Physics 207: Lecture 26, Pg 1 Lecture 26 Goals: Chapter 18 Chapter 18  Understand the molecular basis for pressure and the ideal- gas law.  Predict the molar specific heats of gases and solids.  Understand how heat is transferred via molecular collisions and how thermally interacting systems reach equilibrium.  Obtain a qualitative understanding of entropy, the 2 nd law of thermodynamics Assignment Assignment  HW11, Due Tuesday, May 5 th  For Tuesday, Read through all of Chapter 19

Physics 207: Lecture 26, Pg 2 Macro-micro connection Mean Free Path If a “real” molecule has N coll collisions as it travels distance L, the average distance between collisions, which is called the mean free path λ is The mean free path is independent of temperature The mean time between collisions is temperature dependent

Physics 207: Lecture 26, Pg 3 And the mean free path is… l Some typical numbers VacuumPressurePressure (Pa)MoleculesMolecules / cm 3 MoleculesMolecules/ cm 3 mean free path Ambient pressure * * nm Medium vacuum – mm Ultra High vacuum – – km

Physics 207: Lecture 26, Pg Molecular Speed (m/s) # Molecules O 2 at 1000°C O 2 at 25°C Distribution of Molecular Speeds A “Maxwell-Boltzmann” Distribution

Physics 207: Lecture 26, Pg 5 Macro-micro connection l Assumptions for ideal gas:  # of molecules N is large  They obey Newton’s laws  Short-range interactions with elastic collisions  Elastic collisions with walls (an impulse…..pressure) l What we call temperature T is a direct measure of the average translational kinetic energy l What we call pressure p is a direct measure of the number density of molecules, and how fast they are moving (v rms )

Physics 207: Lecture 26, Pg 6 Kinetic energy of a gas 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 The molecules in an ideal gas at 10°C have a root-mean-square (rms) speed v rms. At what temperature T 2 (in degrees Celsius) will the molecules have twice the rms speed, 2v rms ? (A) T 2 = 859°C (B) T 2 = 20°C (C) T 2 = 786°C

Physics 207: Lecture 26, Pg 7 l Consider a fixed volume of ideal gas. When N or T is doubled the pressure increases by a factor of If T is doubled, what happens to the rate at which a single molecule in the gas has a wall bounce? (B) x2(A) x1.4(C) x4 2. If N is doubled, what happens to the rate at which a single molecule in the gas has a wall bounce? (B) x1.4(A) x1(C) x2 Exercise

Physics 207: Lecture 26, Pg 8 A macroscopic “example” of the equipartition theorem l Imagine a cylinder with a piston held in place by a spring. Inside the piston is an ideal gas a 0 K. l What is the pressure? What is the volume? l Let U spring =0 (at equilibrium distance) l What will happen if I have thermal energy transfer?  The gas will expand (pV = nRT)  The gas will do work on the spring l Conservation of energy  Q = ½ k x 2 + 3/2 n R T (spring & gas)  and Newton  F piston = 0 = pA – kx  kx =pA  Q = ½ p (Ax) + 3/2 n RT  Q = ½ p V + 3/2 n RT (but pV = nRT)  Q = ½ nRT + 3/2 RT (25% of Q went to the spring) +Q ½ nRT per “degree of freedom”

Physics 207: Lecture 26, Pg 9 Degrees of freedom or “modes” l Degrees of freedom or “modes of energy storage in the system” can be: Translational for a monoatomic gas (translation along x, y, z axes, energy stored is only kinetic) NO potential energy l Rotational for a diatomic gas (rotation about x, y, z axes, energy stored is only kinetic) l Vibrational for a diatomic gas (two atoms joined by a spring-like molecular bond vibrate back and forth, both potential and kinetic energy are stored in this vibration) l In a solid, each atom has microscopic translational kinetic energy and microscopic potential energy along all three axes.

