Saturday, October 4, 2008

Bohr’s Model and Physics of the Atom - July-Dec Revision

Revision - Points of the chapter

Early Atomic Models

The idea that all matter is made of very small indivisible particles is very old.

Robert Boyle’s study of compression and expansion of air brings out the idea that air is made of tiny particles with lot of empty space between the particles.

The smallest unit of an element which carries all the properties of the element is called an atom.

Experiments on discharge tube, measurement of e/m by Thomson etc. established the existence of negatively charged electrons in the atoms.

Because atoms are electrically neutral, a search for the positive charge inside the atom was started.

Thomson’s Model of the atom

Thomson (1898) suggested that the atom is a positively charged solid sphere in which electrons are embedded in sufficient number to create a neutral atom. This model of the atom could explain why only negatively charged particles are being emitted when a metal is heated. This model was also useful to explain the formation of ions and ionic compounds.

Lenard’s Suggestion

Lenard observed that cathode rays are pssing through thin material without any deviation. According to him, this was so because, there is a lot of empty space in atoms. Hence the positive charged particles are also tiny like electrons.

Rutherford’s Model of the Atom

Rutherford experimented with alpha rays or particles. When he bombarded gold foils with alpha particles, Many went without deviation, some had some deviation and some were deflected by more than 90 and came back. Hence he made a conclusion that there was a particle with a mass equivalent to alpha particle inside the atom. The mass of an atom is concentrated in this particle.

The size of this particle was also estimated by Rutherford. Its linear size is 10 fermi ( 1 fermi is equal to 1 femtometre = 10^-15 m).

Rutherford proposed that the atom contains a positively charged tiny particle called nucleus. It contains the entire mass of the atom. Outside this nucleus, at some distance, electrons move around. The positive of charge of nucleus is exactly equal to the negative charge of the electrons of the atom.

Because electrons are very very light compared to the nucleus, due to heat only electrons come out.

Movement of electrons is to brought in and the coulomb force between the nucleus and the electron is assumed to provide only centripetal force to make the electron rotate in a circular motion.

Hydrogen Spectra

If hydrogen gas is enclosed in a sealed tube and heat to high temperatures, it emits radiation. If this radiation is passed through a prsim, components are different wavelengths get deviated by different amounts and we get the hydrogen spectra on a screen. In the spectra of hydrogen atom, it is observed that light of wavelength 656.3 nm and then light of wave length 486.1 nm are present. Hydrogen atoms do not emit any radiation between 656.3 nm and 486.1 nm. Similarly radiation is observed at 434.1 nm and 4202.nm.

In the invisible region also, there is radiation emitted by the hydrogen atom at discrete wavelengths.

The wavelengths nicely fit the equation

1/λ = R [1/n² - 1/m²]

where R = 1.09737*10^7 m-1.
n and m are integers with m>n.

The spectrum in the ultraviolet region is called Lyman series and you get the series by setting n = 1.

The hydrogen spectrum in the visible region is called Balmer series and you get the series by setting n = 2.

The hydrogen spectrum in infrared region is called Paschen series and you get the series by setting n= 3.

Difficulties with Rutherford’s Model

Rutherford’s model assumes that the electron rotates around the nucleus. Maxwell’s equations of a electromagnetism show that accelerated electron must continuously emit electromagnetic radiation. But a hydrogen does not emit radiation at ground level energy or normal energy. It emits radiation only when heated. Also, if it emits radiation it will lose energy and the radius of its circular motion will decreases and finally it will fall into the nucleus. Hence, the atomic model proposed by Rutherford needs modification. Bohr proposed such modifications.

Bohr’s Model

Bohr’s postulates

1. The electrons revolve around the nucleus in circular orbits.
2. the orbit of the electron around the nucleus can be only some special values of radius. In these special radii orbits, the electron does not radiate energy as expected from Maxwell’s laws. These orbits are called stationary orbits.
3. The energy of the atom has a definite value when electrons are in a given stationary orbit. But the if more energy is provided to the atom, the electron can jump from one stationary orbit to another stationary orbit of higher energy. If it jumps from an orbit of higher energy (E2) to an orbit of lower energy (E1), it emits a photon of radiation. The energy of the emitted photon will be E2 – E1.

The wave length of the emitted radiation is given by the Einstein-Planck equation

E2-E1 = hυ = hc/λ

4. In stationary orbits, the angular momentum l of the electron about the nucleus is an in integral multiple of the Planck constant h divided by 2 π.

l = nh/2 π

This assumption is called Bohr’s quantization rule.

Energy of an Hydrogen Atom

Bohr’s postulated can be used to find the allowed energies of the hydrogen atom when its single electron is in various stationary orbits. The methodology can be used any hydrogen like ions which have only one electron.. Therefore it is valid for He+, Li++, Be+++ etc.

