2. Equilibrium Thermodynamics
Based on the idealized concept of equilibrium, which can be
attained only in infinite time
Relies on the concept of reversibility
Perfect reversible path is only imaginary
Deals with Isolated or closed systems
All natural systems are open systems
Attainment of equilibrium or reversibility is out of question
in open systems
Time invariant state attained in open systems is steady state
and not state of equilibrium
Incompetent to deal with open systems
3. Non Equilibrium Thermodynamics
• Thermodynamics of open systems
• Owes its origin to Onsanger’s reciprocity relations, also
known as Onsanger’s Thermodynamics. Also have been
given the status of Fourth law of thermodynamics
4. Entropy in Non equilibrium
• According to second law entropy of an Isolated system can
not be reduced (System + Environment)
• For a process to take place in within an Isolated system, if
entropy increases in the forward direction it should
decrease in the backward direction
• A
𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑆
B
•
𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑆
• Backward reaction thus is impossible and forward reaction
is irreversible
5. Entropy in Non equilibrium
• ds = 𝑑 𝑒S + 𝑑𝑖 S =
𝑑𝑄
𝑇
+ 𝑑𝑖 S
• 𝑑 𝑒S: Entropy supplied to the system by it’s environment
• 𝑑𝑄: Heat supplied to the system by it’s environment
• 𝑑𝑖 S: Entropy produced inside the system by irreversible
processes.
• 𝑑 𝑒S may be zero or negative but 𝑑𝑖 S can’t be
• For reversible systems 𝑑𝑖 S=0
• ds =
𝑑𝑄
𝑇
• For ir reversible systems 𝑑𝑖 S> 0
• ds >
𝑑𝑄
𝑇
6. Lawsof Non equilibriumThermodynamics
• Nonequilibrium systems have flow of matter or energy, caused by
certain agencies such as gradient of temperature, gradient of electrical
potential, gradient of concentration. Known as driving forces or forces or
affinities.
• The thermodynamic forces are symbolized by 𝑋𝑖 (i=1,2,3………)
• Irreversible phenomena caused by these forces is known as fluxes, or
flows or current and are symbolized by 𝐽𝑖 (i=1,2,3…….)
• Normally temp. gradient will lead into heat flow and conc. Gradient will
lead into mass flow, but temp. gradient and conc gradient both can cause
mass flow. In general
• 𝐽𝑖 = 𝐽𝑖 (𝑋1 , 𝑋2 … . .)
7. Lawsof Non equilibriumThermodynamics
• If values of 𝑋1 , 𝑋2 … are not very large, using Tayler’s expansion
theorem and neglecting squares and higher order terms
• 𝐽𝑖= 𝐽=1
𝑛 𝜕𝐽 𝑖
𝜕𝑋 𝑗
0 𝑋𝑗+………………
• General linear phenomenological relation can be written as
• 𝐽𝑖= 𝑗=1
𝑛
𝐿𝑖𝑗 𝑋𝑗…………………………..1.
• Where
• 𝐿𝑖𝑗 = [
𝜕𝐽 𝑖
𝜕𝑋 𝑗
]0
8. Phenomenological Equations
• Onsanger extended his concept to include all thermodynamical flows
and forces. For a system having n flows and n forces the
Phenomenological equations can be written as
𝑱 𝟏= 𝑳 𝟏𝟏 𝑿 𝟏 + 𝑳 𝟏𝟐 𝑿 𝟐 +………….+ 𝑳 𝟏𝒏 𝑿 𝒏
𝑱 𝟐= 𝑳 𝟐𝟏 𝑿 𝟏 + 𝑳 𝟐𝟐 𝑿 𝟐 +………….+ 𝑳 𝟐𝒏 𝑿 𝒏
.
.
.
𝑱 𝟏= 𝑳 𝒏𝟏 𝑿 𝟏 + 𝑳 𝒏𝟐 𝑿 𝟐 +………….+ 𝑳 𝒏𝒏 𝑿 𝒏
𝑳 𝟏𝟏, 𝑳 𝟏𝟐 𝒆𝒕𝒄. are called phenomenological coefficients. The evaluation of these
coefficients is one of the major objectives in irreversible thermodynamics. The
phenomenological equations can also be written as
𝑋𝑖= 𝑗=1
𝑛
𝑅𝑖𝑗 𝐽𝑗 i=1,2……….3.
𝑅𝑖𝑗have characteristics of generalized resistances or frictions
𝐿𝑖𝑗 have characteristics of generalized conductance or mobilities
2
9. Phenomenological Coefficients
• Type of phenomenological relation 1 or 2 depends upon the convenience
of analysis or physical reality
• Linear Phenomenological relations hold good for sufficiently slow
processes occuring when the system is not too far from equilibrium.
