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Chapter 16. Thermal Properties of Matter. Macroscopic Description of Matter. State Variables. State variable = macroscopic property of thermodynamic system Examples: pressure p volume V temperature T mass m. State Variables.
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Chapter 16 Thermal Propertiesof Matter
State Variables • State variable = macroscopic property of thermodynamic system • Examples: pressure p volume V temperature T mass m
State Variables • State variables: p, V, T, m • I general, we cannot change one variable without affecting a change in the others • Recall: For a gas, we defined temperature T (in kelvins) using the gas pressure p
Equation of State • State variables: p, V, T, m • The relationship among these:‘equation of state’ • sometimes: an algebraic equation exists • often: just numerical data
Equation of State • Warm-up example: • Approximate equation of state for a solid • Based on concepts we already developed • Here: state variables are p, V, T Derive the equation of state
The ‘Ideal’ Gas • The state variables of a gas are easy to study: • p, V, T, mgas • often use: n = number of ‘moles’ instead of mgas
Moles and Avogadro’s Number NA • 1 mole = 1 mol = 6.02×1023 molecules = NA molecules • n = number of moles of gas • M = mass of 1 mole of gas • mgas = n M Do Exercise 16-53
The ‘Ideal’ Gas • We measure: the state variables (p, V, T, n) for many different gases • We find: at low density, all gases obey the same equation of state!
Ideal Gas Equation of State • State variables: p, V, T, n pV = nRT • p = absolute pressure (not gauge pressure!) • T = absolute temperature (in kelvins!) • n = number of moles of gas
Ideal Gas Equation of State • State variables: p, V, T, n pV = nRT • R = 8.3145 J/(mol·K) • same value of R for all (low density) gases • same (simple, ‘ideal’) equation Do Exercises 16-9, 16-12
Ideal Gas Equation of State • State variables: p, V, T, and mgas= nM • State variables: p, V, T, and r = mgas/V Derive ‘Law of Atmospheres’
Non-Ideal Gases? • Ideal gas equation: • Van der Waals equation: Notes
Ideal Gas Equation pV = nRT • n = number of moles of gas = N/NA • R = 8.3145 J/(mol·K) • N = number of molecules of gas • NA = 6.02×1023 molecules/mol
Ideal Gas Equation • k = Boltzmann constant = R/NA = 1.381×10-23 J/(molecule·K)
Ideal Gas Equation pV = nRT pV = NkT • k = R/NA • ‘ RT per mol’ vs. ‘kT per molecule’
Assumptions • gas = large number N of identical molecules • molecule = point particle, mass m • molecules collide with container walls= origin of macroscopic pressure of gas
Kinetic Model • molecules collide with container walls • assume perfectly elastic collisions • walls are infinitely massive (no recoil)
Elastic Collision • wall: infinitely massive, doesn’t recoil • molecule: vy: unchanged vx : reverses direction speed v : unchanged
Kinetic Model • For one molecule: v2 = vx2 + vy2 + vz2 • Each molecule has a different speed • Consider averaging over all molecules
Kinetic Model • average over all molecules: (v2)av= (vx2 + vy2 + vz2)av = (vx2)av+(vy2)av+(vz2)av = 3 (vx2)av
Kinetic Model • (Ktr)av= total kinetic energy of gas due to translation • Derive result:
Kinetic Model • Compare to ideal gas law: pV = nRT pV = NkT
Kinetic Energy • average translational KE is directly proportional to gas temperature T
Kinetic Energy • average translational KE per molecule: • average translational KE per mole:
Kinetic Energy • average translational KE per molecule: • independent of p, V, and kind of molecule • for same T, all molecules (any m) have the same average translational KE
Kinetic Model • ‘root-mean-square’ speed vrms:
Molecular Speeds • For a given T, lighter molecules move faster • Explains why Earth’s atmosphere contains alomost no hydrogen, only heavier gases
Molecular Speeds • Each molecule has a different speed, v • We averaged over all molecules • Can calculate the speed distribution, f(v)(but we’ll just quote the result)
Molecular Speeds f(v) = distribution function f(v) dv = probability a molecule has speed between v and v+dv dN = number of molecules with speed between v and v+dv = N f(v) dv
Molecular Speeds • Maxwell-Boltzmann distribution function
Molecular Speeds • At higher T:more molecules have higher speeds • Area under f(v) = fraction of molecules with speeds in range: v1 < v < v1 or v > vA
Molecular Speeds • average speed • rms speed
Molecular Collisions? • We assumed: • molecules = point particles, no collisions • Real gas molecules: • have finite size and collide • Find ‘mean free path’ between collisions
Molecular Collisions • Mean free path between collisions:
Announcements • Midterms: • Returned at end of class • Scores will be entered on classweb soon • Solutions available online at E-Res soon • Homework 7 (Ch. 16): on webpage • Homework 8 (Ch. 17): to appear soon
Heat Capacity Revisited DQ = energy required to change temperature of mass m by DT c = ‘specific heat capacity’ = energy required per (unit mass × unit DT)
Heat Capacity Revisited • Now introduce ‘molar heat capacity’ C C = energy per (mol × unit DT) required to change temperature of n moles by DT
Heat Capacity Revisited • important case:the volume V of material is held constant • CV = molar heat capacity at constant volume
CV for the Ideal Gas • Monatomic gas: • molecules = pointlike(studied last lecture) • recall: translational KE of gas averaged over all molecules (Ktr)av = (3/2) nRT
CV for the Ideal Gas • Monatomic gas: (Ktr)av = (3/2) nRT • note: your text just writesKtr instead of (Ktr)av • Consider changing T by dT
CV for the Ideal Gas • Monatomic gas: (Ktr)av = (3/2) nRT d(Ktr)av = n (3/2)R dT • recall: dQ = n CV dT • so identify: CV= (3/2)R
In General: If (Etot)av = (f/2) nRT Then d(Etot)av = n (f/2)R dT But recall: dQ = n CV dT So we identify: CV= (f/2)R
A Look Ahead (Etot)av = (f/2) nRT CV= (f/2)R Monatomic gas:f = 3 Diatomic gas:f = 3, 5, 7