Assume that the nucleus has a positive charge Ze ( there are z protons each with positive charge e).

By equating the coulomb force acting between Ze and e to the centripetal acceleratin mv²/r, we get r the radius at which the electron revolves.

r = Ze²/4π ε0

From Bohr’s quantization rule,

mvr = nh/2 π
where n is a positive integer

Eliminating v from both the equations we get

r = ε0h²n²/πmZe²

We get expression for v as

v = Ze²/2 ε0hn

Hence allowed radii are proportinal to n² and for each value of n = 1,2,3…we allowed orbits.

The smallest radius orbit will have n = 1.

As we have expression for v, we can give an expression for kinetic energy when electron is in nth orbit is

K = ½ mv² = mZ²e4/8 ε0²h²n²

The potential energy of the atom is

V = - Ze²/4π ε0r = -mZ²e4/4ε0²h²n²

The expression for potential energy is obtained by assuming the potential energy to be zero when the nucleus and the electron are widely separated.

The total energy of the atom is

E = K+V = - mZ²e4/8 ε0²h²n²

When an atom is nth stationary orbit, it is said to be in the nth energy state.

In giving an expression for the total energy of the atom, kinetic energy of the electron and potential energy of theelectron-nucleus pair are considered. Kinetic energy of the nucleus is assumed to be negligible.

Radii or different orbits.

For hydrogen, z =1 and we get r1 as 53 picometre ( 1pm = 10^-12 m) or 0.053 nm. This length is called the Bohr radius and is a convenient unit for measuring lengths in atomic physics. It is denoted by me as the symbol a0 (In HC Verma a different symbol is given. I am using this symbol as a convenience).

In terms of Bohr radius the second allowed radius is 4 a0 and third is 9 a0 and so on. In general nth orbit of hydrogen atom is n²a0.

For a hydrogen like ion with Z protons in the nucleus,

rn = radius of ‘n’ th orbit = n²a0/Z

Energy at Ground and Excited states

From the total energy equation or expression we can find for hydrogen total energy when electron is in the state n = 1 (which is the ground state) as E1 = -13.6 eV. The radius corresponding to this energy is 53 picometre or 0.053 nm.

As in the energy equation only n changes with orbits, En is proportional to n².

There En = E1/n² = -13.6eV/n²

Note that the energy is expressed in negative units, so that larger magnitude means lower energy.

The state of an atom with the lowest energy is called its ground state.

The states with higher energies are called excited states.
Energy of hydrogen atom in the ground state is -13.6 eV.
Energy of hydrogen atom in the next excited state, that is n = 2 state is -3.4 eV.

Emission by Hydrogen Atom and Hydrogen spectra

When heated some atoms in the hydrogen become excited and when electrons jump from higher energy levels to lower energy levels in those excited atoms, photons with specific wavelengths are emitted or radiated.

If an electron jumps from mth orbit to nth orbit (m>n), the energy of the atom gets reduced from Em to En. The wavelength of the emitted radiation will be

1/ λ = (Em – En)/hc = RZ²{1/n² - 1/m²]

where R is the Rydberg constant.
R = 1.0973*10^7 m-

In terms of the Rydberg constant total energy of the atom in the nth state is E = -RhcZ²/n²

For hydrogen atom, when n =1, E = -Rhc and we know its value is -13.6 eV.
Energy of 1 rydberg means -13.6 eV.
Rhc = 13.6eV

Series structures

Lyman series:
All transitions to n =1 state from higher state give the radiation in Lyman series.

Jumping of the electron from n =2 to n =1 gives
1/ λ = R[1 – 1/2²] = R(1 – ¼) which will give λ = 121.6 nm.

Jumping of the electron from n = ∞ to n = 1 gives

1/ λ = R[1 – 1/∞²] = R(1 -0) which gives λ = 91.2 nm.

Balmer seires
All transitions to n = 2 from higher states given radiations within the range of 656.3 nm and 365.0 nm. These wavelengths fall in the visible region.

Paschen series
The transitions or jumps to n = 3 from higher energy levels give Paschen series in the range 1875 nm to 822 nm.

Ionization potential

The energy of the hydrogen atom in ground state is -13.6 eV. If we supply more than 13.6 eV to the hydrogen atom, the electron and the nucleus get separated and electron moves with some kinetic energy independently (Remember plasma in nuclear fusion).
The minimum energy needed to ionize an atom is called ionization energy. The potential difference through which an electron should be accelerated to acquire this much energy is called ionization potential.

Thus ionization energy of hydrogen atom in ground state is 13.6 eV and ionization potential is 13.6 V.

