Entropy

1. Entropy
Degree of disorder

2. Introduction
• Entropy, the measure of a system's thermal energy per unit temperature that is unavailable for doing useful
work.
• Entropy is a measure of molecular disorder, or molecular randomness. As a system becomes more
disordered, the positions of the molecules become less predictable and the entropy increases.
• The entropy of a substance is lowest in the solid phase and highest in the gas phase. Unlike energy, entropy
is a no conserved property, and there is no such thing as a conservation of entropy principle.
• Entropy traces out its origin –molecular movement interpretation-Rudolf Clausius in 1850
• It can be visualized due to the process of expansion, heating, mixing and reaction.
• Entropy is associated with heat and temperature.
• Disorder can be pointed out in three different types. They are:
 Positional disorder-whether the atoms are free to move or not
 Vibrational disorder(thermal disorder) -whether the atoms vibrate about an average position
 Configurational disorder -this refers to the distribution of different atoms or sites in lattice.
 It is a Greek word which means transformation and denoted by letter S

3. Introduction --cont
• According to the second law of thermodynamics, the entropy of a system can only decrease if the entropy of another
system increases.
• It is easier to define the change in entropy, as it is dependent on the amount of heat transfer in a process. It is the ratio of
the amount of heat transfer during a process to its absolute temperature.
• Consider a reversible process in which the amount of heat transfer in a small segment in the process is dQ and the
absolute temperature is T, then the change in entropy in the small segment is given by
• 𝛿𝑆 =
𝛿𝑄
𝑇
• Between two states in a process
• 𝛿𝑆 =
𝛿𝑄
𝑇

4. Properties of Entropy
• It is a thermodynamic function.
• It is a state function. It depends on the state of the system and not the path that is followed.
• It is represented by S but in the standard state, it is represented by S°.
• Its SI unit The unit for the entropy is kJ/K and the specific entropy is given in kJ/kg K.
• Entropy is an extensive property which means that it scales with the size or extent of a system.
• It increases when heat is supplied irrespective of the fact whether temperature changes or not.
• It decrease when heat is removed whether temperature changes or not.
• It remains unchanged in all adiabatic frictionless processes.
• It increases if temperature of heat is lowered without work being done as in a throttling process.

5. CLAUSIUS THEOREM ,INEQUALITY and Entropy
• The Clausius theorem (1855) states that for a thermodynamic system (e.g. heat engine or heat pump) exchanging heat with
external reservoirs and undergoing a thermodynamic cycle,
𝛿𝑄
𝑇𝑠𝑢𝑟𝑟
≤ 0
• with a little history thrown in.
• Carnot proposed that heat must always be wasted in order for a heat engine to produce net work, but he did not quantify how
much heat had to be wasted.
• in ~ 1850 Rudolf Clausius confirmed the existing theories:
• Energy is conserved (1st law) was quantified as 𝑄 – 𝑊 = 𝛥𝑈
• Heat flows naturally from hot to cold (not quantified) and added an important contribution:
• Clausius knew that some heat had to be rejected from a reversible (Carnot) heat engine (QL), as Carnot proposed, and he knew
that
• 𝑄𝐿 = 𝑊𝑛𝑒𝑡 – 𝑄𝐻 (1ST LAW)
• but no principle fixed the absolute amount of rejected heat.
• Clausius then observed that for reversible heat engines, the ratio of the heat input to the rejected heat was consistently equal to
the ratio of the absolute temperatures of the high and low temperature reservoirs:
•
𝑄𝐻
𝑄𝐿
=
𝑇𝐻
𝑇𝐿
(1)
• If and only if the temperature of the high (TH) and low (TL) temperature reservoirs are always expressed in degrees Kelvin.

