Physics 313
Professor Lee Carkner
Lecture 25

 Final is Tuesday, May 18, 9am
 75 minutes worth of chapters 9-12
 45 minutes worth of chapters 1-8
 Same format as other tests (multiple choice and short answer)
 Worth 20% of grade
 Three formula sheets given on test (one for
Ch 9-12 and previous two)
 Bring pencil and calculator

 Set escape velocity equal to maximum
Maxwell velocity
 (2GM/R) ½ = 10(3kT/m) ½
 m = (150KTR/GM)
 Planetary atmospheres
 Earth: m > 9.5X10
-27 kg (NH
3
, O
2
)
 Jupiter: m > 1.4X10
-28 kg (He, NH
3
, O
2
)
 Titan: m > 5.6X10
-26 kg (None)
 Moon: m > 2.2X10
-25 kg (None)

Two identical metal blocks, one at 100 C and one at 120 C, are placed together. Which transfers the most heat?
 Two objects at different temperatures will exchange heat until they are at the same temperature
 Zeroth Law: Two systems in thermal equilibrium with a third are in thermal equilibrium with each other

 Heat:
Q = mc D T = mc(T f
-T i
)
 Conduction: dQ/dt = -KA(dT/dx)
 Radiation
Q/t = -KA(T
1
-T
2
)/x dQ/dt = A es (T env
4 -T 4 )

How would you make a tube of mercury into a
Celsius thermometer? A Kelvin thermometer?
 Thermometers defined by the triple point of water
 A system at constant temperature can have a range of values for the other variables
 Isotherm

 Thermometers
T (X) = 273.16 (X/X
TP
)
 Temperature scales
T (R) = T (F) + 459.67
T (K) = T (C) + 273.15
T (R) = (9/5) T (K)
T (F) = (9/5) T (C) +32

If the temperature of an ideal gas is doubled while the volume stays the same, what happens to the pressure?
 Equation of state detail how properties change with temperature
 Increasing T will generally increase the force and displacement terms

 General Relations: dx = (  x/  y) z dy + (  x/  z) y dz
(  x/  y) z
(  x/  y) z
 Specific Relations:
= 1/(  y/  x)
(  y/  z) x
(  z/  x) z y
= -1
 Volume Expansivity: b = (1/V)(dV/dT)
P
 Isothermal Compressibility: k =-(1/V)(dV/dP)
T
 Linear Expansivity: a = (1/L)(dL/dT)
 Young’s modulus: Y = (L/A)(d t /dL)
T t

How much work is done in an isobaric compression of a gas at 1 Pa from 2 to 1 m 3 ?
 The work done a system is the product of a force term and a displacement term
 No displacement, no work
 Compression is positive, expansion is negative
 Work is area under PV (or XY) curve
 Work is path dependant

dW = -PdV
W =  PdV
 For ideal gas P = nRT/V
 Examples:
 Isothermal ideal gas:
W = -nRT  (1/V) dV = -nRT ln (V f
/V i
)
 Isobaric ideal gas:
W = -P  dV = -P(V f
-V i
)

Rank the following processes in order of increasing internal energy:
Adiabatic compression
Isothermal expansion
Isochoric cooling
 Energy is conserved
 Internal energy is a state function, work and heat are not

D U = U f
-U i
= Q+W dU = dQ +dW dU = CdT - PdV

 If the volume of an ideal gas is doubled and the pressure is tripled isothermally, how does the internal energy change?
lim (PV) = nRT
(dU/dP)
T
(dU/dT)
V dQ = C
V
C
P
= (dU/dV)
T
= C
V
= C
+ nR dT+PdV = C
P
V
= 0 dT-VdP

 Can an adiabatic process keep constant
P, V, or T?
PV g = const
TV g -1 = const
T/P ( g -1)/ g = const
W = (P f
V f
- P i
V i
)/ g -1

 If the rms velocity of gas molecules doubles what happens to the temperature and internal energy
(1/2)mv 2 = (3/2)kT
U = (3/2)NkT
T = mv 2 /3k

 If the heat entering an engine is doubled and the work stays the same what happens to the efficiency?
 Engines are cycles
 Change in internal energy is zero
 Composed of 4 processes h = W/Q
H
= (Q
H
Q
H
-Q
L
)/Q
H
= W + Q
L
= 1 - Q
L
/Q
H