Binding energy

Binding energy of a system is defined as the energy released when its constituents are brought from infinity to form the system. Now we know that binding energy of a hydrogen atom is 13.6 eV. The energy is zero when the electron and nucleus at infinite distance. When the electron is brought into n = 1 orbit, the energy becomes -13.6 eV and hence 13.6 eV is released which is the binding energy.

Excitation potential

The energy needed to take the atom form its ground state to an excited state is called the excitation energy of that excited state.

As the hydrogen atom’s ground state energy is -13.6 eV and its energy when electron is in n =2 orbit is -3.4 eV, we have to supply 10.2 eV to excite a hydrogen atom to its first excited state which is electron in n = 2 orbit.

The potential through which an electron should be accelerated to acquire the excitation energy is the excitation potential.
The excitation potential needed bring hydrogen to its first excited state is 10.2 V.

Limitations of Bohr’s Model

Maxwell’s theory of electromagnetism is not replaced or refuted but it is arbitrarily assumed that in certain orbits, electrons get the licence to disobey the laws of electromagnetism and are allowed not to radiate energy.

The Wave Function of an Electron

Quantum mechanics describes the spectra in a much better way than Bohr’s model.

Electron has a wave character as well as a particle character. The wave function of the electron ψ(r,t ) is obtained by solving Schrodinger’s wave equation. The probability of finding an electron is high where | ψ(r,t )|² is greater. Not only the information about the electron’s position but information about all the properties including energy etc. that we calculated using the Bohr’s postulates are contained in the wave function of ψ(r,t).

Quantum Mechanics of the Hydrogen Atom

The wave function of the electron ψ(r,t) is obtained from the Schrodinger’s equation

-(h²/8π²m) [∂²ψ /∂x² + ∂²ψ /∂y² + ∂²ψ/∂z²] - Ze²ψ/4πε0r = E ψ

(x.y,z ) refers to a point with the nucleus as the origin and r is the distance of this point from the nucleus.
E refers to the energy.
Z is the number of protons.

There are infinite number of functions ψ(r,t) which satisfy the equations.

These functions may be characterized by three parameters n,l, and ml.

For each combination of n,l, and ml there is an associated unique value of E of the atom of the ion.

The energy of the wave function of characterized by n,l, and ml depends only on n and may be written as

En = - mZ²e4/8 ε0²h²n²

These energies are identical with Bohr’s model energies.

The paramer n is called the principal quantum number, l the orbital angular momentum quantum number and ml. The magnetic quantum number.

When n = 1, the wave function of the hydrogen atom is

ψ(r) = ψ100 = √(Z³/ π a0²) *(e-r/ a0)

ψ100 denotes that n =1, l = 0 and ml = 0

a0 = Bohr radius

In quantum mechanics, the idea of orbit is invalid. At any instant the wve function is spread over large distances in space, and wherever ψ≠ 0, the presence of electron may be felt.

The probability of finding the electron in a small volume dV is | ψ(r)| ² dV
We can calculate the probability p(r)dr of finding the electron at a distance between r and r+dr from the nucleus.

In the ground state for hydrogen atom it comes out to be

P(r) = (4/ a0)r²e -2r/ a0

The plot of P(r) versus r shows that P(r) is maximum at r = a0 Which the Bohr’s radius.

But when we put n =2, the maximum probability comes at two radii one near r = a0 and the other at r = 5.4 a0. According to Bohr model all electrons should be at r = 4 a0.

Nomenclature in Atomic Physics

An interesting property of electrons is that each electron has a spin angular momentum. It is characterized by ms and it can take values of +1/2 or -1/2.

Therefore a wave function is described by four characteristics n,l, ml and ms.

Any particular wave function described particular values of the above four characteristics or quantum numbers is termed a quantum state.

For n =1, l = 0 and ml = 0. Hence there will be two quantum states.

For n = 2 there 8 quantum states.

In general there are 2n² quantum states.

The quantum states corresponding to a particular n are together called a major shell.

n =1 shell is called K shell, n = 2 is called L shell and n = 3 shell is called M shell etc.

Pauli’s exclusion principle says that there cannot be more than one electron in any quantum state.

It is customary to use the symbols s,pd,f etc. to denote the value of the orbital angular momentum quantum number l corresponding to the value of l = 0,1,2,3 etc. respectively. These are called subshell for a given shell.

For an atom having many electrons, the quatum states are gradually filled from lower energy to higher energy to form the ground state of the atom.

Protons and neutrons also obey Pauli principle. They also have quantum numbers even though we do not study them in the current syllabus.

Any particle that obeys Pauli exclusion principle is called a fermion.
Electrons, protons, and neutrons are all fermions.

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