Newton's laws of motion proved to be a boon to the Physicists in the beginning of 19th Classical mechanics Century.
The branch of science that is based on Newton's laws of motion is called as Classical mechanics.
This laws were found to explain the properties of All the macroscopic particles moving with speeds much less than the speed of light. However, when applied to microscopic particles like electron moving with velocities of the order of that of the light, these laws failed to explain the behavior.
Classical mechanics can be applied to only macroscopic particles as the position as well as momentum can be predicted accurately for these particles. But this is not possible in case of microscopic particles. According to Heisenberg's Uncertainty Principle, it is impossible to determine the position as well as velocity of a microscopic particle accurately.
Classical mechanics assumes that the energy is absorbed or emitted in a continuous manner. But according to Planck's theory, the transition of energy is in the form of packets called as quanta.
Four important phenomenon which could not be explained on the basis of classical mechanics are discussed in details.
1. Black body radiation
When light falls on any body, part of the light is absorbed and part is reflected. However, the body which absorbs the entire radiations incident on it is called as perfectly black body.
Perfectly black body is not a good absorber of radiation but is also a good emitter of radiations. A good approximation to black body is a hollow sphere blackened on the inside with lamp black. It has a small aperture one side and small hump on the opposite wall so that the light should get reflected number of times inside the body.
Stefan studied the radiations from black body and found that the amount of energy radiated by a perfectly black body per unit area per unit time is directly proportional to fourth power of absolute temperature.
That is, E α T⁴ or E= σT⁴
Where,σ is called as Stefan's constant and the law is called as Stefan-Boltzmann's fourth power law. This energy emitted by a black body does not consist of a single wavelength but it is uniformly distributed along a spectrum. At different temperatures, the spectrum obtained is shown as in Figure.
Important features of the spectrum are summarized below-
At a particular temperature, the energy corresponding to all the wavelengths is not equal.
For each temperature, the maximum energy corresponds to a specific wavelength. This wavelength is denoted by λm and the enerɡy Em.
The area under the curve at any temperature gives the total amount of energy emitted per unit area per second. This is in accordance with Stefan's law.
With increase in temperature, the maxima becomes sharper and shifts towards lower value of wavelength and also the area under curve increases. It means that, with increase in temperature, the total energy emitted increases and λm decreases.
The wavelength corresponding to maxima is inversely proportional to absolute temperature.
That is, λm α 1 ∴ λm T= Constant
T
This is called as Wien's displacement law. This law explains why the colour of hot body changes on heating.
In order to explain the black body radiation, Rayleigh and Jeans used classical mechanics.
According to Wein's radiation law,
Eλ = 8πhc e −hc / kTλ ------------(1)
λ⁵
According to Rayleigh-Jeans law, the equation is
Eλ = 8π kT -----------(2)
λ⁴
However, both of these equations failed to explain the entire spectrum of black body radiation. Wein's law is obeyed at only shorter wavelength while Rayleigh-Jean's law is obeyed at only longer wavelengths.
Another important limitation of the classical mechanics lies in the failure to explain Ultraviolet Catastrophe. It has been observed that, large amount of energy is emitted by black body in high frequency region (lower wavelength). This is called as Ultraviolet Catastrophe. It can not be explained on the basis of classical mechanics.
It was accurately explained by Max Planck on the basis of quantum theory. According to this theory-
Energy is emitted or absorbed in the form of small packets called as quanta.
Energy of each quantum is given by hʋ.
Thus, the total energy emitted or absorbed is integral multiple of hʋ.
Based on these concepts, Planck deduced expression for the energy radiated by a black body at wavelength 'λ' as-
Eλ = 8πhc 1 ------------(3)
λ⁵ hc / kTλ
e – 1
This expression is called as Planck's Radiation law.
At shorter wavelength:
hc / kTλ
When λ is small,e <<1. Hence, equation (3) becomes,
Eλ = 8πhc e −hc / kTλ ------------(4)
λ⁵
This is Wein's law. It is obeyed at shorter wavelengths.
At longer wavelength:
When λ is larger, the binomial expansion of
hc / kTλ
e can be written as-
hc / kTλ
e =1+ hc . + _ _ _ _ _
kTλ
Putting in eqn.(3) we get,
Eλ = 8πhc [1/(1+hc / kTλ +_ _ _)-1]
λ⁵
Neglecting the higher terms we get,
Eλ = 8πhc [1/(hc / kTλ )]=8πhc [ kTλ/hc]=8πkT
λ⁵ λ ⁵ λ⁴
-----------(5)
This is Rayleigh-Jean's law. It is obeyed at longer wavelengths.
