INTRODUCTORY THERMAL PHYSICS
1 INTRODUCTION 1 2 ZEROTH LAW OF THERMODYNAMICS THE CONCEPT OF EMPIRICAL TEMPERATURE 8 3 THE CONCEPT OF INTERNAL ENERGY AND THE FIRST LAW OF THERMODYNAMICS 14 4 MATHEMATICAL INTERLUDE 25 5 THE RESTRAINT STATEMENT OF THE SECOND LAW OF THERMODYNAMICS THE CONCEPT OF ENTROPY 34 6 THE GENERAL STATEMENT OF THE SECOND LAW OF THERMODYNAMICS THE FUNDAMENTAL EQUATION OF THERMODYNAMICS THE THIRD LAW OF THERMODYNAMICS AND THE SCALING OF THE ENTROPY 55 7 STATISTICAL DESCRIPTION OF THERMAL PHENOMENA 65 8 EQUATIONS OF STATE 107 9 RESPONSE FUNCTIONS OF THERMODYNAMIC SYSTEMS 116 10 SOME GENERAL CONSEQUENCES OF THE LAWS OF THERMODYNAMICS 129 11 CHARACTERISTIC FUNCTIONS 137 12 THE STATISTICAL APPROACH IN EQUILIBRIUM THERMODYNAMICS 153 13 FLUCTUATIONS AND STABILITY OF THE EQUILIBRIUM STATES OF THERMODYNAMIC SYSTEMS 188 14 EQUILIBRIUM BETWEEN PHASES 198 CONTENTS 1 INTRODUCTION 1 1 MACROSCOPIC AND MICROSCOPIC THEORIES PHYSICS IS A SCIENCE DEALING WITH EXPERIMENTAL FACTS THAT IS WITH EVENTS WHICH CAN BE SYSTEMATICALLY REPRODUCED BY PREPARING EACH TIME ANALOGOUS ENVIRONMENTAL CONDITIONS THE EXPERIMENTAL FACTS ARE FURTHER ORGANIZED IN VARIOUS THEORIES ATTEMPTING TO FIT OUR MENTAL REASONING ON THE OBTAINED EXPERIMENTAL RESULTS PHYSICAL THEORIES CAN BELONG ESSENTIALLY TO TWO IMPORTANT CATEGORIES MICROSCOPIC OR MACROSCOPIC GENERALLY SPEAKING A MACROSCOPIC THEORY MAKES USE OF A RESTRAINT SET OF PARAMETERS NAMELY THE ONES WHICH ARE MORE OR LESS AVAILABLE TO OUR DIRECT SENSES ON THE CONTRARY A MICROSCOPIC THEORY MAY INVOLVE QUITE ABSTRACT REPRESENTATIONS NEVER MET IN OUR CURRENT SENSITIVE EXPERIENCE AND THEREFORE NEEDS A MUCH HIGHER NUMBER OF PHYSICAL PARAMETERS USUALLY IT IS USEFUL TO ATTACH BOTH A MACROSCOPIC AND A MICROSCOPIC POINT OF VIEW TO EVERY FIELD OF PHYSICS ONE PARTICULARLY STRIKING EXAMPLE IN THIS SENSE IS THE PHYSICS OF THERMAL PHENOMENA IT CAN BE DESCRIBED BOTH MACROSCOPICALLY IN THE SO CALLED FIELD OF THERMODYNAMICS AND MICROSCOPICALLY IN STATISTICAL PHYSICS THOSE TWO DIFFERENT THEORIES ARE ACTUALLY NOT COMPLETELY DISTINCT SINCE THEY SHARE THE SAME STARTING POINTS AND THE SAME PURPOSES THERE ARE ALSO SEVERAL SHARED PHYSICAL PARAMETERS AND CONCEPTS WHICH ALLOW A CLOSE INTERACTION BETWEEN THE TWO THERMAL THEORIES 1 2 MEASURABLE AND HIDDEN PARAMETERS IT IS NOW A VERY WELL ESTABLISHED FACT THAT EVERY MACROSCOPIC BODY THAT IS A BODY WHOSE PRESENCE MAY BE EVIDENCED THROUGH USUAL