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THERMODYNAMICS
1
THERMODYNAMICS CONTENTS
1. Introduction
2. Maxwell’s thermodynamic equations
2.1 Derivation of Maxwell’s equations
3. Function and derivative
3.1 Differentiation
Partial Differentiation
4. Cyclic Rule
5. State Function and its characteristics
6. Thermodynamic co-efficients
7. Reversible PV work
8. Reversible and Irreversible Process
9. Heat capacity
10. The First Law of thermodynamics
11. Enthalpy
12. Thermodynamic equations of state
12.1 First Thermodynamic equation of state
12.2 Second Thermodynamic equation of state
12.3 Some Important Relations
13. Reversible isothermal process for ideal gas
14. Irreversible isothermal expansion of gas
15. Reversible adiabatic process for ideal gas
15.1 Work done on reversible expansion of an ideal gas
16. Work done on irreversible expansion of an ideal gas
17. Comparison Between the final volume and final Pressure of
reversible isothermal and adiabatic process
18. Joule Thomson Experiment
18.1 Calculation of Joule Thomson coefficient for ideal gas
18.2 Calculation of Joule Thomson coefficient for real gas
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18.3 Concept of inversion temperature
18.3.1 The Case for gas cooling
18.3.2 The Case for gas heating
18.3.3 The Case where gas neither cools or heat
18.4 Relation between iT and
19. Carnot Cycle
19.1 Characteristics of Carnot Cycle
19.2 Processes in Carnot Engine
20. Concept of Refrigerators
21. Entropy
21.1 Entropy change of an ideal gas for a reversible process
21.2 Entropy change in Mixing of Solids
21.3 Entropy change in Mixing of ideal Gases
22. Phase Transformation
22.1 Reversible phase transformation
22.2 Irreversible Phase Transformation
23. Phase Diagram
23.1 One-Component Systems
23.2 Two-Component Systems
23.3 Three-Component Systems
24. Activity and Activity Coefficient
24.1 Activity
24.2 Activity Coefficient
25. Debye-HückelTheory
26. Clausius-Clapeyron Equation
27. Third law of thermodynamics
28. The Kinetic theory of gases
28.1 Derivation of Kinetic gas equation
28.2 Kinetic Energy of 1 mole of gas
28.3 Kinetic energy for 1 molecule
29. Deduction of various gas laws from kinetic gas equation
29.1 Boyle’s law
29.2 Charle’s law
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29.3 Avogadro’s law
29.4 Graham’s law of diffusion
30. Maxwell’s distribution of molecules kinetic Energies
30.1 Types of Molecular velocities
30.1.1 Most probable speeds
30.1.2 Average Speed
30.1.3 Root mean Square velocity
30.1.4 Relation between different types of Speeds
31. Collision diameter
31.1 Collision number
31.2 Collision frequency
31.3 Mean free path
32. Degrees of freedom
32.1 Translational degree of freedom
32.2 Rotational degrees of freedom
32.3 Vibrational degrees of freedom
33. Principle of Equipartition of Energy
34. Real gases: Vander Waals equation
35. Partition Function
35.1 Physical significance of q
35.2 Translational Partition Function
35.3 Rotational Partition Function
35.4 Vibrational Partition Function
35.5 Electronic Partition Function
35.6 Canonical Ensemble partition
36. Relation between Partition function and thermodynamic
functions
36.1 Internal Energy
36.2 Heat capacity
36.3 Entropy and partition function
36.4 Work function (A) and partition function
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CHAPTERI
INTRODUCTION
Thermodynamics is a macroscopic science that studies the
interrelationships of the various equilibrium properties of a system and
the changes in equilibrium properties in processes.
Thermodynamics is the study of heat, work, energy and the changes
they produce in the states of systems. It is sometimes defined as the
study of the relation of temperature to the macroscopic properties of
matter.
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CHAPTER2
MAXWELL’S THERMODYNAMIC EQUATION
The four Maxwell’s thermodynamic equations are as follows;
dH TdS VdP …(2a)
dG VdP SdT …(2b)
dA PdV SdT …(2c)
dU TdS PdV …(2d)
2.1 DERIVATION OF MAXWELL’S EQUATIONS
Thermodynamic coordinates are S, P, V, T
Thermodynamic Potential are G, H, A, U
[1] H comes in between S and P,so H S P
For partial differential is used.