In short, Plank's quantum theory can explain the phenomenon of black body radiation over entire spectrum of wavelength.
2. Photoelectric effect
When a beam of light with frequency equal to or greater than and a particular value (threshold frequency) is allowed to strike the surface of metal ,electron are ejected instantaneously from the surface of metal. This phenomenon is called as photoelectric effect and ejected electrons are called as photoelectrons.
some characteristic of photoelectric effects are given below
the electrons are ejected only if the frequency of incident light is equal to or greater than minimum value called as threshold frequency (ν₀).
The electrons are ejected instantaneously there is no time lag between incidence of light and ejection of electron.
Kinetic energy of ejected electrons depend upon frequency of incident light.
The number of ejected electrons depends upon the intensity of incident radiation.
According to classical mechanics, the kinetic energy of ejected electron depends upon the intensity and not on the frequency of light. Hence if light of sufficient high intensity is allowed to incident on the metal, electrons show have limited irrespective of the frequency of light. But this is not the case. Hence classical mechanics fails to explain these properties of photoelectric effect.
According to Plank's quantum theory, is quantum of light called photon has energy equal to hv. When the photon hits the metal surface its transfer its energy to the electron. Energy equal to threshold value (hvo) is used for ejection of electron and remaining energy is converted to kinetic energy.
hv=hvo +½ mv²
Where, hvo=ø is called as work function,
K.E=hv-hvo
This equation shows that kinetic energy of the ejected electron varies linearly with frequency of incident radiation.
Thus an increase in frequency increases the kinetic energy and an increase in intensity increases the number of photoelectrons.
3. Heat capacity of solids
The amount of heat required to raise the temperature of 1g of a solid through 1°C is called as heat capacity of that solid. Dulong and Petit found that for all the monoatomic solids, the molar heat capacity is same with value of about 25 JK ̄ ¹ mol ̄ ¹.
According to classical mechanics, every molecule can be considered as an oscillator with vibrational energy 3kT (k=Boltzmann constant).
Thus, total vibrational energy of one mole of oscillators (NA atoms) is given by-
E=N A (3kT)=3RT ∴ (δE/δT)ʋ =3R
From the definition of molar heat capacity at constant volume,
(δE/δT)ʋ =Cʋ
∴ Cʋ =3R= 3×8.314=24.9 JK ̄ ¹ mol ̄ ¹
This shows that classical mechanics can very well explain the observed value of heat capacity of solids. However, at low temperature when T→0, the heat capacity is lower than 3R. That is Cʋ<3R. This observation can be explained on the basis of quantum theory. Einstein applied quantum mechanics to the oscillators and suggested that the energy of all oscillators is not same, but they have energy values integral multiples of hʋ₀. The mean enerɡy of oscillator is ɡiven by-
Ē = 3hʋ₀ . ------------(1)
hʋ₀ / kT
e −1
For one mole of solid, Ē= N A Ē=3NA hʋ₀ ------(2)
hʋ₀ / kT
e −1
hʋ₀ / kT
∴ Cʋ =(δE/δT)ʋ =3NA k(hʋ₀ / kT)² e . ----(3)
hʋ₀ / kT
( e −1)²
hʋ₀ / kT
At low temperature,hʋ₀ / kT>>1 or e >>1 and hence
hʋ₀ / kT
Cʋ=3NA k(hʋ₀ / kT)² e .
hʋ₀ / kT
( e )²
- hʋ₀ / k
=3NA k(hʋ₀ / kT)² e -----------(4)
This expression shows that the value of heat capacity depends on temperature.
At hiɡh temperature,hʋ₀ / kT<<1, hence we can
hʋ₀ / kT
apply binomial expansion to e
Cʋ=3NA k(hʋ₀ / kT)² (1+ (hʋ₀ / kT)+ ----------) .
{(1+(hʋ₀ / kT)+ ------) – 1}²
=3NA k (hʋ₀ / kT)² + (hʋ₀ / kT)³
{(hʋ₀ / kT)+ -----}²
Neglecting higher powers of hʋ₀/ kT as T is very high, we get,
Cʋ=3NA k (hʋ₀ / kT)² =3NA k=3R ------------(5)
(hʋ₀ / kT)²
This is the value obtained from classical mechanics.
Thus, quantum mechanics can explain behavior of solids at low as well as high temperatures.
