THEMODYNAMICS OF SOLIDS (ME 460/MSC 405) Spring Semester, 2018

Transcription

1 THEMODYNAMICS OF SOLIDS (ME 460/MSC 405) Spring Semester, 2018 Ranga Dias TU/TH 4:50-6:05 PM Morey 504 LECTURE 1: 23 rd January 2018

2 Some semantics We introduce here classical thermodynamics. The word thermo-dynamic, used first by Thomson (later Lord Kelvin), has Greek origin, and is translated as the combination of : θℇρµη, therme: heat, and δυναµις, dynamis: power Thermodynamics: the science that deals with heat and work and those properties of matter that relate to heat and work. One of the main goals of this course will be to formalize the relationship between heat, work, and energy

3 THEMODYNAMICS and STATISTICAL MECHANICS Thermodynamics/ Stat Mech, being what many consider to be one of the most intellectually difficult subjects of all, is noted for its prevalence of suicides and suicide a7empts by a significant number of its founders!! Deep thinking Depression!!

4 Ludwig Boltzmann : 1906 hanged himself Paul Ehrenfest (Phase Transitions): 1933 shot his daughter and himself Gilbert Lewis (Chemical Thermodynamics): 1946 took cyanide after not getting the Nobel prize Percy Bridgman (Equation of State): 1961 shot himself Now it is our turn to study THERMODYNAMICS Perhaps it will be wise to approach the subject Cautiously David Goodstein in States of Matter, 1975, Dover N.Y.

5 Arnold Sommerfeld Sommerfeld was asked why he had never written a book on thermodynamics having written books on most other areas of Physics a d Engineering. He replied: Thermodynamics is a funny subject. The first time you go through it, you don't understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don't understand it, but by that time you are so used to it, it doesn't bother you anymore.

6 The subject of thermal physics involves studying assemblies of large numbers of atoms What do we mean by a large number? Speed of light 300 million ms -1 Earth s population is 7.6 billion people Age of the universe 14 bullion years But even these large numbers pale into insignificance compared with the numbers involved in thermodynamics Atoms/ molecules in order of x kg of Nitrogen gas contains approximately 2 x10 25 N 2 molecules Lets try to make a prediction about the motion of the molecules in this amount of gas

7 In one year, there are about 3.2x10 7 seconds So that a 3 GHz personal computer can count molecules at a rate of ~ year 1,if it counts one molecule every computer clock cycle 0.2 billion years just for this computer to count all the molecules in 1kg of N 2 gas HOPELESS TASK!!! Hence, to make progress in thermodynamics it is necessary to make approximations and deal with the statistical properties of molecules, i.e. to study how they behave on AVERAGE

8 What is a mole? But also a name (first coined about a century ago from the German Molekul [molecule]) representing a certain numerical quantity of stuff

9 What is a mole? It functions in the same way as the word dozen, which describes a certain number of eggs (12) The mole: A mole is defined as the quantity of matter that contains as many objects (for example, atoms, molecules, formula units, or ions) as the number of atoms in exactly 12 g (=0.012 kg) of 12 C. A mole of atoms is equivalent to an Avogadro number N A of atoms. The Avogadro number, expressed to 4 significant figures, is N A = x10 23

10 One can write N A as 6.022x10 23 mol 1 as a reminder of its definition, but N A is dimensionless, as are moles. They are both numbers. By the same logic, one would have to define the eggbox number as 12 dozen 1 1 mole of carbon is x10 23 atoms of carbon 1 mole of Nitrogen gas is x10 23 molecules of N 2 1 mole of NaCl contains x10 23 NaCl formula units, etc. A mole of eggs would make an omelette with about half the mass of the Moon

11 The molar mass The molar mass of a substance is the mass of one mole of the substance. Thus the molar mass of carbon is 12 g The mass m of a single molecule or atom is therefore the molar mass of that substance divided by the Avogadro number molar mass = mn A Lets check 1 kg of Nitrogen gas