H S P
And for complete differential (d) is used
dH ds dp
[2] Now S is pointing toward T and arrow is away from S positive sign
comes along with T ,P points toward V and arrow is away so positive
sign comes i.e.
dH TdS VdP
Similarly, the relations can be derived for derive for ,dG dA and dU
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Relationship between thermodynamic coordinates
Case 1:
PV ST
S P
V T
Taking partial differential on both the sides
S P
V T
T V
S P
V T …(2.1a)
{Since, S points toward T and P points toward V, the sign on the equation (a) is
positive.}
Case 2:
PV ST
S V
P T
Taking partial differential on both the sides
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T P
S V
P T … (2.1b)
{Since, S points toward T thus, it is positive whereas arrow is on the V, the sign
on the V is negative.}
Case 3:
PV ST
V T
S P
Taking partial differential on both the sides
P S
V T
S P …(2.1c)
Case 4:
PV ST
P T
S V
Taking partial differential on both the sides
V S
P T
S V
…(2.1d)
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CHAPTER3
FUNCTION AND DERIVATIVE
Function is a rule that relates 2 or more variables. If 3z x y
Then z is a function of x and y because the change in the value of x and
y, changes the value of z.
Example: In ideal gas equation P is a function of T and V. T is a
function of P and V.V is a function of T and P.
Derivative is a measure of how a function changes as its input changes
in simple words, derivative is as how much one quantity is changing in
response to change in some of the quantity.
3.1 DIFFERENTIATION
The process of finding a derivative is called differentiation.
It is a method to compute the rate at which dependent output y
changes w.r.t change in independent input x.
Formulas for differential are as follows:
0da
dx
d au du
adx dx
1
n
nd x
nxdx
ax
axd e
aedx
1dln ax
dx x
sin
cosd ax
a axdx
cossin
d axa ax
dx
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d u v du du
dx dx dx
d uv du duu v
dx dx dx
1
2 1/ d uvd u v dv du
uv vdx dx dx dx
3.2 PARTIAL DIFFERENTIATION
In thermodynamics, we usually deal with functions of two or more
variables.
To find partial derivative z
x
y
we take ordinary derivative of z with
respect to x while regarding y as constant. For example, if 2 3 + eyxz x y ,
then 3 eyxzxy y
x
2
y
; also e2 yxzy x x
y
23x
y x
z zdz dx dy
x y
In this equation, dz is called the total differentialof ,z x y . An analogous
equation holds for the total differential of a function of more than two
variables. For example, if , , ,z z r s t then
, , ,s t r t r s
z z zdx dr ds dt
r s t
Partial differentiation of ideal gas
For 1 mole of gas
PV RT
Differentiate the above equation with respect to T at constant V
i.e.
V
P
T
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Where,
P is a function, is an operator and V is constant.
V
V
RTP V
T T
RT
V
Differentiate the ideal gas equation with respect to V at constant T
2
T
T
RTP V
V V
RT
V
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CHAPTER4
CYCLIC RULE
The triple product rule, known variously as we cyclic chain rule, cyclic
rule or Euler’s chain rule. It is a formula relates partial differential of 3
independent variables.
1
T p V
P V T
V T P ….(4.1)
Where, ,P V and T are independent related by cyclic rule.
Examples:
1. Prove that cyclic rule is valid for ideal gas?
Solution:
Ideal gas equation is given by
; ;
PV RT
RT RT PVP V T
V P R
Differentiate P with respect to V at constant T
...(i)
2
T
T
RT
P V
V V
RT
V
Differentiate V with respect to T at constant P
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...(ii)
P
P
RT
V P
T T
R
P
Differentiate T with respect to P at constant V
...(iii)
V
V
PV
T R
P P
V
R
Cyclic rule is given by
1
T P V
P V T
V T P
Substitute (i), (ii) and (iii) in equation (iv)
2
[ ]
1
T P V
P V T RT R V
V T P V P R
RT
VP
RTPV RT
RT
2. For 1 mol of ideal gas the value of
V P T
P V V
T T P is
1a
2 2b RP
1c
2 2d RP
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Solution:
Ideal gas equation is given by
; ;
PV RT
RT RT PVP V T
V P R
Differentiate P with respect to T at constant V
...(i)
V
V
RT
P V
T T
R
V
Differentiate V with respect to T at constant P
...(ii)
P
P
RT
V P
T T
R
P
Differentiate V with respect to P at constant T
...(iii)
2
T
T
RT
V P
P P
RT
P
...(iv)
V P T
P V V
T T P
Substitute (i), (ii) and (iii) in equation (iv)
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2
3
2
3
2
2
2
V P T
P V V R R RT
T T P V P P
R T
VPP
R T
RTP
R
P
Hence, the correct option is (b)
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