4. Atomic spectra of hydrogen
It is excellent evidence in favour of quantization of energy. When the emission spectrum of hydrogen was observed,it was found to be a line spectra with a large number of lines at different wavelengths (or wave numbers). It means that the emission of energy from a hydrogen atom is not taking place in a continuous manner as expected by classical mechanics. The line spectra is due to de-excitation of electron from various energy levels.
It is expected that the electron in hydrogen atom should be present in the ground state all the time. However, the number of hydrogen atoms taken as sample is so large that some of them contain electrons in various excited levels. These electrons come to ground state by emitting energy equal to the difference between energies of the two states. Since the energy is quantized, we get discontinuous line spectra instead of continuous band spectra.
Consider an electron jumping from higher energy level with principal quantum no. n₂ to a lower energy level with principal quantum number n₁. Let the energies of the two levels are E₂and E₁ respectively. The energy emitted due to electronic transition is given by -
∆E=E₂ – E₁
∴∆E=( _ 1 2π² me⁴) _ ( _ 1 2π² me⁴ )
(4πɛ₀)² n₂² h² (4πɛ₀)² n₁² h²
∴ hc = 1 2π² me⁴ (1 _ 1 )
λ (4πɛ₀)² h² (n₁² n₂²)
∴ 1 = 1 2π² me⁴ (1 _ 1 )
λ (4πɛ₀)² h³ (n₁² n₂²)
∴ v̄= Rʜ (1 _ 1 )
(n₁² n₂²)
Where,Rʜ is called as Rydberg's constant having value 109677 cm ̄ ¹ and v̄ is called as wave number in cm ̄ ¹
There are six series of lines obtained in the atomic spectra of hydrogen. They are-
Lyman series: This series is obtained when electron jumps from higher energy level to level n=1. It lies in the ultraviolet region of the spectrum. Hence, for Lyman series-
∴ v̄= Rʜ (1 _ 1 ) Where,n₂=2,3,4-----
(1² n₂²)
Balmer series: This series is obtained when electron jumps from higher energy level to level n=2. It lies in the visible region of the spectrum. Hence, for Balmer series-
∴ v̄= Rʜ (1 _ 1 ) Where,n₂=3,4,5-----
(2² n₂²)
Paschen series: This series is obtained when electron jumps from higher energy level to level n=3. It lies in the IR region of the spectrum. Hence, for Paschen series-
∴ v̄= Rʜ (1 _ 1 ) Where,n₂=4,5,6-----
(3² n₂²)
Brackett series: The series is obtained when electron jumps from higher energy level to level n=4. It lies in the IR region of the spectrum. Hence, for Brackett series-
∴ v̄= Rʜ (1 _ 1 ) Where,n₂=5,6,7-----
(4² n₂²)
Pfund series: The series is obtained when electron jumps from higher energy level to level n=5. It lies in the IR region of the spectrum. Hence, for Pfund series-
∴ v̄= Rʜ (1 _ 1 ) Where,n₂=6,7,8-----
(5² n₂²)
Humphrey series: This series is obtained when electron jumps from higher energy level to level n=6. It lies in the IR region of the spectrum. Hence, for Humphrey series-
∴ v̄= Rʜ (1 _ 1 ) Where,n₂=7,8,9-----
(6² n₂²)
Very short answer questions on classical mechanics
Explain black body?
Answer: when light falls on anybody apart of light absorbed and path is reflected.However, the body which absorb the entire radiation incident on it is called as perfectly black body .
What is Classical mechanics?
Answer: The branch of science that is based on Newton's laws of motion is called as Classical mechanics.
Define photoelectric effect?
Answer: When a beam of light of sufficiently high frequency is allowed to strike a metal surface, electrons are ejected from the metal surface. This phenomenon is called as photoelectric effect.
What is photoelectrons?
Answer: When a beam of light of sufficiently high frequency is allowed to strike a metal surface, electrons are ejected from the metal surface. This phenomenon is called as photoelectric effect. The ejected electrons are called as photoelectrons.
Define 'quanta'.
Answer: According to Planck's theory, the transition of energy is in the form of packets called as quanta.
Write any two important properties of photoelectric effect.
Answer: 1] The electrons are ejected only if the frequency of incident light is equal to or greater than minimum value called as threshold frequency (ν0).
2] The electrons are ejected instantaneously there is no time lag between incidence of light and ejection of electron.
What is heat capacity of solids?
Answer: The amount of heat required to raise the temperature of 1g of a solid through 1°C is called as heat capacity of that solid.
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