Hydrogen atom - Structure of matter

Question 1

Explain the structure of the hydrogen atom.

Answer 1

The simplest system composed of nucleons and electrons is the hydrogen atom.

In this system, one electron moves in the central electric field of one proton.

The distance from the nucleus, at which the electron appears with the highest probability, can be estimated from the relationshipe of uncertainty.

Question 2

How can the uncertainty of momentum be found?

Answer 2

If the electron moves at a distance r from the nucleus, the the uncertainty just equals r.

From the following equation we get for the uncertainty of momentum, p.

∆p = h/r

Question 3

How can kinetic energy be found?

Answer 3

The total energy of the electron in the field of the atom is given by a sum of its kinetic and potential energies.

By using the equation for the uncertainty of momentum in the equation below, we can get the kinetic energy E_k.

E_k = p²/(2m_e) = ћ²/(2m_er²)

Where m_e is the mass of electron.

Question 4

How can the potential energy of an electron by found?

Answer 4

The potential energy E_p of an electron with the charge -e in the force field of a proton with the charge +e at the distance r is given by:

E_p = - 1/(4 πε₀) * e²/r

Where ε₀ is the permittivity of vacuum.

Question 5

How can the total energy of an electron in the field of one proton be found?

Answer 5

The potential energy of electron in the field of the nuclus is negitive.

It reaches the highest (zero) value at ‘infinite’ distance from the nucleus, where the force action of chrages of electron and nucleus is negligible.

Thus the toaly energy, E, of an electron in the field of one proton is:

E = E_k + E_p = (ћ² / (2m_er²)) -( 1/(4πε₀)) * (e²/r)

Question 6

How can the total electron energy be calculated and plotted on a curve?

Answer 6

Values of the total electron energy can be calculated and plotted as a function of the distance r from the nucleus.

This curve of the total energy manifests a minimum value for a certain distance r₀.

It holds generally in physics that each system is stable at the minimum value of its energy.

Therefore, the higest probability of the electron appearance is just at this distance.

Electron is stable state does not emit energy.

Distance r₀ calculated from the equation:

E = E_k + E_p = ( / (2m_er²)) -( 1/(4πε₀)) * (e²/r)

By uding dE/dr = 0 (the extreme of function can be found from the condition that its first derivitibe equals zero) is given by:

r₀ = (4πε₀ћ²)/(m_ee²)

Question 7

How can Bohr’s radius be found?

Answer 7

By using the equation

r₀ = (4πε₀ћ²)/(m_ee²) for the minimum value for a certain distance r₀.

The values can be substituted as follows:

Electron mass m_e = 9.1*10^-31 kg

Permittivity of vacuum ε₀ = 8.8*10^-12 F.m^-1

Electron charge e = 1.6*10^-19 C

ћ = 1.05*10^-34 Js

We get r₀ = 5.29*10^-11 m.

This distance is called the Bohr’s radius.

Question 8

How can the energy of the hydrogen atom basic state equation be found?

Answer 8

After substituting back for r₀ (r₀ = 5.29*10^-11) into the equation:

E = E_k + E_p = ( ћ/ (2m_er²)) -( 1/(4πε₀)) * (e²/r)

The total electron energy can be found.

We get the energy of the hydrogen atom basic state equation:

E = - (1/(32π²ε₀²)) * ((m_ee⁴)/ћ²)

Question 9

Explain Schrodinger’s equation.

Answer 9

The equation E = - (1/(32π²ε₀²)) * ((m_ee⁴)/ћ²)

corresponds to the solution of the Schrodinger’s equation for an electron in the field of a proton for n=1.

The state with n=1 is the basic state, states with n = 2, 3, … are excited states into which electron mat transit after absorption of energy.

Substituting numerical values for the quantities in equation r_n = n²r₀, we get subsequently for n=1, 2, 3, … up to infinity, the following energy values:

E₁ = -12.6 eV

E₂ = -3.38 eV

E₃ = -1.5 eV etc

up to E_∞ = 0

It can be demonstrated that to the each energy level value E_n corresponds to the most probable distance r_n from the nucleus given by th equation:

r_n = n²r₀

The most probable distance from the nucleus increases with n².

Question 10

Based on the wave theory of matter, how can the electron wavelengths dependance on its momentum be defined as?

Answer 10

Based on the wave theory of matter, the electron wavelength depends on its momentum and is defined by the equation:

λ = h/p = h / (√(2mE))

It can be shown that it equals the value of path 2πr₀ = 3.3*10^-10 m.

This can be generalised to the statement that electron can move around nucleus for an infinitely long tim ewithout emission of energy if its path is an integer multiple of DeBroglie’s wave length. That is is:

nλ = 2πr_n

Where n is principle quantum number.

During transtions from higher into lower energy levels, the hydrogen atom emits photons that possess discrete (line) spectrum.