thermodynamics · the first law of thermodynamics 11. enthalpy 12. thermodynamic equations of state...

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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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Page 1: Thermodynamics · The First Law of thermodynamics 11. Enthalpy 12. Thermodynamic equations of state 12.1 First Thermodynamic equation of state

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

Page 4: Thermodynamics · The First Law of thermodynamics 11. Enthalpy 12. Thermodynamic equations of state 12.1 First Thermodynamic equation of state

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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)