows that KE can be expressed in terms of the given parameters e (elementary charge), me (mass of the electron), and m (representing either mass or a given constant in a given problem).