SENSES IS ACTUALLY A SYSTEM OF A HUGE NUMBER OF THE ORDER OF 1023 OF INTERACTIVE MICROSCOPIC UNITS OR PARTICLES UPON THE CASE THESE MAY BE ATOMS MOLECULES ELECTRONS EVEN VIRTUAL PARTICLES LIKE PHONONS WHICH RESULT FROM THE QUANTIZATION OF THE ATOMIC LATTICE VIBRATIONS IN A CRYSTAL LATTICE OR PHOTONS THAT IS CONFINED ELECTROMAGNETIC FIELD PARTICLES THESE UNITS ARE FAR FROM BEING STILL THEY ARE IN A PERMANENT RAPID AND APPARENTLY DISORDERED MOTION THE PHENOMENA EMERGING FROM THE MOTION AND RECIPROCAL INTERACTION OF THE MICROSCOPIC UNITS OF PHYSICAL BODIES ARE USUALLY TERMED AS THERMAL PHENOMENA IT IS ALMOST OBVIOUS THAT THE MICROSCOPIC DESCRIPTION OF A MACROSCOPIC BODY WOULD IMPLY A GIGANTIC NUMBER OF PARAMETERS FOR EXAMPLE THE CARTESIAN COMPONENTS OF THE POSITION AND VELOCITY OF EVERY MICROSCOPIC UNIT ON THE OTHER HAND THE DIRECT STUDY OF SUCH A SYSTEM CAN BE PERFORMED ONLY BY MACROSCOPIC INSTRUMENTS BEING ESSENTIALLY MACROSCOPIC ENTITIES THESE INSTRUMENTS EVEN THE MOST SENSITIVE OF THEM ARE CERTAINLY UNABLE TO RESOLVE THE ENORMOUS COMPLEXITY OF THE 1 MICROSCOPIC STRUCTURE MACROSCOPIC BODIES ANY PHYSICAL APPARATUS IS A RELATIVELY MACROSCOPIC ENTITY WHOSE OPERATION IS USUALLY HINDERED BY CERTAIN INERTIA DURING THE MEASURING ACT FOR EXAMPLE A MEMBRANE PRESSURE SENSOR PLUNGED IN A GAS CANNOT FEEL THE INDIVIDUAL COLLISIONS OF THE MOLECULES WITH ITS MEMBRANE EVERY MOLECULAR COLLISION RESULTS IN A MOMENTUM TRANSFER TOWARDS THE MEMBRANE BUT THOSE EVENTS REPEAT SO RAPIDLY THAT THEY CANNOT BE FOLLOWED BY DEFORMATIONS OF THE MEMBRANE IN THE SAME WAY THE MOLECULAR COLLISIONS ARE VERY LOCALIZED EVENTS SO THAT THE MOMENTUM TRANSFER TO THE MEMBRANE IS NOT AT ALL HOMOGENEOUS ON ALL THE MICROSCOPIC REGIONS OF ITS SURFACE BUT THE ELASTIC PROPERTIES OF THE MEMBRANE ARE NEVER IDEAL AND IT CANNOT FOLLOW THE SPATIAL INHOMOGENEITIES OF THE FORCES APPLIED BY THE GAS MOLECULES CONSEQUENTLY THE MEMBRANE WILL UNDERGO A CONSTANT AND UNIFORM AVERAGE DEFORMATION ANOTHER POSSIBLE EXAMPLE MAY BE A TEMPERATURE SENSOR PLUNGED IN A CERTAIN AMOUNT FLUID THE MOLECULES OF THE FLUID HIT THE SENSOR S SURFACE IN A RAPID SEQUENCE TRANSFERRING EACH TIME A CERTAIN AMOUNT OF ENERGY TO IT BUT USUALLY THE COLLISION FREQUENCY IS SO HIGH AND ITS SPATIAL VARIATIONS SO LOCALIZED THAT THE SENSOR CAN CATCH ONLY A SPACE AND TIME AVERAGE OF ITS ENERGETIC INCOME ONE CAN THUS CONCLUDE THAT THE PHYSICAL INSTRUMENTS CAN MEASURE ONLY CERTAIN TEMPORAL