Physics 207: Lecture 26, Pg 10 Degrees of freedom or “modes” l A monoatomic gas only has 3 degrees of freedom (just K, kinetic) l A typical diatomic gas has 5 accessible degrees of freedom at room temperature, 3 translational (K) and 2 rotational (K) At high temperatures there are two more, vibrational with K and U l A monomolecular solid has 6 degrees of freedom 3 translational (K), 3 vibrational (U)

Physics 207: Lecture 26, Pg 11 The Equipartition Theorem l The equipartition theorem tells us how collisions distribute the energy in the system. Energy is stored equally in each degree of freedom of the system. l The thermal energy of each degree of freedom is: E th = ½ Nk B T = ½ nRT l A monoatomic gas has 3 degrees of freedom l A diatomic gas has 5 degrees of freedom l A solid has 6 degrees of freedom l Molar specific heats can be predicted from the thermal energy, because

Physics 207: Lecture 26, Pg 12 Exercise l A gas at temperature T is mixture of hydrogen and helium gas. Which atoms have more KE (on average)? (A) H(B) He (C) Both have same KE l How many degrees of freedom in a 1D simple harmonic oscillator? (A) 1 (B) 2 (C) 3 (D) 4 (E) Some other number

Physics 207: Lecture 26, Pg 13 The need for something else: Entropy You have an ideal gas in a box of volume V 1. Suddenly you remove the partition and the gas now occupies a larger volume V 2. (1)How much work was done by the system? (2) What is the final temperature (T 2 )? (3) Can the partition be reinstalled with all of the gas molecules back in V 1 P V1V1 P V2V2

Physics 207: Lecture 26, Pg 14 Exercises Free Expansion and Entropy You have an ideal gas in a box of volume V 1. Suddenly you remove the partition and the gas now occupies a larger volume V 2. (1)How much work was done by the system? (A) W > 0 (B) W =0 (C) W < 0 P V1V1 P V2V2

Physics 207: Lecture 26, Pg 15 Exercises Free Expansion and Entropy You have an ideal gas in a box of volume V 1. Suddenly you remove the partition and the gas now occupies a larger volume V 2. (2) What is the final temperature (T 2 )? (A) T 2 > T 1 (B) T 2 = T 1 (C) T 2 < T 1 P V1V1 P V2V2

Physics 207: Lecture 26, Pg 16 Free Expansion and Entropy You have an ideal gas in a box of volume V 1. Suddenly you remove the partition and the gas now occupies a larger volume V 2. (3) Can the partition be reinstalled with all of the gas molecules back in V 1 (4) What is the minimum process necessary to put it back? P V1V1 P V2V2

Physics 207: Lecture 26, Pg 17 Free Expansion and Entropy You have an ideal gas in a box of volume V 1. Suddenly you remove the partition and the gas now occupies a larger volume V 2. (4) What is the minimum energy process necessary to put it back? Example processes: A. Adiabatic Compression followed by Thermal Energy Transfer B. Cooling to 0 K, Compression, Heating back to original T P V1V1 P V2V2

Physics 207: Lecture 26, Pg 18 Exercises Free Expansion and the 2 nd Law What is the minimum energy process necessary to put it back? Try: B. Cooling to 0 K, Compression, Heating back to original T Q 1 = n C v  T out and put it where…??? Need to store it in a low T reservoir and 0 K doesn’t exist Need to extract it later…from where??? Key point: Where Q goes & where it comes from are important as well. P V1V1 P V2V2

Physics 207: Lecture 26, Pg 19 Modeling entropy l I have a two boxes. One with fifty pennies. The other has none. I flip each penny and, if the coin toss yields heads it stays put. If the toss is “tails” the penny moves to the next box. l On average how many pennies will move to the empty box?

Physics 207: Lecture 26, Pg 20 Modeling entropy l I have a two boxes, with 25 pennies in each. I flip each penny and, if the coin toss yields heads it stays put. If the toss is “tails” the penny moves to the next box.  On average how many pennies will move to the other box?  What are the chances that all of the pennies will wind up in one box?

