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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

Chapter 16

Thermal Propertiesof Matter

state variables
State Variables
  • State variable = macroscopic property of thermodynamic system
  • Examples: pressure p

volume V

temperature T

mass m

state variables1
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
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 state1
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 ‘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 n a
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 gas1
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
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 state1
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 state2
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
Non-Ideal Gases?
  • Ideal gas equation:
  • Van der Waals equation:

Notes

ideal gas equation
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 equation1
Ideal Gas Equation
  • k = Boltzmann constant = R/NA = 1.381×10-23 J/(molecule·K)
ideal gas equation2
Ideal Gas Equation

pV = nRT

pV = NkT

  • k = R/NA
  • ‘ RT per mol’ vs. ‘kT per molecule’
assumptions
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
Kinetic Model
  • molecules collide with container walls
  • assume perfectly elastic collisions
  • walls are infinitely massive (no recoil)
elastic collision
Elastic Collision
  • wall:

infinitely massive, doesn’t recoil

  • molecule:

vy: unchanged

vx : reverses direction

speed v : unchanged

kinetic model1
Kinetic Model
  • For one molecule: v2 = vx2 + vy2 + vz2
  • Each molecule has a different speed
  • Consider averaging over all molecules
kinetic model2
Kinetic Model
  • average over all molecules:

(v2)av= (vx2 + vy2 + vz2)av

= (vx2)av+(vy2)av+(vz2)av

= 3 (vx2)av

kinetic model3
Kinetic Model
  • (Ktr)av= total kinetic energy of gas due to translation
  • Derive result:
kinetic model4
Kinetic Model
  • Compare to ideal gas law:

pV = nRT

pV = NkT

kinetic energy
Kinetic Energy
  • average translational KE is directly proportional to gas temperature T
kinetic energy1
Kinetic Energy
  • average translational KE per molecule:
  • average translational KE per mole:
kinetic energy2
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 model5
Kinetic Model
  • ‘root-mean-square’ speed vrms:
molecular speeds
Molecular Speeds
  • For a given T, lighter molecules move faster
  • Explains why Earth’s atmosphere contains alomost no hydrogen, only heavier gases
molecular speeds1
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 speeds2
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 speeds3
Molecular Speeds
  • Maxwell-Boltzmann distribution function
molecular speeds4
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 speeds5
Molecular Speeds
  • average speed
  • rms speed
molecular collisions
Molecular Collisions?
  • We assumed:
  • molecules = point particles, no collisions
  • Real gas molecules:
  • have finite size and collide
  • Find ‘mean free path’ between collisions
molecular collisions2
Molecular Collisions
  • Mean free path between collisions:
announcements
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 revisited1
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 revisited2
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 revisited3
Heat Capacity Revisited
  • important case:the volume V of material is held constant
  • CV = molar heat capacity at constant volume
c v for the ideal gas
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

c v for the ideal gas1
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
c v for the ideal gas2
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
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
A Look Ahead

(Etot)av = (f/2) nRT

CV= (f/2)R

Monatomic gas:f = 3

Diatomic gas:f = 3, 5, 7

c v for the ideal gas3
CV for the Ideal Gas
  • What about gases with other kinds of molecules?
  • diatomic, triatomic, etc.
  • These molecules are not pointlike
c v for the ideal gas4
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

c v for the ideal gas5
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?
equipartition of energy
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

equipartition of energy1
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

check a known case
Check a known case
  • Monatomic gas:
  • only has translational KEin 3 directions: vx, vy, vz
  • f = 3 degrees of freedom

(Ktr)av = (f/2) nRT = (3/2) nRT

c v for the ideal gas6
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

a look ahead1
A Look Ahead

(Etot)av = (f/2) nRT

CV= (f/2)R

Monatomic gas:f = 3

Diatomic gas:f = 3, 5, 7

c v for the ideal gas7
CV for the Ideal Gas
  • Diatomic gas:

low temperature

  • only translational KEin 3 directions: vx, vy, vz
  • f = 3 degrees of freedom

(Etot)av = (f/2) nRT = (3/2) nRT

c v for the ideal gas8
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

c v for the ideal gas9
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 = 3+2+2 =7 degrees of freedom

(Etot)av = (f/2) nRT = (7/2) nRT

summary of c v for ideal gases
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)

c v for solids
CV for Solids
  • Each atom in a solid can vibrate about its equilibrium position
  • Atoms undergo simple harmonic motion in all 3 directions
c v for solids1
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
c v for solids2
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
c v for solids3
CV for Solids
  • f = 3 + 3 = 6 degrees of freedom

(Etot)av = (f/2) nRT

= 3 nRT

CV= (f/2)R = 3 R

phase changes
Phase Changes
  • ‘phase’ = state of matter = solid, liquid, vapor
  • during a phase transition : 2 phases coexist
  • at the triple point : all 3 phases coexist
announcements1
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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