Recap - Terms / Laws, Derivations - State Functions, Exact Differentials, Maxwell Thermodynamic Relations, Joule Thomson coefficient, Inversion Temperature

1. UNIT I – THERMODYNAMICS
OVERVIEW
SP | MSc – I | Jan 09th

2. INTRODUCTION

3. • Thermodynamics implies flow of heat. It deals with energy changes accompanying
all types of physical and chemical processes.
• The primary objective of chemical thermodynamics is the establishment of a
criterion for determination of the feasibility or spontaneity of a given
transformation.In this manner, chemical thermodynamics is typically used to predict
the energy exchanges that occur in the following processes:
a. Chemical relations
b. Phase changes
c. The formation of solutions
• The laws of thermodynamics apply to only to matter in bulk and not to individual
atoms or molecules i.e. only to macroscopic systems.
Recapitulation

4. Some basic terms:
• System - Isolated/Closed/Open
• Surroundings
• Macroscopic properties
• Homogenous & Heterogenous systems
• State of the system
• State variables
• Thermodynamic equilibrium
• Extensive & intensive properties
• Thermodynamic processes –
Adiabatic/Isothermal/Isobaric
• Reversible & Irreversible processes
• Nature of heat and work

5. LAWS OF THERMODYNAMICS:
• Most identities in chemical thermodynamics arise from application of the first and
second laws of thermodynamics, particularly the law of conservation of energy, to
these state functions.
• The three laws of thermodynamics:
a. The energy of the universe is constant.
b. In any spontaneous process, there is always an increase in entropy of the universe.
c. The entropy of a perfect crystal (well ordered) at 0 Kelvin is zero.

6. STATE FUNCTIONS
&
EXACT DIFFERENTIALS

7. • A function which depends upon the initial and final state (equilibrium state) of the
system and not on the way or path used to carry it out is called a state function.
• It describes the equilibrium state of the system and thus also describes the type of
the system.
• Thermodynamic functions like pressure, volume, temperature, energy, etc. are state
functions. The change in values of these quantities do not depend on how the
changes are carried out.
• If χ is any thermodynamic property of a homogenous system of a constant
composition then its value is completely determined by the three thermodynamic
variables like pressure, volume and temperature.
• They are related to each other by state equations. Thus any two of the three
variables can sufficiently explain the properties of the system and can be used to find
out the third variable.
State function

8. Example:
• If the initial state A of the system has coordinates (Pi , Vi , Ti ) possessing internal energy Ei
• Let us assume that work Wa is done on the system to compress it adiabatically to state B,
where the coordinates are (Pf , Vf , Tf ) having energy Ef
• Consider another process, where the initial state A and final state B are the same i.e.
internal energies of initial state Ei and final state Ef respectively are same, but the
compression is not adiabatic. Here the quantity of energy which enters the system as
heat is qb and work done on it is Wb (𝑊
𝑎 ≠ 𝑊𝑏)
• Thus even though (Wa & Wb) and (qa & qb)
are different the total internal energy
remains the same irrespective of the path
followed.
• Therefore, it is a state function.

9. • Based on Euler’s theorem of exactness
• The differential of a certain quantity which is a state function is called exact differential
state function.
• E.g. Volume of the definite amount of any substance depends upon temperature(T)
and pressure (P) then mathematically,
V = f(P, T) … (for a definite amount)
• If temperature and pressure are given, the value of volume is thereupon fixed.
• If temperature and pressure are changed, the volume will changed accordingly.
• Hence T and P are called independent variables and V is called dependent variable.
• dV can be estimated provided the derivatives of the function V w.r.t T and P are known
• Derivatives  rate of change of dependent variables with independent variables.
Exact differentials

10. • Let (
𝜕𝑉
𝜕𝑇
)P  rate of change of volume w.r.t temperature at constant pressure
(
𝜕𝑉
𝜕𝑃
)T  rate of change of volume w.r.t pressure at constant temperature
• The change in volume due to infinitesimal change of temperature (𝜕𝑇) at constant
pressure is given by:
dV = (
𝜕𝑉
𝜕𝑇
)P . dT
• The change in volume due to infinitesimal change of pressure (𝜕𝑃) at constant
temperature is given by:
dV = (
𝜕𝑉
𝜕𝑃
)T . dP
• If both the temperature and pressure are simultaneously then the total change in volume is
given by:
dV = (
𝜕𝑉
𝜕𝑇
)P . dT + (
𝜕𝑉
𝜕𝑃
)T . dP
• This is called total differential of the volume V .
Derivation

11. • The individual terms are called partial differentials.
• Mathematically the condition for exactness is given by:
ⅆ
ⅆ𝑃
[(
𝜕𝑉
𝜕𝑇
)P ] T =
ⅆ
ⅆ𝑇
[(
𝜕𝑉
𝜕𝑃
)T ] P
thus
ⅆ2𝑉
ⅆ𝑃.ⅆ𝑇
=
ⅆ2𝑉
ⅆ𝑇.ⅆ𝑃
… (1)
• This relation is applicable for any state function φ
• Any other path function γ for equation (1) is not valid and dγ is then called inexact
differential.

