THERMODYNAMICS CONTENTS

Transcription

1 1. Introduction HERMODYNAMICS CONENS. Maxwell s thermodynamic equations.1 Derivation of Maxwell s equations 3. Function and derivative 3.1 Differentiation 4. Cyclic Rule artial Differentiation 5. State Function and its characteristics 6. hermodynamic co-efficients 7. Reversible work 8. Reversible and Irreversible rocess 9. Heat capacity 10. he First Law of thermodynamics 11. Enthalpy 1. hermodynamic equations of state 1.1 First hermodynamic equation of state 1. Second hermodynamic equation of state 1.3 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 ressure of reversible isothermal and adiabatic process 18. Joule homson Experiment 18.1 Calculation of Joule homson coefficient for ideal gas 18. Calculation of Joule homson coefficient for real gas 1

2 18.3 Concept of inversion temperature he Case for gas cooling he Case for gas heating he Case where gas neither cools or heat 18.4 Relation between i and 19. Carnot Cycle 19.1 Characteristics of Carnot Cycle 19. rocesses in Carnot Engine 0. Concept of Refrigerators 1. Entropy 1.1 Entropy change of an ideal gas for a reversible process 1. Entropy change in Mixing of Solids 1.3 Entropy change in Mixing of ideal Gases. hase ransformation.1 Reversible phase transformation. Irreversible hase ransformation 3. hase Diagram 3.1 One-Component Systems 3. wo-component Systems 3.3 hree-component Systems 4. Activity and Activity Coefficient 4.1 Activity 4. Activity Coefficient 5. Debye-Hückelheory 6. Clausius-Clapeyron Equation 7. hird law of thermodynamics 8. he Kinetic theory of gases 8.1 Derivation of Kinetic gas equation 8. Kinetic Energy of 1 mole of gas 8.3 Kinetic energy for 1 molecule 9. Deduction of various gas laws from kinetic gas equation 9.1 Boyle s law 9. Charle s law

3 9.3 Avogadro s law 9.4 Graham s law of diffusion 30. Maxwell s distribution of molecules kinetic Energies 30.1 ypes of Molecular velocities Most probable speeds Average Speed Root mean Square velocity Relation between different types of Speeds 31. Collision diameter 31.1 Collision number 31. Collision frequency 31.3 Mean free path 3. Degrees of freedom 3.1 ranslational degree of freedom 3. Rotational degrees of freedom 3.3 ibrational degrees of freedom 33. rinciple of Equipartition of Energy 34. Real gases: ander Waals equation 35. artition Function 35.1 hysical significance of q 35. ranslational artition Function 35.3 Rotational artition Function 35.4 ibrational artition Function 35.5 Electronic artition Function 35.6 Canonical Ensemble partition 36. Relation between artition function and thermodynamic functions 36.1 Internal Energy 36. Heat capacity 36.3 Entropy and partition function 36.4 Work function (A) and partition function 3

4 CHAERI INRODUCION hermodynamics is a macroscopic science that studies the interrelationships of the various equilibrium properties of a system and the changes in equilibrium properties in processes. hermodynamics 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. 4

5 CHAER MAXWELL S HERMODYNAMIC EQUAION he four Maxwell s thermodynamic equations are as follows; dh ds d (a) dg d Sd (b) da d Sd (c) du ds d (d).1 DERIAION OF MAXWELL S EQUAIONS hermodynamic coordinates are S,,, hermodynamic otential are G, H, A, U [1] H comes in between S and,so H S For partial differential is used. H S And for complete differential (d) is used dh ds dp [] Now S is pointing toward and arrow is away from S positive sign comes along with, points toward and arrow is away so positive sign comes i.e. dh ds d Similarly, the relations can be derived for derive for dg, da and du 5

6 Relationship between thermodynamic coordinates Case 1: S S aking partial differential on both the sides S S (.1a) {Since, S points toward and points toward, the sign on the equation (a) is positive.} Case : S S aking partial differential on both the sides 6

7 S (.1b) {Since, S points toward thus, it is positive whereas arrow is on the, the sign on the is negative.} Case 3: S S aking partial differential on both the sides S S (.1c) Case 4: S S aking partial differential on both the sides S S (.1d) 7

8 CHAER3 FUNCION AND DERIAIE Function is a rule that relates or more variables. If z 3x y hen 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 is a function of and. is a function of and. is a function of and. 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 DIFFERENIAION he 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 dx 0 d au du a dx dx n n d x dx d e dx ax d ln ax dx nx ae 1 x ax 1 d sinax dx a cosax d cosax a sinax dx 8

9 HERMODYNAMICS d u v du du dx dx dx d uv du du u v dx dx dx 1 / d uv d u v dv 1du uv v dx dx dx dx 3. ARIAL DIFFERENIAION In thermodynamics, we usually deal with functions of two or more variables. o find partial derivative z we take ordinary derivative of z with x y respect to x while regarding y as constant. For example, if then z x 3 xy y y e yx z z dz dx dy x y y x z ; also 3y x xe y x yx z x y 3 +e yx, 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 artial differentiation of ideal gas For 1 mole of gas z z z dx dr ds dt r s t s, t r, t r, s R Differentiate the above equation with respect to at constant i.e. 9

10 Where, is a function, is an operator and is constant. R R Differentiate the ideal gas equation with respect to at constant R R 10

11 CHAER4 CYCLIC RULE he 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. p 1 Where, and, are independent related by cyclic rule. Examples:.(4.1) 1. rove that cyclic rule is valid for ideal gas? Solution: Ideal gas equation is given by R R R ; ; R Differentiate with respect to at constant R R...(i) Differentiate with respect to at constant 11

12 R R...(ii) Differentiate with respect to at constant R R...(iii) Cyclic rule is given by 1 Substitute (i), (ii) and (iii) in equation (iv) R R R R R R 1. For 1 mol of ideal gas the value of a 1 b R c 1 d R is [ R ] 1

13 Solution: Ideal gas equation is given by R R R ; ; R Differentiate with respect to at constant R R...(i) Differentiate with respect to at constant R R...(ii) Differentiate with respect to at constant R R...(iii)...(iv) Substitute (i), (ii) and (iii) in equation (iv) 13

14 R R R 3 R 3 R R R Hence, the correct option is (b) 14