Question

The total energy of the electron in the hydrogen atom in the ground state is \[{{ - 13}}{{.6 eV}}\]. The K.E of the electron is:
A. \[{{13}}{{.6 ev}}\]
B. Zero
C. \[{{ - 13}}{{.6 eV}}\]
D. \[{{6}}{{.8 eV}}\]

Answer 1

Hint: Total energy of an electron is the sum of kinetic as well as potential energy. When we know the total energy of the electron, we can calculate the kinetic energy by substituting the value of kinetic energy into potential energy to bring the equation in one variable term.

Complete step by step answer:
Kinetic energy of an electron can be defined as the energy that it possesses due to their motion and it can be calculated by the following formula.
\[{{Total \;energy = Kinetic \;energy + Potential \;energy}}\]…………………………… (1)
“m” mass electron with charge “e” revolving around the nucleus charge \[{{Ze}}\] and tangential velocity “v” in radius of the orbit “r”.
According to Coulomb's law, the kinetic energy of an electron can be calculated by the following formula.
\[{{Kinetic\; energy\; of\; electron = }}\dfrac{{{{kZ}}{{{e}}^{{2}}}}}{{{{2r}}}}\]
Here,
\[{{K}}\]= Electrostatic constant
\[{{Z}}\] = Atomic number
\[{{e}}\] = Charge of the electron
\[{{r}}\] = Radius of orbit.
The potential energy is the negative of kinetic energy of an electron.
\[{{Potential \;energy\; of\; electron = - }}\dfrac{{{{kZ}}{{{e}}^{{2}}}}}{{{r}}}\]
Where, \[{{Z}}\] is an atomic number of atoms.
\[{{P}}{{.E = - 2(K}}{{.E)}}\]
Substitute the equation (1)
\[{{ = - 2(K}}{{.E) + K}}{{.E = - K}}{{.E}}\]
From the given,
The total energy of the electron in the hydrogen atom in the ground state = \[{{ - 13}}{{.6 eV}}\]
\[{{Kinetic\; energy = - K}}{{.E}}\]
\[{{ = - ( - 13}}{{.6 eV) = 13}}{{.6 eV}}\]

So, the correct answer is Option A.

Additional information:
The energies allowed for the electron in hydrogen atom is ${{{E}}_{{n}}}{{ = }}\dfrac{{{{ - 13}}{{.6 eV}}}}{{{{{n}}^{{2}}}}}$ .
The n used here is the principle quantum number.
The calculated value of \[{{{E}}_{{n}}}\]is the possible value of the total energy of the hydrogen atom i.e. potential energy and kinetic energy combined.
In the case of hydrogen atoms, the energy level totally depends on the principle quantum number. The energy levels are degenerate in hydrogen atom i.e. the electrons in hydrogen atom may be present in different states, having different wave functions, marked by different sets of quantum numbers but still having the same energy..

Note: The energy eigenfunctions denotes the stationary state of the electrons. We cannot locate the electron as well as calculate its energy at the same time. If we happen to calculate or know the electron’s energy, we can only predict the probabilities as to where that electron is present. If we calculate the position of the electron then we might lose the information about the energy of the electron.