12 The thermodynamic limit How to deal with average quantities Imagine that you are sitting inside a tiny hut with a flat roof The raindrops arrive randomly, so sometimes two arrive close together, but sometimes there is quite a long gap between raindrops. Each raindrop transfers its momentum to the roof and exerts an impulse on it If we know the mass and terminal velocity of a raindrop, we could estimate the force on the roof of the hut

13 The thermodynamic limit How to deal with average quantities F F F

14 The thermodynamic limit (1) The force, on average, gets bigger as the area of the roof gets bigger. This is not surprising because a bigger roof catches more raindrops (2) The fluctuations in the force get smoothed out and the force looks like it stays much closer to its average value. In fact, the fluctuations are still big but, as the area of the roof increases, they grow more slowly than the average force does The force grows with area, so it is useful to consider the pressure which is defined as Pressure = force / area

15 The thermodynamic limit 1. The average pressure due to the falling raindrops will not change as the area of the roof increases 2. But the fluctuations in the pressure will decrease In fact, we can completely ignore the fluctuations in the pressure Area Infinity This is precisely analogous to the limit we refer to as the thermodynamic limit

16 The thermodynamic limit Consider now the molecules of a gas which are bouncing around in a container Pressure (force per unit area) exerted on the walls of the container Molecules in a container of gas is extremely large, fluctuations can be ignored Pressure of the gas appears to be completely uniform Pressure in the thermodynamic limit

17 The thermodynamic limit Suppose that the container of gas has volume V, that the temperature is T, the pressure is p and the kinetic energy of all the gas molecules adds up to U. V = V/2 U = U/2 What about the P and T? Variables which scale with the system size, like V and U, are called extensive variables. Those which are independent of system size, like p and T, are called intensive variables

18 The ideal gas Experiments on gases show that the pressure P of a volume V of gas depends on its temperature T P 1/V (constant T) V T (constant P) P T (constant V) Boyle s Law Charles Law Gay-Lussac s Law PV T PV Nk B T We have stated this law purely as an empirical law, but we will derive this later

19 Why do we call it ideal? The ideal gas (i) we assume that there are no intermolecular forces, so that the molecules are not attracted to each other (ii) we assume that molecules are point-like and have zero size These are idealized assumptions This does describe gases quite well under quite a wide range of conditions

20 Combinatorial problems Problem : Let us imagine that a certain system contains ten atoms. Each of these atoms can exist in one of two states, according to whether it has zero units or one unit of energy. These units of energy are called quanta of energy. How many distinct arrangements of quanta are possible for this system if you have at your disposal (a) 10 quanta of energy; (b) 4 quanta of energy?

21 Solution : Combinatorial problems Empty circles signifies an atom with zero quanta of energy Filled circles signifies an atom with one quantum of energy Number of ways of arranging r quanta among n atoms First quantum can be assigned to any of the n atoms Second quantum can be assigned any of the remaining atoms (n-1 ) r th quantum can be assigned any of the remaining atoms (n-r+1 )

22 Solution : Combinatorial problems Ω guess = n x (n 1) x (n 2) x... x (n r + 1) Ω guess = n! / (n r)! However, this assumes that we have labelled the quanta as the first quantum, the second quantum etc Ω = n! / (n r)! r!

23 Combinatorial problems Solution : Ω = n! / (n r)! r! r = 10 r = n = 100 and r = 40 ; Ω = n = 1000 and r = 400 ; Ω =

24 Solution : Combinatorial problems n = x lnω = ln(n!) ln((n r)!) ln(r!) Stirling s formula: ln n! n ln(n) n ln n! n ln (n) n + 1/2 ln (2πn) Only gives a significant advantage when n is not too large

25 A definition of heat Heat is energy in transit Experiments suggest that heat spontaneously transfers from a hotter body to a colder body when they are in contact, and not in the reverse direction. However, there are circumstances when it is possible for heat to go in the reverse direction The in transit part of our definition is very important. Though you can add heat to an object, you cannot say that an object contains a certain quantity of heat 1W=1J s 1