Question

Total energy of an electron in the ground state of the hydrogen atom is$-13.6eV$. Its total energy, when a hydrogen atom is in the first excited state, is:
A. $+13.6eV$
B. $+3.4eV$
C. $-3.4eV$
D. $-54.4eV$

Answer 1

Hint: According to Bohr's postulate, when an electron revolves around the nucleus, it possesses kinetic energy due to motion and potential energy due to the electric field. Therefore we can say that the centripetal force is equal to electrostatic force.

Complete step by step solution:
The total energy in ${{n}^{th}}$the state is given by,
${{E}_{n}}=\dfrac{-m{{e}^{4}}}{8\varepsilon _{0}^{2}{{h}^{2}}}\dfrac{1}{{{n}^{2}}}$
Where,
M= mass of an electron
-e= charge on electron
+e= charge on nucleus
V= velocity of the electron
R= radius of a circular orbit
Value of the term $\dfrac{-m{{e}^{4}}}{8\varepsilon _{0}^{2}{{h}^{2}}}=-13.6$
Given that total energy of an electron in the ground state that is n=1 is given by,
${{E}_{n}}=\dfrac{-m{{e}^{4}}}{8\varepsilon _{0}^{2}{{h}^{2}}}\dfrac{1}{{{n}^{2}}}=-13.6\dfrac{1}{{{1}^{2}}}=-13.6eV$
When a hydrogen atom is in the first excited state, the value of n=2. Therefore total energy is given by,
${{E}_{n}}=\dfrac{-m{{e}^{4}}}{8\varepsilon _{0}^{2}{{h}^{2}}}\dfrac{1}{{{n}^{2}}}=-13.6\dfrac{1}{{{2}^{2}}}=\dfrac{-13.6eV}{4}=-3.4eV$
Answer- (C)

Note: From energy equation, we can say that the total energy depends on a number of orbit i.e. N. It also depends on the mass of the electron. An atomic number of hydrogen is one. In the hydrogen spectrum, the value of n in the ground state is 1 then the value of n in the first state is 2, then the second state n=3 and so on. The negative sign in value indicates that the electron is bound to the nucleus due to electrostatic force.