THERMODYNAMICS
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 Ti 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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THERMODYNAMICS
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 HSP
For partial differential is used.
HSP
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
SP VT
Taking partial differential on both the sides
SP VT
SP …(2.1a) VT TV
{Since, S points toward T and P points toward V, the sign on the equation (a) is positive.}
Case 2:
SV PT
Taking partial differential on both the sides
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SV … (2.1b) PT TP
{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
VT SP
Taking partial differential on both the sides
VT …(2.1c) SP PS
Case 4:
PT SV
PT Taking partial differential on both the sides SV VS …(2.1d) 7
THERMODYNAMICS
CHAPTER3 FUNCTION AND DERIVATIVE
. Function is a rule that relates 2 or more variables. If z3 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:
da 0 dx
d au du a dx dx
dx n nxn 1 dx
de ax aeax dx
dln ax 1 dx x
dsin ax acos ax dx dcos ax asin ax dx 8
THERMODYNAMICS d u v du du dx dx dx d uv du du uv dx dx dx
1 d u/ v d uv dv du uv21 v dx dx dx dx
3.2 PARTIAL DIFFERENTIATION
. In thermodynamics, we usually deal with functions of two or more variables. z . To find partial derivative we take ordinary derivative of z with x y respect to x while regarding y as constant. For example, if z x23 y +eyx ,
z 3 yx z 22 yx then 2xy ye ; also 3y x xe x y y x zz . dz dx dy xyy x
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
z z z dx dr ds dt r s,,, t s r t t r s
Partial differentiation of ideal gas
For 1 mole of gas
PV RT
Differentiate the above equation with respect to T at constant V
P i.e. T V
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Where,
P is a function, is an operator and V is constant.
RT P V TT V V RT V
Differentiate the ideal gas equation with respect to V at constant T
RT P V VV T T RT V 2
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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.
PVT 1 ….(4.1) VTP T p V Where, PV, 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 PV PVT;; VPR Differentiate P with respect to V at constant T
RT P V VVT T RT ...(i) V 2 Differentiate V with respect to T at constant P
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RT V P TTP P R ...(ii) P Differentiate T with respect to P at constant V
PV T R PPV V V ...(iii) R
Cyclic rule is given by PVT 1 VTP TPV
Substitute (i), (ii) and (iii) in equation (iv)
P V T RT R V 2 VTPVPR TPV RT VP RT []PV RT RT 1 2. For 1 mol of ideal gas the value of PVV is TTP VPT a 1 bR 2 2 P c 1 dR2 2 P 12
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Solution: Ideal gas equation is given by PV RT RT RT PV PVT;; VPR Differentiate P with respect to T at constant V
RT P V TTV V R ...(i) V Differentiate V with respect to T at constant P
RT V P TTP P R ...(ii) P Differentiate V with respect to P at constant T
RT V P PPT T RT ...(iii) P 2 PVV ...(iv) TTP VPT
Substitute (i), (ii) and (iii) in equation (iv)
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P V V R R RT 2 TTPVPP VPT RT3 2 VPP RT3 RTP 2 R 2 P 2 Hence, the correct option is (b)
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