 Otto
 Adiabatic, Isochoric h = 1 - (T
1
/T
2
)
 Diesel
 Adiabatic, isochoric, isobaric h = 1 - (1/ g )(T
4
-T
1
)/(T
3
-T
2
)
 Rankine (steam)
 Adiabatic, isobaric
 Stirling
 Isothermal, isochoric

 Transfer heat from low to high T with the addition of work
 Operates in cycle
 Transfers heat with evaporation and condensation at different pressures
K = Q
L
/W
K = Q
L
/(Q
H
-Q
L
)

 Is an ice cube melting at room temperature a reversible process?
 Kelvin-Planck
 Cannot convert heat completely into work
 Clausius
 Cannot move heat from low to high temperature without work

 What two processes make up a Carnot cycle? How many temperatures is heat transferred at?
 Adiabatic and isothermal h = 1 - T
L
 Most efficient cycle
/T
H
 Efficiency depends only on the temperature

 The second law of thermodynamics can be stated:
 Engine cannot turn heat completely into work
 Heat cannot move from low to high temperatures without work
 Efficiency cannot exceed Carnot efficiency
 Entropy always increases

 Entropy change is zero for all reversible processes
 All real processes are irreversible
 Can compute entropy for an irreversible process by replacing it with a reversible process that achieves the same result
 Entropy change of system + entropy change of surroundings = entropy change of universe
(which is > 0)

 Can integrate dS to find D S dS = dQ/T
D S =  dQ/T (integrated from T i
 Examples: to T
 Heat reservoir (or isothermal process)
D S = Q/T
 Isobaric
D S = C
P ln (T f
/T i
) f
)

 Can plot phases and phase boundaries on a
PV, PT and PTV diagram
 Saturation
 condition where substance can change phase
 Critical point
 above which substance can only be gas
 where ( d P/ d V) =0 and ( d 2 P/ d V 2 ) = 0
 Triple point
 where fusion, sublimation and vaporization curves intersect

c c
P
V
= (dQ/dT)
= (dQ/dT)
P
(per mole)
T
(per mole) b = (1/V)(dV/dT) k = -(1/V)(dV/dP)
 c
P
, c
V temperature and then level off
P
T and b are 0 at 0 K and rise sharply to the Debye
 c
P and c
V end up near the Dulong and Petit value of 3R
 k is constant at a finite value at low T and then increases linearly

 Legendre Transform: df = udx +vdy g= f-ux dg = -xdu+vdy
 Useful theorems:
( d x/ d y) z
( d x/ d y)
( d y/ d z) x f
( d y/ d z)
( d z/ d x) y
=-1 f
( d z/ d x) f
=1 dU = -PdV +T dS dH = VdP +TdS dA = - SdT - PdV dG = V dP - S dT
( d T/ d V)
S
( d T/ d P)
( d S/ d V)
S
T
( d S/ d P)
T
= - ( d P/ d S)
V
= ( d V/ d S)
P
= ( d P/ d T)
V
= -( d V/ d T)
P

 Entropy
T dS = C
V
T dS = C
P
 Internal Energy dT + T ( d P/ d T)
V dT - T( d V/ d T)
P dV dP
( d U/ d V)
( d U/ d P)
T
 Heat Capacity
T
= T ( d P/ d T)
V
= -T ( d V/ d T)
P
- P( d
- P
V/ d P)
T
C
P
- C
V
= -T( d V/ d T) c
P
- c
V
P
2 (
= Tv b 2 / k d P/ d V)
T

 Can plot on PT diagram
 Isenthalpic curves show possible final states for an initial state m = (1/c
P
- v] = slope
 Inversion curve separates heating and cooling region
P
)[T(dv/dT) m = 0
 Total enthalpy before and after throttling is the same
 For liquefaction: h i
= yh
L
+ (1-y)h f

 Any first order phase change obeys:
(dP/dT) = (s f -s i )/(v f - v i )
= (h f - h i )/T (v f -v i )
 dP/dT is slope of phase boundary in PT diagram
 Can change dP/dT to D P/ D T for small changes in P and T

 For a steady flow open systems mass and energy are conserved:
S in
S m in
= S m out
[Q + W + m q ] = S out
[Q + W + m
 Where q is energy per unit mass or: q ] q = h + ke +pe (per unit mass)
 Chemical potential = m = ( d U/ d n) m i
= m f
 For open systems in equilibrium: