Lecture 8
• State functions and exact differential
Ch. 2 The First Law
• Let’s consider a system undergoing changes from U(Ti, Vi) to U(Tf, Vf).
Path 1: adiabatic,
Path 2: non-adiabatic,
0
,
0
q
w
0
,
0
q w
2
1 path
path U
U
f
i dU U
(state function)
• Thus dU is an exact differential.
• In general, an exact differential is an infinitesimal quantity that, when integrated, gives a path-independent result.
• For the inexact differential, such as dq (or q) and dw (or w) , a path-dependent quantity is obtained by an integration.
f
path
i dq
q ,
• In a closed system, the internal energy can be regarded as a function of of V, T and P.
• But, only two of the variables are independent. phase rule (Ch. 6)
• Among U = U(V,T), U = U(p,T) and U = U(p,V), we can
arbitrarily choose U = U(V,T) for the purpose of our discussion.
T dT dV U
V dU U
V T
dT C
V dV
dU U V
T
same dimension with p dT
C dV
dU U
• When there are no interactions between the molecules, the U is independent of their separation (or volume).
• For an ideal gas, T = 0
• For attraction-dominant real gases, T > 0
• For repulsion-dominant real gases, T < 0
T
T V
U
Internal pressure:
• Joule observed no change in temperature in his set-up.
• No work for free expansion: w = 0
• Because the temperature of the bath did not change, no energy entered or left the gas, so q =0.
• Consequently, within accuracy of his experiment, U = 0.
T
T V
U
• The Joule experiment reveals that U does not change much when a gas expands isothermally. T = 0
• But the above equation (T = 0) is true only for ideal gases.
• The experimental error is due to the large heat capacity of the apparatus.
V p
T p
T C V T
U
dT C
dV
dU T V Dividing by dT at constant p,
T p
V
V
1
By definition, expansion coefficient
• The expansion coefficient ()is a measure of fractional change in volume when the temperature is increased by a small amount.
• A large value of means that the V of the sample responds strongly to changes in temperature.
(See Ex 2.8.)
V T
p
C T V
U
T p
V V
( )
• For a closed system, the temperature dependence of U at constant p:
• For a perfect gas, T = 0, so then:
V p
T C
U
V V
T C
U
(for only ideal gas) (in general)
• For a perfect gas,
T nR nR U
T U
T U T
pV U
T U T
C H C
p p
p p
p p
V p
• The general relation between Cp and CV is:
T V
p
C TV
C
2
where the isothermal compressibility (T) is defined as:
T
T p
V V
1
• The T is a measure of fractional change in volume when the pressure is increased by a small amount.
• The negative sign in the definition ensures that the compressibility is a positive quantity.
(See page 69 for the derivation.)
p
VC
Cp V T or
• For the perfect gas, is zero and is nR. TV pV
T pV
nR dT
dT p
nR V
T p nRT V
T V V
p p
1 1
1
1
p
VC
Cp V
T• In a closed system, the enthalpy can be also regarded as a function of of V, T and P.
• But, only two of the variables are independent. phase rule (Ch. 6)
• Among H = H(V,T), H = H(p,T) and H = H(p,V), we can
arbitrarily choose H = H(p,T) for the purpose of our discussion.
T dT dp H
p dH H
T p
dT C
p dp
dH H p
T
• Now we want to express the red-box term with recognizable
1 )
, (
relation chain
Euler The
y x
z y
z z
x x
y
y x z z
dT C
p dp
dH H p
T
dT C dp p C
T
dT C T dp
H p
dT T C dp H
T T
dH p
p p
H
p H p
p
p H
1
dT C dp C
dH p p
where the Joule-Thomson coefficient () is defined as:
p H
T
1 )
, (
relation chain
Euler The
p T
H p
H H
T T
p
T p H H
p H
T
• The Joule-Thomson coefficient () is related to the liquefaction of gases.
• How can we measure the change in temperature by changing pressure at constant enthalpy (called isenthalpic) ?
•Joule and Thomson let a gas expand
through a porous barrier from one constant pressure to another, and monitored the T by the adiabatic expansion.
• They observed a lower T on the low-p side, and the T was proportional to the p.
• This cooling by isenthalpic expansion is
• The apparatus of Joule and Thomson can be simplified to a throttling process.
• Both upstream and downstream play roles of a piston to maintain constant pressures either side of the throttle.
• Because the Joule-Thomson expansion is a adiabatic process,
ΔU w q 0 and
• The work done on the gas on the left side:
i i i
i V pV
p
w1 (0 )
• The work done on the gas on the right side:
f f f
f V p V
p
w2 ( 0)
• The total work:
f f i
iV p V
p w
w
w 1 2
f f i
i i
f U w pV p V
ΔU U
• Therefore, the Joule-Thomson expansion occurs at constant enthalpy.
• In this experiment, T/ p is measured at constant H, and by taking the limitation of small p, can be obtained.
• Currently, the is indirectly obtained by measuring the isothermal Joule-Thomson coefficient (T).
i i i
f f
f p V U pV
U
0
and
H ΔH
Hf i
p H
T
H
The slope of H-p curve at constant T.
H p p
H T
T T
H p
T H
T T
p p
H
1
T Cp
• To measure , the gas is flowed through a porous plug inside a thermally insulated container with an electric heater.
• The steep pressure drop (p) is measured, and the cooling effect is exactly offset by
the electric heater.
• The energy provided by the heater (H=qp) is monitored.
T
T p
H
dT C dp
dH T p
1 )
, (
relation chain
Euler The
p T
H p
H H
T T
p
T p H H
• Real gas have nonzero Joule-Thomson coefficients.
• The sign of depends on the condition of gas the identity of the gas, the relative magnitude of the attractive and repulsive intermolecular forces and temperature.
• > 0 cooling by isenthalpic adiabatic expansion; attraction-dominant conditions
• < 0 heating by isenthalpic adiabatic expansion; repulsion-dominant conditions
• The boundary at a given P is the inversion temperature (TI) of the gas at that pressure.
• For an ideal gas, = 0 no change in T by isenthalpic adiabatic expansion; no
p H
T
• The mean kinetic energy of molecules in a gas is proportional to the temperature.
• In a attraction-dominant condition, when a gas expands, some kinetic energy of them must be converted to potential energy to reach greater separation. cooling
• In a repulsion-dominant condition, when a gas expands, some potential energy of them must be converted to kinetic energy to reach greater separation. heating
Comparison of T & T
T
T p
H
T
T V
U
Internal Pressure Isothermal Joule-Thomson Coeff.
• For an ideal gas, T = 0 and = 0 (T = 0)
• For an attraction-dominant gas, T > 0 and > 0 (T < 0)
T Cp
p H
T
• A gas typically have two inversion temperatures at a given p.
• The Joule-Thomson effects are utilized to liquefy gases.
Linde Refrigerator
• Reading: page 76 ~ 84
• Problem Set: Discussions 2.2, 2.6
Exercises 2.8b, 22b, 29b, 30b
• 4월 6일 휴강 4월 13일 보강 (저녁 7시 별232)
• 연습: 4월 6일 (저녁 7시 별232)