Physics 207: Lecture 26, Pg 21 2 nd Law of Thermodynamics l Second law: “The entropy of an isolated system never decreases. It can only increase, or, in equilibrium, remain constant.” l The 2 nd Law tells us how collisions move a system toward equilibrium. l Order turns into disorder and randomness. l With time thermal energy will always transfer from the hotter to the colder system, never from colder to hotter. l The laws of probability dictate that a system will evolve towards the most probable and most random macroscopic state Entropy measures the probability that a macroscopic state will occur or, equivalently, it measures the amount of disorder in a system Increasing Entropy

Physics 207: Lecture 26, Pg 22 Entropy l Two identical boxes each contain 1,000,000 molecules. In box A, 750,000 molecules happen to be in the left half of the box while 250,000 are in the right half. In box B, 499,900 molecules happen to be in the left half of the box while 500,100 are in the right half. l At this instant of time:  The entropy of box A is larger than the entropy of box B.  The entropy of box A is equal to the entropy of box B.  The entropy of box A is smaller than the entropy of box B.

Physics 207: Lecture 26, Pg 23 Entropy l Two identical boxes each contain 1,000,000 molecules. In box A, 750,000 molecules happen to be in the left half of the box while 250,000 are in the right half. In box B, 499,900 molecules happen to be in the left half of the box while 500,100 are in the right half. l At this instant of time:  The entropy of box A is larger than the entropy of box B.  The entropy of box A is equal to the entropy of box B.  The entropy of box A is smaller than the entropy of box B.

Physics 207: Lecture 26, Pg 24 Reversible vs Irreversible l The following conditions should be met to make a process perfectly reversible: 1. Any mechanical interactions taking place in the process should be frictionless. 2. Any thermal interactions taking place in the process should occur across infinitesimal temperature or pressure gradients (i.e. the system should always be close to equilibrium.) l Based on the above answers, which of the following processes are not reversible? 1. Melting of ice in an insulated (adiabatic) ice-water mixture at 0°C. 2. Lowering a frictionless piston in a cylinder by placing a bag of sand on top of the piston. 3. Lifting the piston described in the previous statement by removing one grain of sand at a time. 4. Freezing water originally at 5°C.

Physics 207: Lecture 26, Pg 25 Reversible vs Irreversible l The following conditions should be met to make a process perfectly reversible: 1. Any mechanical interactions taking place in the process should be frictionless. 2. Any thermal interactions taking place in the process should occur across infinitesimal temperature or pressure gradients (i.e. the system should always be close to equilibrium.) l Based on the above answers, which of the following processes are not reversible? 1. Melting of ice in an insulated (adiabatic) ice-water mixture at 0°C. 2. Lowering a frictionless piston in a cylinder by placing a bag of sand on top of the piston. 3. Lifting the piston described in the previous statement by removing one grain of sand at a time. 4. Freezing water originally at 5°C.

Physics 207: Lecture 26, Pg 26 Exercise l A piston contains two chambers with an impermeable but movable barrier between them. On the left is 1 mole of an ideal gas at 200 K and 1 atm of pressure. On the right is 2 moles of another ideal gas at 400 K and 2 atm of pressure. The barrier is free to move and heat can be conducted through the barrier. If this system is well insulated (isolated from the outside world) what will the temperature and pressure be at equilibrium? p,T,V R p,T,V L

Physics 207: Lecture 26, Pg 27 Exercise l If this system is well insulated (isolated from the outside world) what will the temperature and pressure be at equilibrium? At equilibrium both temperature and pressure are the same on both sides.  E Th(Left) +  E Th(Right) = 0 1 x 3/2 R (T-200 K) + 2 x 3/2 R (T-400 K) = 0 (T-200 K) + 2 (T-400 K) = 0 3T = 1000 K T=333 K Now for p….notice P/T = const. = n R / V n L R / V L = n R R / V R n L V R = n R V L V R = 2 V L

Physics 207: Lecture 26, Pg 28 Exercise l If this a system is well insulated (isolated from the outside world) what will the temperature and pressure be at equilibrium? V R = 2 V L and V R + V L = V initial = (1 x 8.3 x 200 / x 8.3 x 400 / 2x10 5 ) V initial = m 3 V R =0.033 m 3 V L = m 3 P R = n R RT / V R = 2 x 8.3 x 333 / = 1.7 atm P l = n L RT / V l = 1 x 8.3 x 333 / = 1.7 atm

Physics 207: Lecture 26, Pg 29 Lecture 26 Assignment rehash Assignment rehash  HW11, Due Tuesday May 5 th  For Tuesday, read through all of Chapter 19!