It will shows the definition of specific heat capacity, quantification techniques and application areas.

1. Specific Heat Capacity
A. S. M. Jannatul Islam
1
Department of Electrical and Electronic Engineering
Khulna University of Engineering & Technology
Khulna-9203

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Contents
☼ Specific heat capacity
☼ Graphical representation
☼ Quantification theories
☼ Dulong Petit’s Law
☼ Einstein Theory
☼ Debye Theory
☼ Summary
☼ Specific heat capacity applications

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Specific heat capacity
The amount of heat energy needed to increase the
temperature of 1 Kg of a substance by 1 K is called
the specific heat capacity.
It is expressed in
Joules per Kg per
Kelvin.
Heat Added
1 Kg
1 K
1ºC
Q

4. 4
The specific heat of a substance can also be defined as the
change in internal energy E with respect to temperature T. Thus,
If the substance is a gas then it is important to specify whether
the gas is being held at constant volume or constant pressure.
For solids the difference is negligible.
The internal Energy, E of a solid is the combination of the total
kinetic energy (due to motion) and potential energy (due to
intermolecular forces) of the molecules in the solid.
Specific heat capacity
C = (∂E/∂T)

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Specific heat capacity

6. 6
Specific heat capacity
Experimental Curves for Specific heat capacity of
solids

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Dulong
and
Petit’s
Law
Einstein’s
Theory
Debye’s
Theory
Quantification Theories
Specific heat capacity can be quantified by the following
Laws and Theories.
Each of which has its own advantages and limitations.

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 It states that solids have a constant heat capacity. It was explained
by considering every atoms as independent classical oscillator
vibrating independently of each other.
 The average energy of an atom per degree of freedom is kT. So for
N oscillators/atoms with 3 degree of freedom, the energy is
E=3NkT.
Dulong and Petit’s Law
Where k is the Boltzmann constant and T is the absolute
Temperature
𝑪𝑽 =
𝝏𝑬
𝝏𝑻 𝑽
= 𝟑𝑵𝒌

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Dulong and Petit’s Law
 It is a good approximation for the measured values for solids at room
temperatures or high temperature.
 At low temperatures the Dulong and Petit value is not a good
approximation
Dulong and Petit’s Law Curve

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Motivation to Other’s Theory
The shape of the curve for T near
zero is thus of interest. It appears
to be proportional to a power of
T, say T² or T³.
With decrease in temperature,
heat capacity also decreases and is
zero at 0 K

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Einstein’s Theory
 In this model, solid is considered as the collection of individual
atoms/oscillators vibrating independently of each other. But
instead of classical oscillator, Einstein considered quantum
oscillators so that they can have only discrete energy values.
 Einstein treated the atoms in a crystal as N simple harmonic
oscillators, all having the same frequency 𝝑. The frequency 𝝑
depends on the strength of the restoring force acting on the atom,
i.e. the strength of the chemical bonds within the solid.
 Since the equation of motion for each atom decomposes into three
independent equations for the x, y and z components of
displacement, and N atom solid is equivalent to 3N harmonic
oscillators, each vibrating independently at frequency 𝝑.

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The heat capacity at constant volume is therefore
𝑪𝑽 =
𝝏𝑬
𝝏𝑻 𝑽
= 𝟑𝑵𝒌
𝒉𝝑
𝒌𝑻
𝟐
𝐞𝐱𝐩(
𝒉𝝑
𝒌𝑻
)
𝐞𝐱𝐩
𝒉𝝑
𝒌𝑻
− 𝟏
𝟐
Considering 𝑻𝒆 =
𝒉𝝑
𝒌
𝑪𝑽 = 𝟑𝑵𝒌
𝑻𝒆
𝑻
𝟐 𝒆𝒙𝒑(
𝑻𝒆
𝑻
)
[𝒆𝒙𝒑
𝑻𝒆
𝑻
− 𝟏]𝟐
Where k is the Boltzmann constant, T is the absolute temperature, 𝝑 is the
vibrating frequency of atoms, h is the plank’s constant and 𝑻𝒆 is the Einstein
Temperature (which is different for each solid, and reflect the rigidity of
lattice)
Einstein’s Theory

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However, the values of specific heat using this model do not
match with the experimental values at very low
temperatures.
 At the high temperature limit, when 𝑻 >> 𝑻𝒆 (and
𝒉𝝑
𝒌𝑻
<<1 ), the
Einstein heat capacity reduces to 𝑪𝑽=3Nk, the Dulong and petit law.
 At the low temperature limit, when T<<𝑻𝒆(and
𝒉𝝑
𝒌𝑻
≫1 ), 𝑪𝑽→0 as T
→0, as required by the third law of thermodynamics.
Einstein’s Theory

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However, the values of specific heat using this model do not
match with the experimental values at very low
temperatures.
Einstein’s Theory

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Debye’s Theory
 Debye improved on Einstein’s theory by treating the coupled
vibrations of the solid in terms of 3N normal modes of vibration of
the whole system, each with its own frequency.
 The lattice vibrations are therefore equivalent to 3N independent
harmonic oscillators with these normal mode frequencies.
 The collective vibrational modes can accept energy only in
discrete amounts (quanta of energy is called phonon).
 Hence there is a continuous range of frequencies that cuts off at a
maximum frequency 𝝑𝑫, which is the characteristic of a particular
solid.
 The normal modes are the frequencies of the standing waves that
are possible in the medium.

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According to Debye Theory, the expression for specific heat is
𝑪𝑽 = 𝟗𝑵𝒌
𝑻
𝑻𝑫
𝟑
𝟎
𝑻𝑫
𝑻 𝒙𝟒𝒆𝒙
(𝒆𝒙 − 𝟏)𝟐
𝒅𝒚
If 𝑻 ≪ 𝑻𝑫then
𝑪𝑽 =
𝟏𝟐𝝅𝟒𝑵𝒌
𝟓
𝑻
𝑻𝑫
𝟑
Where 𝑻𝑫 =
𝒉𝝑𝑫
𝒌
, 𝒙 =
𝒉𝝑
𝒌𝑻
𝒂𝒏𝒅 𝒙𝑫 =
𝒉𝝑𝑫
𝒌𝑻
At low temperature limit,
Debye Model can show accurate values of specific
heat capacity for solids.
Debye’s Theory

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The key difference between Debye and Einstein model is that the Debye
model treats vibrations of the atomic lattice as phonons in a box whereas
Einstein model treats solids as many individual, non-interacting quantum
harmonic oscillators.
Summary

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Applications
 Substances having a small specific heat capacity can be quickly
heated up, it also experience a big change in temperature even
though only small amount of heat is supplied.
 Substances having a small specific heat capacity, are very useful as
material in cooking instruments such as frying pans, pots, kettles
and so on, because, they can be quickly heated up even when small
amount oh heat is supplied.
 Sensitive thermometers also must be made from materials with
small specific heat capacity so that it can detect and show a change
of temperature rapidly and accurately.

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 Substances that have a high specific heat capacity is suitable as a
material for constructing kettle handlers, insulators and oven
covers, because, a high amount of heat will cause only a small
change in temperature aka the material won't get hot too fast!
 Heat storage instruments are very useful and they are usually
made of substances with a high specific heat capacity.
 Water as a cooling agent acts excellent as a cooling agent in
engines. Water is also used in houses in cold climate countries
because as it is heated up (boiled) it tends to retain heat and warm
the house due to its high specific heat capacity.
Applications

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