6. CLAUSIUS THEOREM ,INEQUALITY and Entropy
• Rearranging (1) for a reversible heat engine:
𝑄𝐻
𝑇𝐻
=
𝑄𝐿
𝑇𝐿
(2)
• Clausius then examined irreversible processes and found that the relation (2) did not hold. he reasoned, for
example, if 10 Joules 0f heat flow from a hot object at 350K into a cool room at 300K, then the heat transfer
term for heat leaving the object and heat transferred into the room is the same and
𝑄
𝑇𝐻
<
𝑄
𝑇𝐿
OR
10
350
<
10
300
(3)
• That is, for irreversible processes, the ratio of heat over absolute temperature increases in the direction of
natural heat flow.
• Combining (2) and (3) for a heat engine with one heat input and one heat output
𝑄𝐻
𝑇𝐻
−
𝑄𝐿
𝑇𝐿
≤ 0 (4)
• OR for any sequence of processes with discrete heat transfer terms
𝐾
𝑄
𝑇 𝐾
≤ 0 (5)

7. CLAUSIUS THEOREM ,INEQUALITY and Entropy
• WHERE
𝑄
𝑇 𝐾
=the ratio of the amount of heat transferred in a process to the
temperature of the surroundings where heat is transferred
• Calculating the ratio as the integral of a continuous function of
𝛿𝑄
𝑇
, clausius’
principle for a reversible or irreversible cycle is:
𝛿𝑄
𝑇
≤ 0 (6)
• FOR REVERSIBLE CYCLES:
•
•
𝛿𝑄
𝑇
= 0 AND 𝐾
𝑄
𝑇 𝐾
= 0 (7)
• FOR IRREVERSIBLE CYCLES
•
𝛿𝑄
𝑇
< 0 AND 𝐾
𝑄
𝑇 𝐾
< 0

8. CLAUSIUS THEOREM ,INEQUALITY and Entropy
• change in a cycle, but it is also associated with heat transfer (a path
function). In a paper published in 1865, “On various forms of the laws of
thermodynamics that are convenient for applications,” Clausius named the
heat: temperature ratio entropy, from the Greek word for “transformation”
with the symbol for entropy, the letter “S.” Clausius’ 1865 paper ends with
a bold statement that is the broadest possible application of the laws of
thermodynamics:
• The energy of the universe is constant (1st Law)
• The entropy of the universe tends toward a maximum (2nd Law)
• Clausius’ statement of the 2nd Law of Thermodynamics and his discovery
of entropy as the ratio of two macroscopic components: heat and absolute
temperature, was truly remarkable since the true nature of entropy was
only discovered later when physicists understood the nature of individual
atoms and molecules – the microscopic world.

9. Statements of the Second Law
• Since scientists were working independently, they came up with many theories that were only later shown to be similar. Hence, many
corollaries of the second law of thermodynamics exist. This section lists various statements of the second law.
• Corollary 1: (Carnot principle or Carnot theorem) No engine operating between two given reservoirs can have a greater efficiency
than a reversible engine operating between the same two reservoirs.
• Corollary 2: (Carnot corollary) All reversible engines operating between the same temperature limits have the same efficiency.
• Corollary 3: The efficiency of any reversible engine operating between two reservoirs is independent of the nature of the working
fluid and depends only on
• the temperature of the reservoirs, for any reversible process undergone by the system between states 1 and 2. This property is called
the entropy.
• Corollary 4: It is theoretically impossible to reduce the temperature of a system to absolute zero by a series of finite processes.

10. Statements of the Second Law
• Corollary 5: (The thermodynamic temperature scale) Define the ratio of two temperatures as the ratio of the heat
absorbed by a Carnot engine to the heat rejected when the engine is operated between reservoirs at these temperatures.
• Thus, the equality QL/QH = TL/TH becomes a matter of definition, and the fundamental problem of thermometry, that of
establishing a temperature scale, reduces to a problem in calorimetry.
• Corollary 6: (The inequality of clausius) When a system is carried around a cycle and the heat δQ added to it at every point
is divided by its temperature, the sum of all such quotients is less than zero for irreversible cycles and in the limit is equal
to zero for reversible cycles:
𝛿𝑄
𝑇
≤ 0
• C orollary 7: There exists a property (denoted by s) of a system such that a change in its value is equal to 𝑆1 − 𝑆2 = 1
2 𝜕𝑄
𝑇
for any reversible process undergone by the system between states 1 and 2. This property is called the entropy.
• Corollary 8: (Principle of the increase of entropy) In any process whatever between two equilibrium states of a system, the
increase in entropy of the system plus the increase in entropy of its surroundings is equal to or greater than zero.