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Thermal Properties of Matter
Chapter 16 Thermal Properties of Matter
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Macroscopic Description of Matter
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State Variables State variable = macroscopic property of thermodynamic system Examples: pressure p volume V temperature T mass m
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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
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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
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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
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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
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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
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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!
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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
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Ideal Gas Equation of State
State variables: p, V, T, n pV = nRT R = J/(mol·K) same value of R for all (low density) gases same (simple, ‘ideal’) equation Do Exercises 16-9, 16-12
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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’
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Non-Ideal Gases? Ideal gas equation: Van der Waals equation: Notes
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pV–Diagram for an Ideal Gas
Notes
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pV–Diagram for a Non-Ideal Gas
Notes
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Microscopic Description of Matter
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Ideal Gas Equation pV = nRT n = number of moles of gas = N/NA
R = J/(mol·K) N = number of molecules of gas NA = 6.02×1023 molecules/mol
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Ideal Gas Equation k = Boltzmann constant = R/NA = 1.381×10-23 J/(molecule·K)
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Ideal Gas Equation pV = nRT pV = NkT k = R/NA
‘ RT per mol’ vs. ‘kT per molecule’
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Kinetic-Molecular Theory of an Ideal Gas
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Assumptions gas = large number N of identical molecules
molecule = point particle, mass m molecules collide with container walls = origin of macroscopic pressure of gas
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Kinetic Model molecules collide with container walls
assume perfectly elastic collisions walls are infinitely massive (no recoil)
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Elastic Collision wall: infinitely massive, doesn’t recoil molecule:
vy: unchanged vx : reverses direction speed v : unchanged
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Kinetic Model For one molecule: v2 = vx2 + vy2 + vz2
Each molecule has a different speed Consider averaging over all molecules
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Kinetic Model average over all molecules: (v2)av= (vx2 + vy2 + vz2)av
= (vx2)av+(vy2)av+(vz2)av = 3 (vx2)av
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Kinetic Model (Ktr)av= total kinetic energy of gas due to translation
Derive result:
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Kinetic Model Compare to ideal gas law: pV = nRT pV = NkT
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Kinetic Energy average translational KE is directly proportional to gas temperature T
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Kinetic Energy average translational KE per molecule:
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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
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Kinetic Model ‘root-mean-square’ speed vrms:
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Molecular Speeds For a given T, lighter molecules move faster
Explains why Earth’s atmosphere contains alomost no hydrogen, only heavier gases
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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)
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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
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Molecular Speeds Maxwell-Boltzmann distribution function
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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
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Molecular Speeds average speed rms speed
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Molecular Collisions? We assumed:
molecules = point particles, no collisions Real gas molecules: have finite size and collide Find ‘mean free path’ between collisions
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Molecular Collisions
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Molecular Collisions Mean free path between collisions:
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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
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Heat Capacity Revisited
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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)
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Heat Capacity Revisited
Now introduce ‘molar heat capacity’ C C = energy per (mol × unit DT) required to change temperature of n moles by DT
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Heat Capacity Revisited
important case: the volume V of material is held constant CV = molar heat capacity at constant volume
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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
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CV for the Ideal Gas Monatomic gas: (Ktr)av = (3/2) nRT
note: your text just writes Ktr instead of (Ktr)av Consider changing T by dT
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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
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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
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A Look Ahead (Etot)av = (f/2) nRT CV = (f/2)R Monatomic gas: f = 3
Diatomic gas: f = 3, 5, 7
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CV for the Ideal Gas What about gases with other kinds of molecules?
diatomic, triatomic, etc. These molecules are not pointlike
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CV for the Ideal Gas Diatomic gas: molecules = ‘dumbell’ shape
its energy takes several forms: (a) translational KE (3 directions) (b) rotational KE (2 rotation axes) (c) vibrational KE and PE Demonstration
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(Etot)av = (Ktr)av + (Krot)av + (Evib)av
CV for the Ideal Gas Diatomic gas: Etot = Ktr + Krot + Evib (Etot)av = (Ktr)av + (Krot)av + (Evib)av we know: (Ktr)av = (3/2) nRT what about the other terms?
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Equipartition of Energy
Can be proved, but we’ll just use the result Define: f = number of degrees of freedom = number of independent ways that a molecule can store energy
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Equipartition of Energy
It can be shown: The average amount of energy in each degree of freedom is: (1/2) kT per molecule i.e. (1/2) RT per mole
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Check a known case Monatomic gas:
only has translational KE in 3 directions: vx, vy, vz f = 3 degrees of freedom (Ktr)av = (f/2) nRT = (3/2) nRT
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CV for the Ideal Gas Diatomic gas:
more forms of energy are available to the gas as you increase its T: (a) translational KE (3 directions) (b) rotational KE (2 rotation axes) (c) vibrational KE and PE
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A Look Ahead (Etot)av = (f/2) nRT CV = (f/2)R Monatomic gas: f = 3
Diatomic gas: f = 3, 5, 7
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CV for the Ideal Gas Diatomic gas: low temperature
only translational KE in 3 directions: vx, vy, vz f = 3 degrees of freedom (Etot)av = (f/2) nRT = (3/2) nRT
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CV for the Ideal Gas Diatomic gas: higher temperature
translational KE (in 3 directions) rotational KE (about 2 axes) f = 3+2 = 5 degrees of freedom (Etot)av = (f/2) nRT = (5/2) nRT
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CV for the Ideal Gas Diatomic gas: even higher temperature
translational KE (in 3 directions) rotational KE (about 2 axes) vibrational KE and PE f = =7 degrees of freedom (Etot)av = (f/2) nRT = (7/2) nRT
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Summary of CV for Ideal Gases
(Etot)av = (f/2) nRT CV = (f/2)R Monatomic: f = 3 (only) Diatomic: f = 3, 5, 7 (with increasing T)
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CV for Solids Each atom in a solid can vibrate about its equilibrium position Atoms undergo simple harmonic motion in all 3 directions
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CV for Solids Kinetic energy : 3 degrees of freedom K = Kx+ Ky + Kz
Kx = (1/2) mvx2 Ky = (1/2) mvy2 Kz = (1/2) mvz2
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CV for Solids Potential energy: 3 degrees of freedom U = Ux+ Uy + Uz
Ux = (1/2) kx x2 Uy = (1/2) ky y2 Uz = (1/2) kz z2
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CV for Solids f = 3 + 3 = 6 degrees of freedom (Etot)av = (f/2) nRT
CV = (f/2)R = 3 R
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Phase Changes Revisited
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Phase Changes ‘phase’ = state of matter = solid, liquid, vapor
during a phase transition : 2 phases coexist at the triple point : all 3 phases coexist
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Do Exercise 16-39 pT Phase Diagram
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pV–Diagram for a Non-Ideal Gas
Notes
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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
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