Lecture 25 Goals: Chapter 18 Understand the molecular basis for pressure and the idealgas


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1 Lecture 5 Goals: Chapter 18 Understand the molecular basis for pressure and the idealgas law. redict 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 nd law of thermodynamics Assignment HW1, Due Tuesday, May 4 th For this Tuesday, Read through all of Chapter 19 hysics 07: Lecture 6, g 1 Macromicro connection Mean Free ath If a molecule, with radius r, averages n collisions as it travels distance L, then the average distance between collisions is L/n, and is called the mean free path The mean free path is independent of temperature The mean time between collisions is temperature dependent hysics 07: Lecture 6, g age 1
2 Some typical numbers The mean free path is Vacuum ressure (a) Molecules / cm 3 Molecules/ m 3 mean free path Ambient pressure * * x 109 m Medium vacuum mm Ultra High vacuum km hysics 07: Lecture 6, g 3 Distribution of Molecular Speeds A MaxwellBoltzmann Distribution 1.4 O at 5 C 1. # Molecules O at 1000 C Fermi Chopper 1 Most probable Mean (Average) Molecular Speed (m/s) hysics 07: Lecture 6, g 4 age
3 Macromicro connection Assumptions for ideal gas: # of molecules N is large They obey Newton s laws Shortrange interactions with elastic collisions Elastic collisions with walls (an impulse..pressure) What we call temperature T is a direct measure of the average translational kinetic energy What we call pressure p is a direct measure of the number density of molecules, and how fast they are moving (v rms ) Relationship between Average Energy per Molecule & Temperature hysics 07: Lecture 6, g 5 Macromicro connection One new relationship 3 k B T = ε avg T = ε 3 k B avg p = 3 pv = Nk B T N V ε avg E = ½ m v v rms = ( v ) avg = 3k BT m hysics 07: Lecture 6, g 6 age 3
4 Kinetic energy of a gas 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, K 1? (A) T 1 = 0 C (B) T 1 = 93 C (C) T 1 = 100 C 1 mv 3 = kbt = ε avg The molecules in an ideal gas at 10 C have a rootmea nsquare (rms) speed v rms. At what temperature T (in degrees Celsius) will the molecules have twice the rms speed, v rms? (A) T = 859 C (B) T = 0 C (C) T = 786 C hysics 07: Lecture 6, g 7 Consider a fixed volume of ideal gas. When N or T is doubled the pressure increases by a factor of. 1 mv = 3 k B T 1. If T is doubled, what happens to the rate at which a single molecule in the gas has a wall bounce (i.e., how does v vary)? (A) x1.4 (B) x (C) x4.. If N is doubled, what happens to the rate at which a single molecule in the gas has a wall bounce? (A) x1 Exercise pv (B) x1.4 = Nk B T (C) x hysics 07: Lecture 6, g 8 age 4
5 A macroscopic example of the equipartition theorem Imagine a cylinder with a piston held in place by a spring. Inside the piston is an ideal gas a 0 K. What is the pressure? What is the volume? Let U spring =0 (at equilibrium distance) What will happen if I have thermal energy transfer? The gas will expand (pv = nrt) The gas will do work on the spring Conservation of energy Q = ½ k x + 3/ n R T (spring & gas) and Newton Σ F piston = 0 = pa kx kx =pa Q = ½ (pa) x + 3/ n RT Q = ½ p V + 3/ n RT (but pv = nrt) Q = ½ nrt + 3/ n RT (5% of Q went to the spring) ½ nrt per degree of freedom +Q hysics 07: Lecture 6, g 9 Degrees of freedom or modes 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 Rotational for a diatomic gas (rotation about x, y, z axes, energy stored is only kinetic) Vibrational for a diatomic gas (two atoms joined by a springlike molecular bond vibrate back and forth, both potential and kinetic energy are stored in this vibration) In a solid, each atom has microscopic translational kinetic energy and microscopic potential energy along all three axes. hysics 07: Lecture 6, g 10 age 5
6 Degrees of freedom or modes A monoatomic gas only has 3 degrees of freedom (x, y, z to give K, kinetic) A typical diatomic gas has 5 accessible degrees of freedom at room temperature, 3 translational (K) and rotational (K) At high temperatures there are two more, vibrational with K and U to give 7 total A monomolecular solid has 6 degrees of freedom 3 translational (K), 3 vibrational (U) hysics 07: Lecture 6, g 11 The Equipartition Theorem 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. The thermal energy of each degree of freedom is: E th = ½ Nk B T = ½ nrt A monoatomic gas has 3 degrees of freedom A diatomic gas has 5 degrees of freedom 5 E th = nrt 3 E th = nrt A solid has 6 degrees of freedom Molar specific heats can be predicted from the thermal energy, because Monoatomic gas Diatomic gas Elemental solid E th = nc T E th = 3nRT 3 5 C V = nrt C nrt V = C V = 3nRT hysics 07: Lecture 6, g 1 age 6