12. MAXWELL
THERMODYNAMIC RELATIONS

13. • Maxwell's relations are a set of equations in thermodynamics which are derivable from the
symmetry of second derivatives and from the definitions of the thermodynamic potentials.
• The structure of Maxwell relations is a statement of equality among the second derivatives
for continuous functions.
• The four most common Maxwell relations are the equalities of the second derivatives of
each of the four thermodynamic potentials, with respect to their thermal natural variable
(temperature T, or entropy S) and their mechanical natural variable (pressure P, or volume V):

14. To prove the equation:
𝜕𝑃
𝜕𝑇 𝑉
=
𝜕𝑆
𝜕𝑉 𝑇
• By definition of Helmholtz free energy,
A = E – TS … (1)
• On differentiating this equation we get,
dA = dE – TdS – SdT … (2)
But w.k.t,
dE = dq – dW
thus dq = TdS and dW = PdV
thus dE = TdS – PdV … (3)
Substitute equation (3) in equation (2)
dA = TdS – PdV – TdS – SdT
dA = – PdV – SdT … (4)
• This indicates that ‘A’ i.e. Helmholtz free energy is the function of temperature and volume
Derivation

15. • Hence, total differential of A can be expressed as:
ⅆ𝐴 =
𝜕𝐴
𝜕𝑇 𝑉
ⅆ𝑇 +
𝜕𝐴
𝜕𝑉 𝑇
ⅆ𝑉 … (5)
• On comparing equation (4) and (5) we get
𝜕𝐴
𝜕𝑇 𝑉
= −𝑆 … (6)
𝜕𝐴
𝜕𝑉 𝑇
= −𝑃 … (7)
• On differentiating equation (6) w.r.t V at constant T and equation (7) w.r.t T at constant V,
ⅆ
ⅆ𝑉
[
𝜕𝐴
𝜕𝑇 𝑉
]T = −
𝜕𝑆
𝜕𝑉 𝑇
ⅆ
ⅆ𝑇
[
𝜕𝐴
𝜕𝑉 𝑇
]V = −
𝜕𝑃
𝜕𝑉 𝑇
… (8)
• and squaring them (since A is an exact differential function)
𝜕2𝐴
𝜕𝑉⋅𝜕𝑇
=
𝜕2𝐴
𝜕𝑉⋅𝜕𝑇

16. • Applying this condition on equation (8) we get,
𝜕𝑃
𝜕𝑇 𝑉
=
𝜕𝑆
𝜕𝑉 𝑇
• Used to establish relationships between thermodynamic parameters such as entropy,
number of ions, temperature, pressure, volume, etc.
• These relationships enormously simplify thermodynamic analysis.
• Used in deducing free expansion, adiabatic as well as isothermal compression
equations.
Applications
:

17. JOULE – THOMSON EFFECT

18. • Joule - Thomson describes the temperature change of a real gas or liquid when it is
forced through a valve or porous plug while keeping it insulated so that no heat is
exchanged with the environment. This procedure is called Joule–Thomson process.
• At room temperature, all gases except hydrogen, helium, and neon cool upon expansion
by the Joule–Thomson process when being throttled through an orifice; these three
gases experience the same effect but only at lower temperatures.
• The equation for J-T effect is given by:
𝜇J-T = −
1
Cp
[
𝜕𝐸
𝜕𝑉 T
𝜕𝑉
𝜕𝑃 𝑇
+
𝜕𝑃𝑉
𝜕𝑃 𝑇
]
or
𝝁J-T = −
𝟏
Cp
𝟐𝒂
𝑹𝑻
− 𝒃
This is a general equation which can be
applied to any gas.

19. Joule – Thomson coefficient
• Joule – Thomson coefficient is defined as the change in temperature with pressure at
constant enthalpy.
𝜇J-T =
𝜕𝑇
𝜕𝑃 𝐻
≈
Δ𝑇
Δ𝑃 𝐻
• Nature of the J-T coefficient
i. A positive value of 𝜇J-T corresponds to cooling. Since ΔP is negative in Joule – Thomson
experiment, to make it positive ΔT should also be a negative quantity i.e. temperature
should drop.
ii. A negative value of 𝜇J-T corresponds to warming on expansion as ΔP is negative and ΔT
would be positive i.e. temperature should increase.
iii. A zero value of 𝜇J-T corresponds to no temperature change in expansion. The
temperature at which neither cooling nor heating is observed is known as Inversion
temperature.

20. • Joule – Thomson effect has major applications in air conditioning and refrigeration.
• It is used in petrochemical industries.
• Cooling effect is used to liquefy gases.
• Only when the Joule – Thomson coefficient of a given gas at a particular temperature is
greater than zero, can the gas be used for cooling (Linde’s process).
Applications

21. INVERSION TEMPERATURE

22. • If the stream of the gas at high pressure is allowed to expand by passing through a
porous plug into vacuum, a region of low pressure under adiabatic condition, it gets cool
appreciably.
• Hydrogen and Helium are exceptions as they get warm from earth under similar
circumstances. But at very low temperatures (below ~ 80 oC for hydrogen and below
~ 240 oC for helium), these gases show cooling behaviour.
• The temperature below which a gas becomes cooling is known as Inversion temperature.
• The higher temperature at which inversion of Joule –
Thomson coefficient takes place is called upper
inversion temperature.
• The lower temperature at which inversion of Joule –
Thomson coefficient takes place is called lower
inversion temperature.
• Given by the formula: 𝑇𝑖 =
2𝑎
𝑅𝑏

23. REFERENCES
• Puri, Sharma, Pathania – Physical Chemistry
• Atkins - 07th edition
• Davis – Introduction to Chemical Thermodynamics

24. THANK YOU