Physics 207: Lecture 25, Pg 1 Lecture 25Goals: Chapters 18, micro-macro connection Chapters 18, micro-macro connection Third test on Thursday at 7:15 pm.

Physics 207: Lecture 25, Pg 2 (m/s) Percentage of molecules Nitrogen molecules near room temperature

Physics 207: Lecture 25, Pg 3 l What is the typical size of an atom or a small molecule? Atomic scale A) mB) mC) m r ≈1 angstrom= m r

Physics 207: Lecture 25, Pg 4 Mean free path l Average distance particle moves between collisions: N/V: particles per unit volume l The mean free path at atmospheric pressure is: λ=68 nm

Physics 207: Lecture 25, Pg 5 Pressure of a gas

Physics 207: Lecture 25, Pg 6 Consider a gas with all molecules traveling at a speed v x hitting a wall. l If (N/V) increases by a factor of 2, the pressure would: A) decreaseB) increase x2C) increase x4 l If m increases by a factor of 2, the pressure would: A) decreaseB) increase x2C) increase x4 l If v x increases by a factor of 2, the pressure would: A) decreaseB) increase x2C) increase x4

Physics 207: Lecture 25, Pg 7 P=(N/V)mv x 2 l Because we have a distribution of speeds: P=(N/V)m(v x 2 ) avg l For a uniform, isotropic system: (v x 2 ) avg =(v y 2 ) avg = (v z 2 ) avg l Root-mean-square speed: (v 2 ) avg =(v x 2 ) avg +(v y 2 ) avg +(v z 2 ) avg =V rms 2

Physics 207: Lecture 25, Pg 8 Microscopic calculation of pressure P=(N/V)m(v x 2 ) avg =(1/3) (N/V)mv rms 2 PV = (1/3) Nmv rms 2

Physics 207: Lecture 25, Pg 9 Micro-macro connection PV = (1/3) Nmv rms 2 PV = Nk B T (ideal gas law) k B T =(1/3) mv rms 2 l The average translational kinetic energy is: ε avg =(1/2) mv rms 2 ε avg =(3/2) k B T

Physics 207: Lecture 25, Pg 10 l The average kinetic energy of the molecules of an ideal gas at 10°C has the value K 1. At what temperature T 1 (in degrees Celsius) will the average kinetic energy of the same gas be twice this value, 2K 1 ? (A) T 1 = 20°C (B) T 1 = 293°C (C) T 1 = 100°C l Suppose that at some temperature we have oxygen molecules moving around at an average speed of 500 m/s. What would be the average speed of hydrogen molecules at the same temperature? (A) 100 m/s (B) 250 m/s (C) 500 m/s (D) 1000 m/s (E) 2000 m/s

Physics 207: Lecture 25, Pg 11 Equipartition theorem l Things are more complicated when energy can be stored in other degrees of freedom of the system. monatomic gas: translation solids: translation+potential energy diatomic molecules: translation+vibrations+rotations

Physics 207: Lecture 25, Pg 12 Equipartition theorem l The thermal energy is equally divided among all possible energy modes (degrees of freedom). The average thermal energy is (1/2)k B T for each degree of freedom. ε avg =(3/2) k B T (monatomic gas) ε avg =(6/2) k B T (solids) ε avg =(5/2) k B T (diatomic molecules) l Note that if we have N particles: E th =(3/2)N k B T =(3/2)nRT (monatomic gas) E th =(6/2)N k B T =(6/2)nRT (solids) E th =(5/2)N k B T =(5/2)nRT (diatomic molecules)

Physics 207: Lecture 25, Pg 13 Specific heat l Molar specific heats can be directly inferred from the thermal energy. E th =(6/2)N k B T =(6/2)nRT (solid) ΔE th =(6/2)nRΔT=nCΔT C=3R (solid) l The specific heat for a diatomic gas will be larger than the specific heat of a monatomic gas: C diatomic =C monatomic +R

Physics 207: Lecture 25, Pg 14 Entropy l A perfume bottle breaks in the corner of a room. After some time, what would you expect? A) B)

Physics 207: Lecture 25, Pg 15 very unlikely probability=(1/2) N l The probability for each particle to be on the left half is ½.

Physics 207: Lecture 25, Pg 16 Second Law of thermodynamics l The entropy of an isolated system never decreases. It can only increase, or in equilibrium, remain constant. l The laws of probability dictate that a system will evolve towards the most probable and most random macroscopic state l Thermal energy is spontaneously transferred from a hotter system to a colder system.