AND OR SPATIAL AVERAGES FOR THE VALUES OF THE PARAMETERS OF THE MICROSCOPIC MOTION THIS IS A DIRECT CONSEQUENCE OF THE GIGANTIC NUMBER OF THE MICROSCOPIC UNITS OF EACH MACROSCOPIC BODY AND OF THEIR APPARENTLY DISORDERED INCESSANT MOTION THUS THE KEY IDEA IS THAT ANY MEASUREMENT REPLACES A GREAT NUMBER OF MICROSCOPIC PARAMETERS BY A REPRESENTATIVE MEAN VALUE WHICH MAY BE LOOKED AT AS A MACROSCOPIC PARAMETER OF THE SYSTEM THIS OBSERVATION IS ESSENTIALLY THE BASIS OF THE CONNECTION BETWEEN THE MICROSCOPIC AND THE MACROSCOPIC DESCRIPTION OF THERMAL PHENOMENA AT THIS POINT WE MAY DISTINGUISH TWO TYPES OF MICROSCOPIC PARAMETERS FOR ONE KIND OF SUCH PARAMETERS THE VALUES OF THE AVERAGES OBTAINED THROUGH THE MEASUREMENT PROCESS STRICTLY DEPEND ON THE EXTERNAL CONDITIONS TO WHICH THE SYSTEM IS SUBJECTED THESE PARAMETERS ARE CALLED MEASURABLE PARAMETERS THEIR REPRESENTATIVE MEAN VALUES ARE NOT IDENTICALLY ZERO FOR THE OTHER TYPE OF MICROSCOPIC PARAMETERS THE MEASUREMENTS GIVE ALWAYS ZERO VALUES OF THE AVERAGES THEY ARE TERMED AS HIDDEN PARAMETERS SINCE THEY DON T SOW UP IN A USUAL MEASUREMENT PROCESS AND NO MACROSCOPIC PARAMETER CAN BE ASSOCIATED WITH THEM AS AN EXAMPLE OF MEASURABLE PARAMETER ONE MAY TAKE THE KINETIC ENERGY OF THE MOLECULAR TRANSLATION IN A CERTAIN MACROSCOPIC AMOUNT OF GAS THE CORRESPONDING MACROSCOPIC PARAMETER WILL BE THE INTERNAL ENERGY ANOTHER EXAMPLE OF MACROSCOPIC PARAMETER RESULTING FROM AN AVERAGE ON A MICROSCOPIC MEASURABLE PARAMETER IS THE GAS PRESSURE ITS MEASURED VALUES ALWAYS RESULT FROM AN AVERAGE OF THE PROJECTION OF THE TRANSLATION MOLECULAR MOMENTUM ON THE DIRECTION NORMAL TO A CERTAIN WALL WHEN ONLY MOLECULES RUNNING TOWARDS THE WALL ARE CONSIDERED NEITHER THE VALUE OF THE INTRODUCTORY THERMAL PHYSICS 2 GAS PRESSURE NOR THE ONE OF ITS INTERNAL ENERGY CAN BE IDENTICALLY ZERO ALSO BOTH OF THEM STRONGLY DEPEND ON THE EXTERNAL CONDITIONS IMPOSED TO THE GAS EXAMPLES OF HIDDEN PARAMETERS MAY BE THE MOMENTUM OF GAS MOLECULES OR THE ORIENTATION OF POLIATOMIC MOLECULES OF A GAS IN THE ABSENCE OF ANY ELECTRIC OR MAGNETIC EXTERNAL FIELD AND AT HIGH TEMPERATURES FOR THE FIRST CASE GIVEN THE ENORMOUS NUMBER OF MOLECULES IN PERMANENT DISORDERED MOTION WITH A VERY HIGH PROBABILITY EACH MOLECULAR MOMENTUM CAN BE PAIRED WITH THE MOMENTUM OF ANOTHER MOLECULE WHICH IS EQUAL IN ABSOLUTE VALUE AND OPPOSITE TO THE FIRST IN THIS WAY THE AVERAGE MOMENTUM MEASURED BY A HYPOTHETIC