7 Exercise A gas at temperature T is an equal mixture of hydrogen and helium gas. Which atoms have more KE (on average)? (A) H (B) He (C) Both have same KE How many degrees of freedom in a 1D simple harmonic oscillator? (A) 1 (B) (C) 3 (D) 4 (E) Some other number hysics 07: Lecture 6, g 13 The need for something else: Entropy V 1 V 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. (1) How much work was done by the system? () What is the final temperature (T )? (3) Can the partition be reinstalled with all of the gas molecules back in V 1? hysics 07: Lecture 6, g 14 age 7
8 V 1 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. (1) How much work was done by the system? V (A) W > 0 (B) W =0 (C) W < 0 hysics 07: Lecture 6, g 15 V 1 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. () What is the final temperature (T )? V (A) T > T 1 (B) T = T 1 (C) T < T 1 hysics 07: Lecture 6, g 16 age 8
9 V 1 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. (3) Can the partition be reinstalled with all of the gas molecules back in V 1 V (4) What is the minimum process necessary to put it back? hysics 07: Lecture 6, g 17 V 1 V 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. (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 hysics 07: Lecture 6, g 18 age 9
10 V 1 V Exercises Free Expansion and the 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. hysics 07: Lecture 6, g 19 Modeling entropy 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. On average how many pennies will move to the empty box? hysics 07: Lecture 6, g 0 age 10
11 Modeling entropy I have a two boxes, with 5 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? hysics 07: Lecture 6, g 1 nd Law of Thermodynamics Second law: The entropy of an isolated system never decreases. It can only increase, or, in equilibrium, remain constant. Increasing Entropy Entropy measures the probability that a macroscopic state will occur or, equivalently, it measures the amount of disorder in a system The nd Law tells us how collisions move a system toward equilibrium. Order turns into disorder and randomness. With time thermal energy will always transfer from the hotter to the colder system, never from colder to hotter. The laws of probability dictate that a system will evolve towards the most probable and most random macroscopic state hysics 07: Lecture 6, g age 11
12 Entropy 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 50,000 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. 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. hysics 07: Lecture 6, g 3 Entropy 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 50,000 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. 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. hysics 07: Lecture 6, g 4 age 1
13 Reversible vs Irreversible The following conditions should be met to make a process perfectly reversible: 1. Any mechanical interactions taking place in the process should be frictionless.. 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.) Based on the above answers, which of the following processes are not reversible? 1. Melting of ice in an insulated (adiabatic) icewater mixture at 0 C.. 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 slowly removing one molecule at a time. 4. Freezing water originally at 5 C. hysics 07: Lecture 6, g 5 Reversible vs Irreversible The following conditions should be met to make a process perfectly reversible: 1. Any mechanical interactions taking place in the process should be frictionless.. 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.) Based on the above answers, which of the following processes are not reversible? 1. Melting of ice in an insulated (adiabatic) icewater mixture at 0 C.. 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. hysics 07: Lecture 6, g 6 age 13
14 Exercise A piston contains two chambers with an impermeable but movable barrier between them. On the left is 1 mole of an ideal gas at 00 K and 1 atm of pressure. On the right is moles of another ideal gas at 400 K and 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 L p,t,v R hysics 07: Lecture 6, g 7 Exercise 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/ R (T00 K) + x 3/ R (T400 K) = 0 (T00 K) + (T400 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 = V L hysics 07: Lecture 6, g 8 age 14
15 Exercise If this a system is well insulated (isolated from the outside world) what will the temperature and pressure be at equilibrium? V R = V L and V R + V L = V initial = (1 x 8.3 x 00 / x 8.3 x 400 / x10 5 ) V initial = m 3 V R =0.033 m 3 V L = m 3 R = n R RT / V R = x 8.3 x 333 / = 1.7 atm l = n L RT / V l = 1 x 8.3 x 333 / = 1.7 atm hysics 07: Lecture 6, g 9 Lecture 6 To recap: HW1, Due Tuesday May 4 th For this Tuesday, read through all of Chapter 19! hysics 07: Lecture 6, g 30 age 15
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