School of Chemical & Biological Engineering, Konkuk University

Lecture 8

• State functions and exact differential

Ch. 2 The First Law

• Let’s consider a system undergoing changes from U(T_i, V_i) to U(T_f, V_f).

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 C_p and C_V 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.)





C

C_p  _V  _T or

• For the perfect gas, is zero and is nR. T^V ^pV

T pV

nR dT

dT p

nR V

T p nRT V

T V V

p p

1 1

1

1   

























 





 



 







 









C

C_p  _V 





• 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

H_{f 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  C_p

• 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=q_p) 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 (T_I) 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  C_p

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)