MACROSCOPIC APPARATUS WOULD ALWAYS VANISH OBVIOUSLY THE SAME CAN BE SAID ABOUT THE ANGULAR ORIENTATION OF THE POLIATOMIC MOLECULES OF THE AFOREMENTIONED GAS IT MUST BE STRESSED HERE THAT DESPITE THE FACT THAT HIDDEN PARAMETERS DON T SHOW UP IN MACROSCOPIC MEASUREMENTS THEIR SIMPLE EXISTENCE BRINGS A DECISIVE INFLUENCE ON THE MACROSCOPIC BEHAVIOR OF THE CONSIDERED SYSTEM 1 3 THE MAIN PURPOSE OF THERMAL PHYSICS THE PHYSICS OF THERMAL PHENOMENA OR SHORTLY THERMAL PHYSICS IS MAINLY CONCERNED WITH THE MACROSCOPIC CONSEQUENCES OF THE EXISTENCE OF HIDDEN PARAMETERS OF SYSTEMS THIS STATEMENT IS ESSENTIALLY CORRECT BUT IT SHOULD NOT BE TAKEN AS EXHAUSTIVE IT IS SUFFICIENT FOR A BEGINNER BUT THE MORE ADVANCED STUDENT MAY EASILY FIND MANY OTHER USEFUL TARGETS OF THIS FIELD OF STUDY GENERALLY IN THERMODYNAMICS ONE TRIES TO TAKE INTO ACCOUNT THE INFLUENCE OF THE HIDDEN PARAMETERS BY DEFINING SOME SPECIFIC THERMODYNAMIC STATE FUNCTIONS AND BY ESTABLISHING THEIR RELATION WITH THE OTHER STATE FUNCTIONS OF THE GIVEN SYSTEM ON THE CONTRARY STATISTICAL PHYSICS STARTS FROM A MICROSCOPIC DESCRIPTION OF THE SYSTEM INTO THE FRAME OF SOME SUITABLE DYNAMICS CLASSICAL OR QUANTUM AND MAKING USE OF THE CONCEPT OF PROBABILITY COMPUTES VARIOUS AVERAGES WHICH CAN BE COMPARED WITH THE RESULTS OF SOME MACROSCOPIC MEASUREMENTS BOTH THE STARTING POINTS AND THE INTERPRETATION OF THE RESULTS OF STATISTICAL PHYSICS IMPLY THE USE OF SOME FUNDAMENTAL THERMODYNAMIC CONCEPTS THIS IS THE REASON FOR WHICH A STUDY OF THERMAL PHYSICS SHOULD BEGIN WITH THE MACROSCOPIC POINT OF VIEW NAMELY WITH THERMODYNAMICS NEVERTHELESS THE MICROSCOPIC POINT OF VIEW PROVES FREQUENTLY ESSENTIAL FOR A THOROUGH UNDERSTANDING OF THERMAL PHENOMENA THUS THE IDEAS OF STATISTICAL PHYSICS WILL BE INTRODUCED AS SOON AS POSSIBLE IN ORDER TO ALLOW THE TREATMENT OF THE MAIN APPLICATIONS FROM BOTH POINTS OF VIEW INTRODUCTORY THERMAL PHYSICS 3 1 4 SOME FUNDAMENTAL CONCEPTS OF THERMODYNAMICS A A THERMODYNAMIC SYSTEM IS A SET OF MACROSCOPIC ENTITIES INCLUDING POSSIBLY RADIATION FIELDS THAT CAN EXCHANGE ENERGY AMONG THEM EVEN IF THEY DON T INTERACT EITHER BY SOME ORDERED MACROSCOPIC WAY OR BY MASS EXCHANGE THIS DEFINITION NEEDS SOME SUPPLEMENTARY EXPLANATIONS FIRST BY MACROSCOPIC ENTITY WE MEAN A REGION OF SPACE THAT CONTAINS A VERY LARGE NUMBER OF THE ORDER OF THE AVOGADRO S NUMBER NA 6 0221 1023 MOL 1 OF IDENTICAL MICROSCOPIC UNITS THEN THE INTERACTION BY SOME ORDERED