Question

Relation betweeb total potential and kinetic energy in electron

Answer 1

Method 1:-In the Bohr model of the hydrogen atom we take the potential energy of the electron to be zero at infinity, so the potential energy becomes negative as the electron approaches the hydrogen atom. However kinetic energy is always positive.

In the Bohr ground state the potential energy is -27.2 eV. Note that as described above this energy is negative. The kinetic energy is +13.6eV, so when we add the two together we get the total energy to be -13.6eV.

The total energy is negative because the electron is bound to the hydrogen atom and to remove the electron we have to put in energy. In fact we have to put in 13.6eV, which is simply the ionisation energy of hydrogen. Since the ground state has an energy of -13.6eV when we put in +13.6eV that makes the total energy zero, and as we said at the start this is the energy of the electron at infinity.

In any system where the force obeys an inverse square law there is a link between the potential and kinetic energy. This link is called the virial theorem and it says that if we call the potential energy V and the kinetic energy T the relationship is:

V=−2T

Note the minus sign in this equation. The total energy E is V+T, and if we add T to both sides of this equation we get:

E=V+T=−T=V/2

That is why the total energy is half the potential energy.

Method 2:-Potential energy is negative because the work done by external force on the electron is negative when it is brought from infinity to the first stationary orbit. KE being positive when added to PE gives us a negative number with less magnitude than that of PE. (TE=−KE=PE2).