Now we are going to start looking at models
for the chemical potential of a given
component in a mixture
• The first model is the ideal gas mixture
• The second model is the ideal solution
• As you study this, think about the differences,
not only mathematical but also the physical
differences of these models
The ideal-gas mixture model
• EOS for an ideal gas
• Calculate the partial molar volume for an ideal
gas component of an ideal gas mixture
For an ideal gas mixture
Vi (T , P)  Vi (T , P)  V (T , P)
ig
ig
and
RT
pi  yi P  yi ig
V
ig
For any partial molar property other than
volume, in an ideal gas mixture:
M (T , P)  M (T , pi )
ig
i
ig
i
H (T , P)  H (T , pi ) 
ig
i
ig
i
dSiig   Rd ln P
at constant T
integrate from pi to P
Partial molar entropy (igm)
S (T , pi )  S (T , P)  R ln yi
ig
i
ig
i
S (T , P)  S (T , pi )  S (T , P)  R ln yi
ig
i
ig
i
ig
i
Partial molar Gibbs energy
Gi ig  H iig  TSiig  H iig  TS iig  RT ln yi
iig  Gi ig  Giig  RT ln yi
Chemical potential of
component i in an
ideal gas mixture
*******************************************************************************
RT
dG  Vi dP 
dP  RTd ln P at constant T
P
integrating between P  1atm and P
ig
i
ig
Giig  i (T )  RT ln P
This is  for a pure component !!!
  i (T )  RT ln P  RT ln yi  i (T )  RT ln yi P
ig
i
Problem
• What is the change in entropy when 0.7 m3 of CO2
and 0.3 m3 of N2, each at 1 bar and 25oC blend to
form a gas mixture at the same conditions? Assume
ideal gases.
We showed that:
Siig  Siig  R ln yi
S   yi S
ig
ig
i
i
  yi S  R  yi ln yi
ig
i
i
S   yi S   R  yi ln yi
ig
ig
i
i
i
i
solution
S  nR yi ln yi
i
n = PV/RT= 1 bar 1 m3/ [R x 278 K]
S = 204.89 J/K
Problem
• What is the ideal work for the separation of an
equimolar mixture of methane and ethane at
175oC and 3 bar in a steady-flow process into
product streams of the pure gases at 35oC and
1 bar if the surroundings temperature Ts =
300K?
1) Read section 5.8 (calculation of ideal work)
2) Think about the process: separation of gases and change of state
First calculate H and S for methane and for ethane changing their state from
P1, T1, to P2T2
Second, calculate H for de-mixing and S for de-mixing from a mixture of ideal
gases
solution
ig
H i   Cp (T )dT
T2
ig
T1
S i  
ig
T2
T1
ig
P2
dT
Cp (T )
 R ln
T
P1
H deig  mix  0
S deig  mix  R  yi ln yi
i
H   yi H iig  H deig  mix
i
S   yi S iig  S deig  mix
i
= -7228 J/mol
= -15.813 J/mol K
Wideal = H – Ts S = -2484 J/mol
Problem
• A vessel, divided into two parts by a partition,
contains 4 mol of N2 gas at 75oC and 30 bar at
one side of the partition and 2.5 mol of Ar at
130oC and 20 bar on the other. If the partition
is removed and the gases mix adiabatically
and completely, what is the change in
entropy? Assume N2 to be an ideal gas with
Cv=(5/2)R and Ar to be an ideal gas with
Cv=(3/2)R