MACROSCOPIC WAY INCLUDES MECHANICAL INTERACTIONS RESULTING IN MACROSCOPIC DEFORMATIONS OF THE SYSTEM OR INTERACTIONS WITH SOME EXTERNAL FIELDS ELECTRIC MAGNETIC GRAVITATIONAL ACTING SYSTEMATICALLY ON ALL THE MICROSCOPIC UNITS OF THE SYSTEM ONE MAY ALREADY OBSERVE THAT THE INTERACTIONS BETWEEN THE COMPONENTS OF A THERMODYNAMIC SYSTEM AND AS IT WILL BE FURTHER DISCUSSED BETWEEN THE THERMODYNAMIC SYSTEMS THEMSELVES REDUCE UNIQUELY TO ENERGY EXCHANGES THIS SPECIAL ROLE OF THE ENERGY IS REALLY FUNDAMENTAL BOTH IN THERMODYNAMICS AND IN STATISTICAL PHYSICS AND IS A CONSEQUENCE OF THE TACIT HYPOTHESIS THAT THERMODYNAMIC SYSTEMS ARE CONSIDERED TO BE FIXED IN THE REFERENCE FRAME OF THE LABORATORY THUS THE VARIOUS INTERACTIONS OF THE THERMODYNAMIC SYSTEM CAN CHANGE ONLY ITS ENERGETIC CONTENT NEVERTHELESS THERE ARE IMPORTANT GENERALIZATIONS WHICH ARE USEFUL MAINLY AT AN ASTRONOMIC SCALE WHICH TAKE INTO ACCOUNT THE VARIATIONS OF ALL THE INTEGRALS OF MOTION OF THE SYSTEM INCLUDING THE TOTAL MOMENTUM OR THE TOTAL ANGULAR MOMENTUM NO SUCH GENERALIZATIONS WILL BE CONSIDERED IN THIS INTRODUCTORY SET OF LECTURES THE DEFINITION OF THE THERMODYNAMIC SYSTEM INCLUDES ANOTHER FUNDAMENTAL IDEA THE POSSIBILITY OF THE ENERGY EXCHANGE BETWEEN SYSTEMS THROUGH SOME SPECIAL MECHANISM DIFFERENT FROM BOTH MACROSCOPIC AND MASS EXCHANGE INTERACTION TYPES THIS NEW TYPE OF INTERACTION USES THE DISORDERED MOTION OF THE MICROSCOPIC UNITS AS AN ENERGY EXCHANGE MECHANISM TINY AMOUNTS OF ENERGY ARE EXCHANGED DURING EACH INTERACTION BETWEEN MICROSCOPIC UNITS THEY RESULT IN A MACROSCOPIC ENERGY TRANSFER WHICH MAY TAKE PLACE ONLY AT THE INTERFACE BETWEEN THE TWO SYSTEMS AND THUS IMPLY THEIR DIRECT PHYSICAL CONTACT THIS TYPE OF ENERGY EXCHANGE IS CALLED THERMAL INTERACTION IT OCCURS SIGNIFICANTLY AND SHOULD BE TAKEN INTO ACCOUNT ONLY IN THERMAL PHENOMENA THEREFORE A THERMODYNAMIC SYSTEM MAY BE DEFINED AS A COLLECTION OF MACROSCOPIC ENTITIES WHICH ARE ABLE TO EXCHANGE ENERGY THROUGH THERMAL INTERACTION THE EXTERIOR OR ENVIRONMENT OF A THERMODYNAMIC SYSTEM IS THE REST OF BODIES NOT INCLUDED IN THE SYSTEM AND WITH WHICH IT IS SUPPOSED TO INTERACT THE REUNION BETWEEN THE SYSTEM AND ITS EXTERIOR REPRESENTS THE THERMODYNAMIC UNIVERSE VERY OFTEN THE NECESSITY DIVIDING THE STUDIED THERMODYNAMIC SYSTEM INTO TWO OR MORE PARTS OCCURS THIS OPERATION IS CALLED THERMODYNAMIC DECOMPOSITION SINCE THE THERMAL INTERACTION IS INTRODUCTORY THERMAL PHYSICS 4