Page 1Physics 207 – Lecture 26Physics 207: Lecture 26, Pg 1Lecture 25 Goals:Goals:••Chapter 18Chapter 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 2ndlaw of thermodynamics••AssignmentAssignment HW12, Due Tuesday, May 4th For this Tuesday, Read through all of Chapter 19Physics 207: Lecture 26, Pg 2Macro-micro connectionMean Free PathIf 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 pathThe mean free path is independent of temperatureThe mean time between collisions is temperature dependentPage 2Physics 207 – Lecture 26Physics 207: Lecture 26, Pg 3The mean free path is… Some typical numbers1-105 km1015– 1011109– 10410-5-10-10Ultra High vacuum0.1 - 100 mm1022-10191016– 1013100-10-1Medium vacuum68 x 10-9m2.7*10252.7*1019105Ambient pressuremean free pathMolecules/ m3Molecules / cm3Pressure (Pa)VacuumPhysics 207: Lecture 26, Pg 4Distribution of Molecular SpeedsA “Maxwell-Boltzmann” DistributionFermi Chopper0 600 1000 1400 18002000.40.60.81.01.21.4Molecular Speed (m/s)# MoleculesO2at 1000°CO2at 25°C121 Most probable2 Mean (Average)Page 3Physics 207 – Lecture 26Physics 207: Lecture 26, Pg 5Macro-micro connection 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) 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 (vrms)Relationship between Average Energy per Molecule& TemperaturePhysics 207: Lecture 26, Pg 6Macro-micro connectionOne new relationshipavg32εVNp =avg32εBkT =mTkvvBrms3)(avg2==avg23ε=TkBE = ½m v 2pV= NkBTPage 4Physics 207 – Lecture 26Physics 207: Lecture 26, Pg 7Kinetic energy of a gas The average kinetic energy of the molecules of an ideal gas at 10°C has the value K1. At what temperature T1(in degrees Celsius) will the average kinetic energy of the same gas be twice this value, 2K1?(A) T1= 20°C(B) T1= 293°C(C) T1= 100°C The molecules in an ideal gas at 10°C have a root-mea n-square (rms) speed vrms. At what temperature T2(in degrees Celsius) will the molecules have twice the rms speed, 2vrms?(A) T2= 859°C(B) T2= 20°C(C) T2= 786°Cavg22321ε== TkvmBPhysics 207: Lecture 26, Pg 8 Consider a fixed volume of ideal gas. When N or T is doubled the pressure increases by a factor of 2.1. 1. If T is doubled, what happens to the rate at which If T is doubled, what happens to the rate at which a single a single moleculemoleculein the gas has a wall bounce (i.e., how does v vary)?in the gas has a wall bounce (i.e., how does v vary)?(B) x2(A) x1.4 (C) x422. If N is doubled, what happens to the rate at which . If N is doubled, what happens to the rate at which a a single moleculesingle moleculein the gas has a wall bounce?in the gas has a wall bounce?(B) x1.4(A) x1 (C) x2ExerciseTkNpVB=12mv2=32kBTPage 5Physics 207 – Lecture 26Physics 207: Lecture 26, Pg 9A 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 Uspring=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 x2+ 3/2 n R T (spring & gas) and Newton Σ Fpiston= 0 = pA – kx kx =pA Q = ½ (pA) x + 3/2 n RT Q = ½ p V + 3/2 n RT (but pV = nRT) Q = ½ nRT + 3/2 n RT (25% of Q went to the spring)+Q½ nRT per “degree of freedom”Physics 207: Lecture 26, Pg 10Degrees 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 spring-like 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.Page 6Physics 207 – Lecture 26Physics 207: Lecture 26, Pg 11Degrees 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 2 rotational (K)At high temperatures there are two more, vibrationalwith K and U to give 7 total A monomolecular solid has 6 degrees of freedom3 translational (K), 3 vibrational (U)Physics 207: Lecture 26, Pg 12The 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:Eth= ½ NkBT = ½ nRT A monoatomic gas has 3 degrees of freedom A diatomic gas has 5 degrees of freedom A solid has 6 degrees of freedom Molar specific heats can be predicted from the thermal energy, becausenRTEth23=nRTEth25=TnCEth∆=∆nRTCV23 gas Monoatomic=nRTCV25 gas Diatomic=nRTCV3 solid Elemental=nRTEth3=Page 7Physics 207 – Lecture 26Physics 207: Lecture 26, Pg 13Exercise 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) 2 (C) 3 (D) 4 (E) Some other numberPhysics 207: Lecture 26, Pg 14The need for something else: EntropyYou have an ideal gas in a box of volume V1. Suddenly you remove the partition and the gas now occupies a larger volume V2.(1) How much work was done by the system?(2) What is the final temperature (T2)?(3) Can the partition be reinstalled with all of the gas molecules back in V1?PV1PV2Page
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