Chemistry

Problem

For a hydrogen atom in the

n

th stationary state: (a) Obtain a general expression for the total energy

E_n

of the electron by combining its kinetic and potential energies. (b) Hence prove that the magnitude of

E_n

varies inversely as

n^{2}

and comment on the sign of the energy obtained. (c) Using your final expression, determine the total energy (in electron-volts) of the electron when

n=3

. Use

m=9.11\times10^{-31}\,\text{kg},\; e=1.60\times10^{-19}\,\text{C},\; h=6.63\times10^{-34}\,\text{J\,s},\; \varepsilon_0=8.85\times10^{-12}\,\text{C}^2\text{N}^{-1}\text{m}^{-2}

. a. (a) Obtain a general expression for the total energy

E_n

of the electron in the

n

th stationary orbit. b. (b) Prove that

|E_n|\propto \dfrac{1}{n^{2}}

and state the physical meaning of the negative sign in

E_n

. c. (c) Calculate the numerical value of

E_3

for hydrogen in electron-volts using the given constants.