However still the range of phenomenon covered are vast.
• Coefficients 𝐿𝑖𝑖,𝐿𝑗𝑗, 𝑅𝑖𝑖,𝑅𝑗𝑗 representing normal expected phenomenon
like heat flow induced by temperature gradient are termed as Straight
Coefficient.
• Coefficients like𝐿𝑖𝑗 , 𝑅𝑖𝑗 representing interaction between two different
types of flows e.g. mass flow induced By Temperature gradient are
termed as Cross Coefficients
10. Onsanger’s Treatment
• Determination of coefficients poses difficulty , even in simplest cases
when there are 2 forces and 2 fluxes number of coefficients are 4. 3
forces and 3 fluxes number of coefficients are 9. Similar number of
experiments need to be performed.
• Onsager’s theorem states that the matrix of phenomenological
coefficients is symmeteric provided that a proper choice is made of
forces and fluxes
• 𝐿𝑖𝑗 = 𝐿𝑗𝑖…………4
• 𝑅𝑖𝑗 = 𝑅𝑗𝑖………….5.
• Equation 4 and 5 are Onsanger’s Reciprocal relations
11. • Proper choice of forces and fluxes must obey the equation
• 𝜎 =
𝑑 𝑖 𝑆
𝑑𝑡
= 𝑖
𝑛
𝐽𝑖 𝑋𝑖………….6
• Where 𝜎 is entropy production
• Equation 6 is written using Gibbs equation
• Tds = dE + Pdv- 𝜇𝑖 d𝐶𝑖……..7
• Equation 6 should be obeyed in every aspect. Changing forces and fluxes
must not affect equation 6
• After proper choice of forces and fluxes linear phenomenological
relations can be written and from them phenomenological coefficients
are evaluated
• This requires attainment of steady state (Time invariant)
12. • Steady state is caused by balancing of fluxes while at equilibrium fluxes
cease to exist
• Equilibrium states are characterized by maximization of entropy at
constant energy and volume
• According to prigogine’s theorem of minimum entropy production,
steady state is state of no net flux and is the same as state of minimum
entropy
• Let us consider a simple system with only two flows and two forces,
entropy production can be written as
• 𝜎 = 𝐽1 𝑋1+ 𝐽2 𝑋2……………………8
• 𝐽1= 𝐿11 𝑋1+ 𝐿12 𝑋2……………………9
• 𝐽2= 𝐿21 𝑋1+ 𝐿22 𝑋2……………………10
13. • 𝜎 = 𝐽1 𝑋1+ 𝐽2 𝑋2……………………8
• 𝐽1= 𝐿11 𝑋1+ 𝐿12 𝑋2……………………9
• 𝐽2= 𝐿21 𝑋1+ 𝐿22 𝑋2……………………10
• Substituting values from equation 9 and 10 in equation 8
• 𝜎 = 𝐿11 𝑋1
2
+(𝐿12+𝐿21) 𝑋1 𝑋2+ 𝐿22 𝑋2
2
……………11
• As per Onsanger’s reciprocity relations
• 𝐿21 = 𝐿12
• Equation 11 can be written as
• 𝜎 = 𝐿11 𝑋1
2
+ 2𝐿21 𝑋1 𝑋2+ 𝐿22 𝑋2
2
……………12
• Imposing restrictions that 𝑋1 is constant and 𝑋2 to adjust differentiating 𝜎
with respect to 𝑋2 and keeping 𝑋1 constant we obtain
•
𝜕𝜎
𝜕𝑋2 𝑋1
= 2(𝐿21 𝑋1+ 𝐿22 𝑋2) = 2𝐽2…………………………………13
• According to the conditions for the steady state if 𝑋2 is unrestricted, the
conjugate flow must vanish and equation 13 can be written as
•
𝜕𝜎
𝜕𝑋2 𝑋1
= 0………………………………..14
14. •
𝜕𝜎
𝜕𝑋2 𝑋1
= 0
• Indicates that the entropy production has an extreme value in the
steady state
• Since 𝜎 is positive this extreme must be a minimum and stationary state
of the system is that state in which the entropy production assumes a
minimal value consistent with the restraints imposed.
• It can be referred from equation that 𝐽2 is 0 similarly if X2 is constant and
X1 is allowed to adjust J1 is 0
• If phenomenological equations are written in inverted form, theorem of
minimum entropy production lead to two stationary forms in which X1
and X2 are 0.
